Cryptography/Print version


Cryptography

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Introduction

Cryptography is the study of information hiding and verification. It includes the protocols, algorithms and strategies to securely and consistently prevent or delay unauthorized access to sensitive information and enable verifiability of every component in a communication.

Cryptography is derived from the Greek words: kryptós, "hidden", and gráphein, "to write" - or "hidden writing". People who study and develop cryptography are called cryptographers. The study of how to circumvent the use of cryptography for unintended recipients is called cryptanalysis, or codebreaking. Cryptography and cryptanalysis are sometimes grouped together under the umbrella term cryptology, encompassing the entire subject. In practice, "cryptography" is also often used to refer to the field as a whole, especially as an applied science. At the dawn of the 21 century in an ever more interconnected and technological world cryptography started to be ubiquitous as well as the reliance on the benefits it brings, especially the increased security and verifiability.

Cryptography is an interdisciplinary subject, drawing from several fields. Before the time of computers, it was closely related to linguistics. Nowadays the emphasis has shifted, and cryptography makes extensive use of technical areas of mathematics, especially those areas collectively known as discrete mathematics. This includes topics from number theory, information theory, computational complexity, statistics and combinatorics. It is also a branch of engineering, but an unusual one as it must deal with active, intelligent and malevolent opposition.

An example of the sub-fields of cryptography is steganography — the study of hiding the very existence of a message, and not necessarily the contents of the message itself (for example, microdots, or invisible ink) — and traffic analysis, which is the analysis of patterns of communication in order to learn secret information.

When information is transformed from a useful form of understanding to an opaque form of understanding, this is called encryption. When the information is restored to a useful form, it is called decryption. Intended recipients or authorized use of the information is determined by whether the user has a certain piece of secret knowledge. Only users with the secret knowledge can transform the opaque information back into its useful form. The secret knowledge is commonly called the key, though the secret knowledge may include the entire process or algorithm that is used in the encryption/decryption. The information in its useful form is called plaintext (or cleartext); in its encrypted form it is called ciphertext. The algorithm used for encryption and decryption is called a cipher (or cypher).

Common goals in cryptography edit

In essence, cryptography concerns four main goals. They are:

  1. message confidentiality (or privacy): Only an authorized recipient should be able to extract the contents of the message from its encrypted form. Resulting from steps to hide, stop or delay free access to the encrypted information.
  2. message integrity: The recipient should be able to determine if the message has been altered.
  3. sender authentication: The recipient should be able to verify from the message, the identity of the sender, the origin or the path it traveled (or combinations) so to validate claims from emitter or to validated the recipient expectations.
  4. sender non-repudiation: The remitter should not be able to deny sending the message.

Not all cryptographic systems achieve all of the above goals. Some applications of cryptography have different goals; for example some situations require repudiation where a participant can plausibly deny that they are a sender or receiver of a message, or extend this goals to include variations like:

  1. message access control: Who are the valid recipients of the message.
  2. message availability: By providing means to limit the validity of the message, channel, emitter or recipient in time or space.

Common forms of cryptography edit

Cryptography involves all legitimate users of information having the keys required to access that information.

  • If the sender and recipient must have the same key in order to encode or decode the protected information, then the cipher is a symmetric key cipher since everyone uses the same key for the same message. The main problem is that the secret key must somehow be given to both the sender and recipient privately. For this reason, symmetric key ciphers are also called private key (or secret key) ciphers.
  • If the sender and recipient have different keys respective to the communication roles they play, then the cipher is an asymmetric key cipher as different keys exist for encoding and decoding the same message. It is also called public key encryption as the user publicly distributes one of the keys without a care for secrecy. In the case of confidential messages to the user, they distribute the encryption key. Asymmetric encryption relies on the fact that possession of the encryption key will not reveal the decryption key.
  • Digital Signatures are a form of authentication with some parallels to public-key encryption. The two keys are the public verification key and the secret signature key. As in public-key encryption, the verification key can be distributed to other people, with the same caveat that the distribution process should in some way authenticate the owner of the secret key. Security relies on the fact that possession of the verification key will not reveal the signature key.
  • Hash Functions are unkeyed message digests with special properties.

Other:

Poorly designed, or poorly implemented, crypto systems achieve them only by accident or bluff or lack of interest on the part of the opposition. Users can, and regularly do, find weaknesses in even well-designed cryptographic schemes from those of high reputation.

Even with well designed, well implemented, and properly used crypto systems, some goals aren't practical (or desirable) in some contexts. For example, the sender of the message may wish to be anonymous, and would therefore deliberately choose not to bother with non-repudiation. Alternatively, the system may be intended for an environment with limited computing resources, or message confidentiality might not be an issue.

In classical cryptography, messages are typically enciphered and transmitted from one person or group to some other person or group. In modern cryptography, there are many possible options for "sender" or "recipient". Some examples, for real crypto systems in the modern world, include:

  1. a computer program running on a local computer,
  2. a computer program running on a 'nearby' computer which 'provides security services' for users on other nearby systems,
  3. a human being (usually understood as 'at the keyboard'). However, even in this example, the presumed human is not generally taken to actually encrypt or sign or decrypt or authenticate anything. Rather, he or she instructs a computer program to perform these actions. This 'blurred separation' of human action from actions which are presumed (without much consideration) to have 'been done by a human' is a source of problems in crypto system design, implementation, and use. Such problems are often quite subtle and correspondingly obscure; indeed, generally so, even to practicing cryptographers with knowledge, skill, and good engineering sense.

When confusion on these points is present (e.g., at the design stage, during implementation, by a user after installation, or ...), failures in reaching each of the stated goals can occur quite easily—often without notice to any human involved, and even given a perfect cryptosystem. Such failures are most often due to extra-cryptographic issues; each such failure demonstrates that good algorithms, good protocols, good system design, and good implementation do not alone, nor even in combination, provide 'security'. Instead, careful thought is required regarding the entire crypto system design and its use in actual production by real people on actual equipment running 'production' system software (e.g., operating systems) -- too often, this is absent or insufficient in practice with real-world crypto systems.

Although cryptography has a long and complex history, it wasn't until the 19th century that it developed anything more than ad hoc approaches to either encryption or cryptanalysis (the science of finding weaknesses in crypto systems). Examples of the latter include Charles Babbage's Crimean War era work on mathematical cryptanalysis of polyalphabetic ciphers, repeated publicly rather later by the Prussian Kasiski. During this time, there was little theoretical foundation for cryptography; rather, understanding of cryptography generally consisted of hard-won fragments of knowledge and rules of thumb; see, for example, Auguste Kerckhoffs' crypto writings in the latter nineteenth century. An increasingly mathematical trend accelerated up to World War II (notably in William F. Friedman's application of statistical techniques to cryptography and in Marian Rejewski's initial break into the German Army's version of the Enigma system). Both cryptography and cryptanalysis have become far more mathematical since WWII. Even then, it has taken the wide availability of computers, and the Internet as a communications medium, to bring effective cryptography into common use by anyone other than national governments or similarly large enterprise.

External links


History

Classical Cryptography edit

The earliest known use of cryptography is found in non-standard hieroglyphs carved into monuments from Egypt's Old Kingdom (ca 4500 years ago). These are not thought to be serious attempts at secret communications, however, but rather to have been attempts at mystery, intrigue, or even amusement for literate onlookers. These are examples of still another use of cryptography, or of something that looks (impressively if misleadingly) like it. Later, Hebrew scholars made use of simple Substitution ciphers (such as the Atbash cipher) beginning perhaps around 500 to 600 BCE. Cryptography has a long tradition in religious writing likely to offend the dominant culture or political authorities.

The Greeks of Classical times are said to have known of ciphers (e.g., the scytale transposition cypher claimed to have been used by the Spartan military). Herodutus tells us of secret messages physically concealed beneath wax on wooden tablets or as a tattoo on a slave's head concealed by regrown hair (these are not properly examples of cryptography per se; see secret writing). The Romans certainly did (e.g., the Caesar cipher and its variations). There is ancient mention of a book about Roman military cryptography (especially Julius Caesar's); it has been, unfortunately, lost.

In India, cryptography was apparently well known. It is recommended in the Kama Sutra as a technique by which lovers can communicate without being discovered. This may imply that cryptanalytic techniques were less than well developed in India ca 500 CE.

Cryptography became (secretly) important still later as a consequence of political competition and religious analysis. For instance, in Europe during and after the Renaissance, citizens of the various Italian states, including the Papacy, were responsible for substantial improvements in cryptographic practice (e.g., polyalphabetic ciphers invented by Leon Alberti ca 1465). And in the Arab world, religiously motivated textual analysis of the Koran led to the invention of the frequency analysis technique for breaking monoalphabetic substitution cyphers sometime around 1000 CE.

Cryptography, cryptanalysis, and secret agent betrayal featured in the Babington plot during the reign of Queen Elizabeth I which led to the execution of Mary, Queen of Scots. And an encrypted message from the time of the Man in the Iron Mask (decrypted around 1900 by Étienne Bazeries) has shed some, regrettably non-definitive, light on the identity of that legendary, and unfortunate, prisoner. Cryptography, and its misuse, was involved in the plotting which led to the execution of Mata Hari and even more reprehensibly, if possible, in the travesty which led to Dreyfus' conviction and imprisonment, both in the early 20th century. Fortunately, cryptographers were also involved in setting Dreyfus free; Mata Hari, in contrast, was shot.

Mathematical cryptography leapt ahead (also secretly) after World War I. Marian Rejewski, in Poland, attacked and 'broke' the early German Army Enigma system (an electromechanical rotor cypher machine) using theoretical mathematics in 1932. The break continued up to '39, when changes in the way the German Army's Enigma machines were used required more resources than the Poles could deploy. His work was extended by Alan Turing, Gordon Welchman, and others at Bletchley Park beginning in 1939, leading to sustained breaks into several other of the Enigma variants and the assorted networks for which they were used. US Navy cryptographers (with cooperation from British and Dutch cryptographers after 1940) broke into several Japanese Navy crypto systems. The break into one of them famously led to the US victory in the Battle of Midway. A US Army group, the SIS, managed to break the highest security Japanese diplomatic cipher system (an electromechanical 'stepping switch' machine called Purple by the Americans) even before WWII began. The Americans referred to the intelligence resulting from cryptanalysis, perhaps especially that from the Purple machine, as 'Magic'. The British eventually settled on 'Ultra' for intelligence resulting from cryptanalysis, particularly that from message traffic enciphered by the various Enigmas. An earlier British term for Ultra had been 'Boniface'.

World War II Cryptography edit

By World War II mechanical and electromechanical cryptographic cipher machines were in wide use, but they were impractical manual systems. Great advances were made in both practical and mathematical cryptography in this period, all in secrecy. Information about this period has begun to be declassified in recent years as the official 50-year (British) secrecy period has come to an end, as the relevant US archives have slowly opened, and as assorted memoirs and articles have been published.

The Germans made heavy use (in several variants) of an electromechanical rotor based cypher system known as Enigma. The German military also deployed several mechanical attempts at a one-time pad. Bletchley Park called them the Fish cyphers, and Max Newman and colleagues designed and deployed the world's first programmable digital electronic computer, the Colossus, to help with their cryptanalysis. The German Foreign Office began to use the one-time pad in 1919; some of this traffic was read in WWII partly as the result of recovery of some key material in South America that was insufficiently carefully discarded by a German courier.

The Japanese Foreign Office used a locally developed electrical stepping switch based system (called Purple by the US), and also used several similar machines for attaches in some Japanese embassies. One of these was called the 'M-machine' by the US, another was referred to as 'Red'. All were broken, to one degree or another by the Allies.

Other cipher machines used in WWII included the British Typex and the American SIGABA; both were electromechanical rotor designs similar in spirit to the Enigma.

Modern Cryptography edit

The era of modern cryptography really begins with Claude Shannon, arguably the father of mathematical cryptography. In 1949 he published the paper Communication Theory of Secrecy Systems in the Bell System Technical Journal, and a little later the book Mathematical Theory of Communication with Warren Weaver. These, in addition to his other works on information and communication theory established a solid theoretical basis for cryptography and for cryptanalysis. And with that, cryptography more or less disappeared into secret government communications organizations such as the NSA. Very little work was again made public until the mid '70s, when everything changed.

1969 saw two major public (i.e., non-secret) advances. First was the DES (Data Encryption Standard) submitted by IBM, at the invitation of the National Bureau of Standards (now NIST), in an effort to develop secure electronic communication facilities for businesses such as banks and other large financial organizations. After 'advice' and modification by the NSA, it was adopted and published as a FIPS Publication (Federal Information Processing Standard) in 1977 (currently at FIPS 46-3). It has been made effectively obsolete by the adoption in 2001 of the Advanced Encryption Standard, also a NIST competition, as FIPS 197. DES was the first publicly accessible cypher algorithm to be 'blessed' by a national crypto agency such as NSA. The release of its design details by NBS stimulated an explosion of public and academic interest in cryptography. DES, and more secure variants of it (such as 3DES or TDES; see FIPS 46-3), are still used today, although DES was officially supplanted by AES (Advanced Encryption Standard) in 2001 when NIST announced the selection of Rijndael, by two Belgian cryptographers. DES remains in wide use nonetheless, having been incorporated into many national and organizational standards. However, its 56-bit key-size has been shown to be insufficient to guard against brute-force attacks (one such attack, undertaken by cyber civil-rights group The Electronic Frontier Foundation, succeeded in 56 hours—the story is in Cracking DES, published by O'Reilly and Associates). As a result, use of straight DES encryption is now without doubt insecure for use in new crypto system designs, and messages protected by older crypto systems using DES should also be regarded as insecure. The DES key size (56-bits) was thought to be too small by some even in 1976, perhaps most publicly Whitfield Diffie. There was suspicion that government organizations even then had sufficient computing power to break DES messages and that there may be a back door due to the lack of randomness in the 'S' boxes.

Second was the publication of the paper New Directions in Cryptography by Whitfield Diffie and Martin Hellman. This paper introduced a radically new method of distributing cryptographic keys, which went far toward solving one of the fundamental problems of cryptography, key distribution. It has become known as Diffie-Hellman key exchange. The article also stimulated the almost immediate public development of a new class of enciphering algorithms, the asymmetric key algorithms.

Prior to that time, all useful modern encryption algorithms had been symmetric key algorithms, in which the same cryptographic key is used with the underlying algorithm by both the sender and the recipient who must both keep it secret. All of the electromechanical machines used in WWII were of this logical class, as were the Caesar and Atbash cyphers and essentially all cypher and code systems throughout history. The 'key' for a code is, of course, the codebook, which must likewise be distributed and kept secret.

Of necessity, the key in every such system had to be exchanged between the communicating parties in some secure way prior to any use of the system (the term usually used is 'via a secure channel') such as a trustworthy courier with a briefcase handcuffed to a wrist, or face-to-face contact, or a loyal carrier pigeon. This requirement rapidly becomes unmanageable when the number of participants increases beyond some (very!) small number, or when (really) secure channels aren't available for key exchange, or when, as is sensible crypto practice keys are changed frequently. In particular, a separate key is required for each communicating pair if no third party is to be able to decrypt their messages. A system of this kind is also known as a private key, secret key, or conventional key cryptosystem. D-H key exchange (and succeeding improvements) made operation of these systems much easier, and more secure, than had ever been possible before.

In contrast, with asymmetric key encryption, there is a pair of mathematically related keys for the algorithm, one of which is used for encryption and the other for decryption. Some, but not all, of these algorithms have the additional property that one of the keys may be made public since the other cannot be (by any currently known method) deduced from the 'public' key. The other key in these systems is kept secret and is usually called, somewhat confusingly, the 'private' key. An algorithm of this kind is known as a public key / private key algorithm, although the term asymmetric key cryptography is preferred by those who wish to avoid the ambiguity of using that term for all such algorithms, and to stress that there are two distinct keys with different secrecy requirements.

As a result, for those using such algorithms, only one key pair is now needed per recipient (regardless of the number of senders) as possession of a recipient's public key (by anyone whomsoever) does not compromise the 'security' of messages so long as the corresponding private key is not known to any attacker (effectively, this means not known to anyone except the recipient). This unanticipated, and quite surprising, property of some of these algorithms made possible, and made practical, widespread deployment of high quality crypto systems which could be used by anyone at all. Which in turn gave government crypto organizations worldwide a severe case of heartburn; for the first time ever, those outside that fraternity had access to cryptography that wasn't readily breakable by the 'snooper' side of those organizations. Considerable controversy, and conflict, began immediately. It has not yet subsided. In the US, for example, exporting strong cryptography remains illegal; cryptographic methods and techniques are classified as munitions. Until 2001 'strong' crypto was defined as anything using keys longer than 40 bits—the definition was relaxed thereafter. (See S Levy's Crypto for a journalistic account of the policy controversy in the US).

Note, however, that it has NOT been proven impossible, for any of the good public/private asymmetric key algorithms, that a private key (regardless of length) can be deduced from a public key (or vice versa). Informed observers believe it to be currently impossible (and perhaps forever impossible) for the 'good' asymmetric algorithms; no workable 'companion key deduction' techniques have been publicly shown for any of them. Note also that some asymmetric key algorithms have been quite thoroughly broken, just as many symmetric key algorithms have. There is no special magic attached to using algorithms which require two keys.

In fact, some of the well respected, and most widely used, public key / private key algorithms can be broken by one or another cryptanalytic attack and so, like other encryption algorithms, the protocols within which they are used must be chosen and implemented carefully to block such attacks. Indeed, all can be broken if the key length used is short enough to permit practical brute force key search; this is inherently true of all encryption algorithms using keys, including both symmetric and asymmetric algorithms.

This is an example of the most fundamental problem for those who wish to keep their communications secure; they must choose a crypto system (algorithms + protocols + operation) that resists all attack from any attacker. There being no way to know who those attackers might be, nor what resources they might be able to deploy, nor what advances in cryptanalysis (or its associated mathematics) might in future occur, users may ONLY do the best they know how, and then hope. In practice, for well designed / implemented / used crypto systems, this is believed by informed observers to be enough, and possibly even enough for all(?) future attackers. Distinguishing between well designed / implemented / used crypto systems and crypto trash is another, quite difficult, problem for those who are not themselves expert cryptographers. It is even quite difficult for those who are.

Revision of modern history edit

In recent years public disclosure of secret documents held by the UK government has shown that asymmetric key cryptography, D-H key exchange, and the best known of the public key / private key algorithms (i.e., what is usually called the RSA algorithm), all seem to have been developed at a UK intelligence agency before the public announcement by Diffie and Hellman in '76. GCHQ has released documents claiming that they had developed public key cryptography before the publication of Diffie and Hellman's paper. Various classified papers were written at GCHQ during the 1960s and 1970s which eventually led to schemes essentially identical to RSA encryption and to Diffie-Hellman key exchange in 1973 and 1974. Some of these have now been published, and the inventors (James Ellis, Clifford Cocks, and Malcolm Williamson) have made public (some of) their work.


Classical Cryptography

Cryptography has a long and colorful history from Caesar's encryption in first century BC to the 20th century.


There are two major principles in classical cryptography: transposition and substitution.

Transposition ciphers edit

Lets look first at transposition, which is the changing in the position of the letters in the message such as a simple writing backwards

    Plaintext: THE PANEL IN THE WALL MOVES
    Encrypted: EHT LENAP NI EHT LLAW SEVOM

or as in a more complex transposition such as:

    THEPAN    
    ELINTH     
    EWALLM
    OVESAA

then take the columns:

    TEEO HLWV EIAE PNLS ATLA NHMA

(the extra letters are called space fillers) The idea in transposition is NOT to randomize it but to transform it to something that is not recognizable with a reversible algorithm (an algorithm is just a procedure, reversible so your correspondent can read the message).

We discuss transposition ciphers in much more detail in a later chapter, Cryptography/Transposition ciphers.

Substitution ciphers edit

The second most important principle is substitution. That is, substituting a Symbol for a letter of your plaintext (or word or even sentence). Slang even can sometimes be a form of cipher (the symbols replacing your plaintext), ever wonder why your parents never understood you? Slang, though, is not something you would want to store a secret in for a long time. In WWII, there were Navajo CodeTalkers who passed along info from unit to unit. From what I hear (someone verify this) the Navajo language was a very exclusive almost unknown and unwritten language. So the Japanese were not able to decipher it.

Even though this is a very loose example of substitution, whatever works works.

Caesar Cipher edit

One of the most basic methods of encryption is the use of Caesar Ciphers. It simply consist in shifting the alphabet over a few characters and matching up the letters.

The classical example of a substitution cipher is a shifted alphabet cipher

   Alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ
     Cipher: BCDEFGHIJKLMNOPQRSTUVWXYZA
    Cipher2: CDEFGHIJKLMNOPQRSTUVWXYZAB
        etc...

Example:(using cipher 2)

          Plaintext: THE PANEL IN THE WALL MOVES
          Encrypted: VJG RCPGN KP VJG YCNN OQXGU

If this is the first time you've seen this it may seem like a rather secure cipher, but it's not. In fact this by itself is very insecure. For a time in the 1500-1600s this was the most secure (mainly because there were many people who were illiterate) but a man (old what's his name) in the 18th century discovered a way to crack (find the hidden message) of every cipher of this type he discovered frequency analysis.

We discuss substitution ciphers in much more detail in a later chapter, Cryptography/Substitution ciphers.

Frequency Analysis edit

In the shifted alphabet cipher or any simple randomized cipher, the same letter in the cipher replaces each of the same ones in your message (e.g. 'A' replaces all 'D's in the plaintext, etc.). The weakness is that English uses certain letters more than any other letter in the alphabet. 'E' is the most common, etc. Here's an exercise count all of each letter in this article. You'll find that in the previous sentence there are 2 'H's,7 'E's, 3 'R's, 3 'S's, etc. By far 'E' is the most common letter; here are the other frequencies [Frequency tables|http://rinkworks.com/words/letterfreq.shtml]. Basically you experiment with replacing different symbols with letters (the most common with 'E', etc.).

      Encrypted: VJG TCKP KP URCKP

First look for short words with limited choices of words, such as 'KP' this may be at, in, to, or, by, etc. Let us select in. Replace 'K' with 'I' and 'P' with 'N'.

      Encrypted: VJG TCIN IN URCIN

Next select VJG this is most likely the (since the huge frequency of 'the' in a normal sentence, check a couple of the preceding sentences).

      Encrypted: THE TCIN IN URCIN

generally this in much easier in long messages the plaintext is 'THE RAIN IN SPAIN'

We discuss many different ways to "attack", "break", and "solve" encrypted messages in a later section of this book, "Part III: Cryptanalysis", which includes a much more detailed section on Cryptography/Frequency analysis.

Combining transposition and substitution edit

A more secure encryption is a transposed substitution cipher.

Take the above message in encrypted form

      Encrypted:VJG RCPGN KP VJG YCNN OQXGU

now spiral transpose it

          VJGRC
          NNOQP
          CAAXG
          YUNGN
          GJVPK

The message starts in the upper right corner and spirals to the center (again the AA is a filler) Now take the columns:

       VNCYG JNAUJ GOANV RQXGP CPGNK

Now this is more resistant to Frequency analysis, see what we did before that started recognizable patterns results in:

       TNCYE HNAUH EOANT RQXEN CNENK

A problem for people who crack codes.


Multiliteral systems edit

The vast majority of classical ciphers are "uniliteral" -- they encrypt a plaintext 1 letter at a time, and each plaintext letter is encrypted to a single corresponding ciphertext letter.

A multiliteral system is one where the ciphertext unit is more than one character in length. The major types of multiliteral systems are:[1]

  • biliteral systems: 2 letters of ciphertext per letter of plaintext
  • dinomic systems: 2 digits of ciphertext per letter of plaintext
  • Triliteral systems: 3 letters of ciphertext per letter of plaintext
  • trinomic systems: 3 digits of ciphertext per letter of plaintext
  • monome-dinome systems, also called straddling checkerboard systems: 1 digit of ciphertext for some plaintext letters, 2 digits of ciphertext for the remaining plaintext letters.
  • biliteral with variants and dinomic with variants systems: several ciphertext values decode to the same plaintext letter (homophonic substitution cipher)
  • Syllabary square systems: 2 letters or 2 digits of ciphertext decode to an entire syllable or a single character of plaintext.[2]


Cryptography in Popular Culture

Books edit

Digital Fortress, by Dan Brown

Films edit

Television series edit

BBC series Spooks, about MI5, with references to GCHQ


Timeline of Notable Events

The desire to keep stored or send information secret dates back into antiquity. As society developed so did the application of cryptography. Below is a timeline of notable events related to cryptography.

BCE edit

  • 3500s - The Sumerians develop cuneiform writing and the Egyptians develop hieroglyphic writing.
  • 1500s - The Phoenicians develop an alphabet
  • 600-500 - Hebrew scholars make use of simple monoalphabetic substitution ciphers (such as the Atbash cipher)
  • c. 400 - Spartan use of scytale (alleged)
  • c. 400BCE - Herodotus reports use of steganography in reports to Greece from Persia (tatoo on shaved head)
  • 100-0 - Notable Roman ciphers such as the Caeser cipher.

1 - 1799 CE edit

  • ca 1000 - Frequency analysis leading to techniques for breaking monoalphabetic substitution ciphers. It was probably developed among the Arabs, and was likely motivated by textual analysis of the Koran.
  • 1450 - The Chinese develop wooden block movable type printing
  • 1450-1520 - The Voynich manuscript, an example of a possibly encrypted illustrated book, is written.
  • 1466 - Leone Battista Alberti invents polyalphabetic cipher, also the first known mechanical cipher machine
  • 1518 - Johannes Trithemius' book on cryptology
  • 1553 - Belaso invents the (misnamed) Vigenère cipher
  • 1585 - Vigenère's book on ciphers
  • 1641 - Wilkins' Mercury (English book on cryptography)
  • 1586 - Cryptanalysis used by spy master Sir Francis Walsingham to implicate Mary Queen of Scots in the Babington Plot to murder Queen Elizabeth I of England. Queen Mary was eventually executed.
  • 1614 - Scotsman John Napier (1550-1617) published a paper outlining his discovery of the logarithm. Napier also invented an ingenious system of moveable rods (referred to as Napier's Rods or Napier's bones) which were a precursor of the slide rule. These were based on logarithms and allowed the operator to multiply, divide and calculate square and cube roots by moving the rods around and placing them in specially constructed boards.
  • 1793 - Claude Chappe establishes the first long-distance semaphore "telegraph" line
  • 1795 - Thomas Jefferson invents the Jefferson disk cipher, reinvented over 100 years later by Etienne Bazeries and widely used a a tactical cypher by the US Army.

1800-1899 edit

  • 1809-14 George Scovell's work on Napoleonic ciphers during the Peninsular War
  • 1831 - Joseph Henry proposes and builds an electric telegraph
  • 1835 - Samuel Morse develops the Morse code.
  • c. 1854 - Babbage's method for breaking polyalphabetic cyphers (pub 1863 by Kasiski); the first known break of a polyaphabetic cypher. Done for the English during the Crimean War, a general attack on Vigenère's autokey cipher (the 'unbreakable cypher' of its time) as well as the much weaker cypher that is today termed "the Vigenère cypher". The advance was kept secret and was, in essence, reinvented somewhat later by the Prussian Friedrich Kasiski, after whom it is named.
  • 1854 - Wheatstone invents Playfair cipher
  • 1883 - Auguste Kerckhoffs publishes La Cryptographie militare, containing his celebrated "laws" of cryptography
  • 1885 - Beale ciphers published
  • 1894 - The Dreyfus Affair in France involves the use of cryptography, and its misuse, re: false documents.

1900 - 1949 edit

  • c 1915 - William Friedman applies statistics to cryptanalysis ( coincidence counting, etc.)
  • 1917 - Gilbert Vernam develops first practical implementation of a teletype cipher, now known as a stream cipher and, later, with Mauborgne the one-time pad
  • 1917 - Zimmermann telegram intercepted and decrypted, advancing U.S. entry into World War I
  • 1919 - Weimar Germany Foreign Office adopts (a manual) one-time pad for some traffic
  • 1919 - Hebern invents/patents first rotor machine design -- Damm, Scherbius and Koch follow with patents the same year
  • 1921 - Washington Naval Conference - U.S. negotiating team aided by decryption of Japanese diplomatic telegrams
  • c. 1924 - MI8 (Yardley, et al.) provide breaks of assorted traffic in support of US position at Washington Naval Conference
  • c. 1932 - first break of German Army Enigma machine by Rejewski in Poland
  • 1929 - U.S. Secretary of State Henry L. Stimson shuts down State Department cryptanalysis "Black Chamber", saying "Gentlemen do not read each other's mail."
  • 1931 - The American Black Chamber by Herbert O. Yardley is published, revealing much about American cryptography
  • 1940 - break of Japan's Purple machine cipher by SIS team
  • December 7, 1941 - U.S. Naval base at Pearl Harbor surprised by Japanese attack, despite U.S. breaks into several Japanese cyphers. U.S. enters World War II
  • June 1942 - Battle of Midway. Partial break into Dec 41 edition of JN-25 leads to successful ambush of Japanese carriers and to the momentum killing victory.
  • April 1943 - Admiral Yamamoto, architect of Pearl Harbor attack, is assassinated by U.S. forces who know his itinerary from decrypted messages
  • April 1943 - Max Newman, Wynn-Williams, and their team (including Alan Turing) at the secret Government Code and Cypher School ('Station X'), Bletchley Park, Bletchley, England, complete the "Heath Robinson". This is a specialized machine for cypher-breaking, not a general-purpose calculator or computer.
  • December 1943 - The Colossus was built, by Dr Thomas Flowers at The Post Office Research Laboratories in London, to crack the German Lorenz cipher (SZ42). Colossus was used at Bletchley Park during WW II - as a successor to April's 'Robinson's. Although 10 were eventually built, unfortunately they were destroyed immediately after they had finished their work - it was so advanced that there was to be no possibility of its design falling into the wrong hands. The Colossus design was the first electronic digital computer and was somewhat programmable. A epoch in machine capability.
  • 1944 - patent application filed on SIGABA code machine used by U.S. in WW II. Kept secret, finally issued in 2001
  • 1946 - VENONA's first break into Soviet espionage traffic from early 1940s
  • 1948 - Claude Shannon writes a paper that establishes the mathematical basis of information theory
  • 1949 - Shannon's Communication Theory of Secrecy Systems pub in Bell Labs Technical Journal, based on work done during WWII.

1950 - 1999 edit

  • 1951 - U.S. National Security Agency founded, subsuming the US Army and US Navy 'girls school' departments
  • 1968 - John Anthony Walker walks into the Soviet Union's embassy in Washington and sells information on KL-7 cipher machine. The Walker spy ring operates until 1985
  • 1964 - David Kahn's The Codebreakers is published
  • June 8, 1967 - USS Liberty incident in which a U.S. SIGINT ship is attacked by Israel, apparently by mistake, though some continue to dispute this
  • January 23, 1968 - USS Pueblo, another SIGINT ship, is captured by North Korea
  • 1969 - The first hosts of ARPANET, Internet's ancestor, are connected
  • 1974? - Horst Feistel develops the Feistel network block cipher design at IBM
  • 1976 - the Data Encryption Standard was published as an official Federal Information Processing Standard (FIPS) for the US
  • 1976 - Diffie and Hellman publish New Directions in Cryptography article
  • 1977- RSA public key encryption invented at MIT
  • 1981 - Richard Feynman proposes quantum computers. The main application he had in mind was the simulation of quantum systems, but he also mentioned the possibility of solving other problems.
  • 1986 In the wake of an increasing number of break-ins to government and corporate computers, the US Congress passes the Computer Fraud and Abuse Act, which makes it a crime to break into computer systems. The law, however, does not cover juveniles.
  • 1988 - First optical chip developed, it uses light instead of electricity to increase processing speed.
  • 1989 - Tim Berners-Lee and Robert Cailliau built the prototype system which became the World Wide Web at CERN
  • 1991 - Phil Zimmermann releases the public key encryption program PGP along with its source code, which quickly appears on the Internet.
  • 1992 - Release of the movie Sneakers (film)|Sneakers, in which security experts are blackmailed into stealing a universal decoder for encryption systems (no such decoder is known, likely because it is impossible).
  • 1994 - 1st ed of Bruce Schneier's Applied Cryptography is published
  • 1994 - Secure Sockets Layer (SSL) encryption protocol released by Netscape
  • 1994 - Peter Shor devises an algorithm which lets quantum computers determine the factorization of large integers quickly. This is the first interesting problem for which quantum computers promise a significant speed-up, and it therefore generates a lot of interest in quantum computers.
  • 1994 - DNA computing proof of concept on toy traveling salesman problem; a method for input/output still to be determined.
  • 1994 - Russian crackers siphon $10 million from Citibank and transfer the money to bank accounts around the world. Vladimir Levin, the 30-year-old ringleader, uses his work laptop after hours to transfer the funds to accounts in Finland and Israel. Levin stands trial in the United States and is sentenced to three years in prison. Authorities recover all but $400,000 of the stolen money.
  • 1994 - Formerly proprietary trade secret, but not patented, RC4 cipher algorithm is published on the Internet
  • 1994 - first RSA Factoring Challenge from 1977 is decrypted as The Magic Words are Squeamish Ossifrage
  • 1995 - NSA publishes the SHA1 hash algorithm as part of its Digital Signature Standard; SHA0 had a flaw corrected by SHA1
  • 1997 - Ciphersaber, an encryption system based on RC4 that is simple enough to be reconstructed from memory, is published on Usenet
  • 1998 - RIPE project releases final report
  • October 1998 - Digital Millennium Copyright Act (DMCA) becomes law in U.S., criminalizing production and dissemination of technology that can circumvent measures taken to protect copyright
  • October 1999 - DeCSS, a computer program capable of decrypting content on a DVD, is published on the Internet
  • 1999: Bruce Schneier develops the Solitaire cipher, a way to allow field agents to communicate securely without having to rely on electronics or having to carry incriminating tools like a one-time pad. Unlike all previous manual encryption techniques -- except the one-time pad -- this one is resistant to automated cryptanalysis. It is published in Neal Stephenson's Cryptonomicon (2000).

2000 and beyond edit

  • January 14, 2000 - U.S. Government announce restrictions on export of cryptography are relaxed (although not removed). This allows many US companies to stop the long running, and rather ridiculous process of having to create US and international copies of their software.
  • March 2000 - President Clinton says he doesn't use e-mail to communicate with his daughter, Chelsea Clinton, at college because he doesn't think the medium is secure.
  • September 6, 2000 - RSA Security Inc. released their RSA algorithm into the public domain, a few days in advance of their US patent 4405829 expiring. Following the relaxation of the U.S. government export restrictions, this removed one of the last barriers to the world-wide distribution of much software based on cryptographic systems. It should be noted that the IDEA algorithm is still under patent and also that government restrictions still apply in some places.
  • 2000 - U.K. Regulation of Investigatory Powers Act 2000|Regulation of Investigatory Powers Act requires anyone to supply their cryptographic key to a duly authorized person on request
  • 2001 - Belgian Rijndael algorithm selected as the U.S. Advanced Encryption Standard after a 5 year public search process by National Institute for Standards and Technology (NIST)
  • September 11, 2001 - U.S. response to terrorist attacks hampered by Communication during the September 11, 2001 attacks|lack of secure communications
  • November 2001 - Microsoft and its allies vow to end "full disclosure" of security vulnerabilities by replacing it with "responsible" disclosure guidelines.
  • 2002 - NESSIE project releases final report / selections
  • 2003 - CRYPTREC project releases 2003 report / recommendations
  • 2004 - the hash MD5 is shown to be vulnerable to practical collision attack
  • 2005 - potential for attacks on SHA1 demonstrated
  • 2005 - agents from the U.S. FBI demonstrate their ability to crack WEP using publicly available tools
  • 2007 - NIST announces w:NIST hash function competition
  • 2012 - proclamation of a winner of the w:NIST hash function competition is scheduled
  • 2015 - year by which NIST suggests that 80-bit keys for symmetric key cyphers be phased out. Asymmetric key cyphers require longer keys which have different vulnerability parameters.


Goals of Cryptography

Crytography is the science of secure communication in the presence of third parties (sometimes called "adversaries").

Modern cryptographers and cryptanalysts work in many areas including

  • data confidentiality
  • data integrity
  • authentication
  • forward secrecy
  • end-to-end auditable voting systems
  • digital currency

Classical cryptography focused on "data confidentiality"—keeping pieces of information secret, i.e. of designing technical systems such that an observer can infer as few as possible - optimally none - information from observing the system. The motivation for this is that the owner of the system wants to prevent the observer from taking advantage (e.g. monetary, influential, emotional) of the possible intelligence.

This secrecy or hiding is achieved by removing contextual information from the system's observable state and/or behaviour, without which the observer cannot gain intelligence about the system.

Examples edit

The term is very often used in conjunction in the context of message exchange between two entities, but of course not restricted to this case.

Hiding System State Alone edit

It may be advantageous for an ATM machine to hide the information as to how much cash is still available in the machine. It may e.g. only disclose the information that no more bank notes are available from it to the holder of a valid debit card.

Hiding Communication Content edit

Two companies doing business with each other may not wish to disclose the information on pricing of their products to third parties tapping into their communications.

Hiding the Fact of Communicating edit

Well known entities with well known fields of activity may wish to hide the fact that they are communicating at all since an observer aware of their fields of activity may already from the fact of some communication happening, be able to infer information.


Symmetric Ciphers

A symmetric key cipher (also called a secret-key cipher, or a one-key cipher, or a private-key cipher, or a shared-key cipher) Shared_secretis one that uses the same (necessarily secret) key to encrypt messages as it does to decrypt messages.

Until the invention of asymmetric key cryptography (commonly termed "public key / private key" crypto) in the 1970s, all ciphers were symmetric. Each party to the communication needed a key to encrypt a message; and a recipient needed a copy of the same key to decrypt the message. This presented a significant problem, as it required all parties to have a secure communication system (e.g. face-to-face meeting or secure courier) in order to distribute the required keys. The number of secure transfers required rises impossibly, and wholly impractically, quickly with the number of participants.

Formal Definition edit

Any cryptosystem based on a symmetric key cipher conforms to the following definition:

  • M : message to be enciphered
  • K : a secret key
  • E : enciphering function
  • D : deciphering function
  • C : enciphered message. C := E(M, K)
  • For all M, C, and K, M = D(C,K) = D(E(M,K),K)

Reciprocal Ciphers edit

Some shared-key ciphers are also "reciprocal ciphers." A reciprocal cipher applies the same transformation to decrypt a message as the one used to encrypt it. In the language of the formal definition above, E = D for a reciprocal cipher.

An example of a reciprocal cipher is Rot 13, in which the same alphabetic shift is used in both cases.

The xor-cipher (often used with one-time-pads) is another reciprocal cipher.[3]

Reciprocal ciphers have the advantage that the decoding machine can be set up exactly the same as the encoding machine -- reciprocal ciphers do not require the operator to remember to switch between "decoding" and "encoding".[4]

Symmetric Cypher Advantages edit

Symmetric key ciphers are typically much less computational overhead Overhead_(computing) than Asymmetric ciphers, sometimes this difference in computing overhead per character can be several orders of magnitude[5]. As such they are still used for bulk encryption of files and data streams for online applications.

Use edit

To set up a secure communication session Session_key between two parties the following actions take place Transport_Layer_Security:

  1. Alice tells Bob (in cleartext) that she wants a secure connection.
  2. Bob generates a single use (session), public/private (asymmetric) key pair (Kpb Kpr).
  3. Alice generates a single use (session) symmetric key. This will be the shared secret (Ks).
  4. Bob sends Alice the public key (Kpb).
  5. Alice encrypts her shared session key Ks with the Public key Kpb Ck := E(Ks, Kpb) and sends it to Bob
  6. Bob decrypts the message with his private key to obtains the shared session key Ks := D(Ck, Kpr)
  7. Now Alice and Bob have a shares secret (symmetric key) to secure communication on this connection for this session
  8. Either party can encrypt a message simply by C := E(M, Ks) and decrypt is by M = D(C,K) = D(E(M,Ks),Ks)

This is the Basis for Diffie–Hellman Diffie_Hellman_key_exchange key exchange Key exchange and its more advanced successors Transport_Layer_Security.

Examples edit


Further Reading edit

Symmetric-key algorithm


 

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Asymmetric Ciphers

In cryptography, an asymmetric key algorithm uses a pair of different, though related, cryptographic keys to encrypt and decrypt. The two keys are related mathematically; a message encrypted by the algorithm using one key can be decrypted by the same algorithm using the other. In a sense, one key "locks" a lock (encrypts); but a different key is required to unlock it (decrypt).

Some, but not all, asymmetric key cyphers have the "public key" property, which means that there is no known effective method of finding the other key in a key pair, given knowledge of one of them. This group of algorithms is very useful, as it entirely evades the key distribution problem inherent in all symmetric key cyphers and some of the asymmetric key cyphers. One may simply publish one key while keeping the other secret. They form the basis of much of modern cryptographic practice.

A Postal Analogy edit

An analogy which can be used to understand the advantages of an asymmetric system is to imagine two people, Alice and Bob, sending a secret message through the public mail. In this example, Alice has the secret message and wants to send it to Bob, after which Bob sends a secret reply.

With a symmetric key system, Alice first puts the secret message in a box, and then locks the box using a padlock to which she has a key. She then sends the box to Bob through regular mail. When Bob receives the box, he uses an identical copy of Alice's key (which he has somehow obtained previously) to open the box, and reads the message. Bob can then use the same padlock to send his secret reply.

In an asymmetric key system, Bob and Alice have separate padlocks. Firstly, Alice asks Bob to send his open padlock to her through regular mail, keeping his key to himself. When Alice receives it she uses it to lock a box containing her message, and sends the locked box to Bob. Bob can then unlock the box with his key and read the message from Alice. To reply, Bob must similarly get Alice's open padlock to lock the box before sending it back to her. The critical advantage in an asymmetric key system is that Bob and Alice never need send a copy of their keys to each other. This substantially reduces the chance that a third party (perhaps, in the example, a corrupt postal worker) will copy a key while it is in transit, allowing said third party to spy on all future messages sent between Alice and Bob. In addition, if Bob were to be careless and allow someone else to copy his key, Alice's messages to Bob will be compromised, but Alice's messages to other people would remain secret, since the other people would be providing different padlocks for Alice to use...

Actual Algorithms - Two Linked Keys edit

Fortunately cryptography is not concerned with actual padlocks, but with encryption algorithms which aren't vulnerable to hacksaws, bolt cutters, or liquid nitrogen attacks.

Not all asymmetric key algorithms operate in precisely this fashion. The most common have the property that Alice and Bob own two keys; neither of which is (so far as is known) deducible from the other. This is known as public-key cryptography, since one key of the pair can be published without affecting message security. In the analogy above, Bob might publish instructions on how to make a lock ("public key"), but the lock is such that it is impossible (so far as is known) to deduce from these instructions how to make a key which will open that lock ("private key"). Those wishing to send messages to Bob use the public key to encrypt the message; Bob uses his private key to decrypt it.

Weaknesses edit

Of course, there is the possibility that someone could "pick" Bob's or Alice's lock. Unlike the case of the one-time pad or its equivalents, there is no currently known asymmetric key algorithm which has been proven to be secure against a mathematical attack. That is, it is not known to be impossible that some relation between the keys in a key pair, or a weakness in an algorithm's operation, might be found which would allow decryption without either key, or using only the encryption key. The security of asymmetric key algorithms is based on estimates of how difficult the underlying mathematical problem is to solve. Such estimates have changed both with the decreasing cost of computer power, and with new mathematical discoveries.

Weaknesses have been found in promising asymmetric key algorithms in the past. The 'knapsack packing' algorithm was found to be insecure when an unsuspected attack came to light. Recently, some attacks based on careful measurements of the exact amount of time it takes known hardware to encrypt plain text have been used to simplify the search for likely decryption keys. Thus, use of asymmetric key algorithms does not ensure security; it is an area of active research to discover and protect against new and unexpected attacks.

Another potential weakness in the process of using asymmetric keys is the possibility of a 'Man in the Middle' attack, whereby the communication of public keys is intercepted by a third party and modified to provide the third party's own public keys instead. The encrypted response also must be intercepted, decrypted and re-encrypted using the correct public key in all instances however to avoid suspicion, making this attack difficult to implement in practice.

A Brief History edit

The first known asymmetric key algorithm was invented by Clifford Cocks of GCHQ in the UK. It was not made public at the time, and was reinvented by Rivest, Shamir, and Adleman at MIT in 1976. It is usually referred to as RSA as a result. RSA relies for its security on the difficulty of factoring very large integers. A breakthrough in that field would cause considerable problems for RSA's security. Currently, RSA is vulnerable to an attack by factoring the 'modulus' part of the public key, even when keys are properly chosen, for keys shorter than perhaps 700 bits. Most authorities suggest that 1024 bit keys will be secure for some time, barring a fundamental breakthrough in factoring practice or practical quantum computers, but others favor longer keys.

At least two other asymmetric algorithms were invented after the GCHQ work, but before the RSA publication. These were the Ralph Merkle puzzle cryptographic system and the Diffie-Hellman system. Well after RSA's publication, Taher Elgamal invented the Elgamal discrete log cryptosystem which relies on the difficulty of inverting logs in a finite field. It is used in the SSL, TLS, and DSA protocols.

A relatively new addition to the class of asymmetric key algorithms is elliptic curve cryptography. While it is more complex computationally, many believe it to represent a more difficult mathematical problem than either the factorisation or discrete logarithm problems.

Practical limitations and hybrid cryptosystems edit

One drawback of asymmetric key algorithms is that they are much slower (factors of 1000+ are typical) than 'comparably' secure symmetric key algorithms. In many quality crypto systems, both algorithm types are used; they are termed 'hybrid systems'. PGP is an early and well-known hybrid system. The receiver's public key encrypts a symmetric algorithm key which is used to encrypt the main message. This combines the virtues of both algorithm types when properly done.

We discuss asymmetric ciphers in much more detail later in the Public Key Overview and following sections of this book.


Random number generation

The generation of random numbers is essential to cryptography. One of the most difficult aspect of cryptographic algorithms is in depending on or generating, true random information. This is problematic, since there is no known way to produce true random data, and most especially no way to do so on a finite state machine such as a computer.

There are generally two kinds of random number generators: non-deterministic random number generators, sometimes called "true random number generators" (TRNG), and deterministic random number generators, also called pseudorandom number generators (PRNG).[10]

Many high-quality cryptosystems use both -- a hardware random-number generator to periodically re-seed a deterministic random number generator.

Quantum mechanical theory suggests that some physical processes are inherently random (though collecting and using such data presents problems), but deterministic mechanisms, such as computers, cannot be. Any stochastic process (generation of random numbers) simulated on a computer, however, is not truly random, but only pseudorandom.

Within the limitations of pseudorandom generators, any quality pseudorandom number generator must:

  • have a uniform distribution of values, in all dimensions
  • have no detectable pattern, ie generate numbers with no correlations between successive numbers
  • have a very long cycle length
  • have no, or easily avoidable, weak initial conditions which produce patterns or short cycles

Methods of Pseudorandom Number Generation edit

Keeping in mind that we are dealing with pseudorandom number generation (i.e. numbers generated from a finite state machine, as a computer), there are various ways to randomly generate numbers.

In C and C++ the function rand() returns a pseudo-random integer between zero and RAND_MAX (internally defined constant), defined with the srand() function; otherwise it will use the default seed and consistently return the same numbers when the program is restarted. Most such libraries have short cycle lengths and are not usable for cryptographic purposes.

"Numerical Recipes in C" reviews several random number generators and recommends a modified version of the DES cypher as their highest quality recommended random number generator. "Practical Cryptography" (Ferguson and Schneier) recommend a design they have named Fortuna; it supersedes their earlier design called Yarrow.

Methods of nondeterministic number generation edit

As of 2004, the best random number generators have 3 parts: an unpredictable nondeterministic mechanism, entropy assessment, and conditioner. The nondeterministic mechanism (also called the entropy source) generates blocks of raw biased bits. The entropy assessment part produces a conservative estimate of the min-entropy of some block of raw biased bits. The conditioner (also called a whitener, an unbiasing algorithm, or a randomness extractor) distills the block of raw bits into a much smaller block of conditioned output bits -- an output block of bits half the size of the estimated entropy (in bits) of the raw biased bits -- eliminating any systematic bias. If the estimate is good, the conditioned output bits are unbiased full-entropy bits even if the nondeterministic mechanism degrades over time. In practice, the entropy assessment is the difficult part.[11]

References edit

Further Reading edit


Hashes

A digest, sometimes simply called a hash, is the result of a hash function, a specific mathematical function or algorithm, that can be described as  . "Hashing" is required to be a deterministic process, and so, every time the input block is "hashed" by the application of the same hash function, the resulting digest or hash is constant, maintaining a verifiable relation with the input data. Thus making this type of algorithms useful for information security.

Other processes called cryptographic hashes, function similarly to hashing, but require added security, in the form or a level of guarantee that the input data can not feasibly be reversed from the generated hash value. I.e. That there is no useful inverse hash function  

This property can be formally expanded to provide the following properties of a secure hash:

  • Preimage resistant : Given H it should be hard to find M such that H = hash(M).
  • Second preimage resistant: Given an input m1, it should be hard to find another input, m2 (not equal to m1) such that hash(m1) = hash(m2).
  • Collision-resistant: it should be hard to find two different messages m1 and m2 such that hash(m1) = hash(m2). Because of the birthday paradox this means the hash function must have a larger image than is required for preimage-resistance.

A hash function is the implementation of an algorithm that, given some data as input, will generate a short result called a digest. A useful hash function generates a fixed length of hashed value.

For Ex: If our hash function is 'X' and we have 'wiki' as our input... then X('wiki')= a5g78 i.e. some hash value.

Applications of hash functions edit

Non-cryptographic hash functions have many applications,[1] but in this section we focus on applications that specifically require cryptographic hash functions:

A typical use of a cryptographic hash would be as follows: Alice poses to Bob a tough math problem and claims she has solved it. Bob would like to try it himself, but would yet like to be sure that Alice is not bluffing. Therefore, Alice writes down her solution, appends a random nonce, computes its hash and tells Bob the hash value (whilst keeping the solution secret). This way, when Bob comes up with the solution himself a few days later, Alice can verify his solution but still be able to prove that she had the solution earlier.

In actual practice, Alice and Bob will often be computer programs, and the secret would be something less easily spoofed than a claimed puzzle solution. The above application is called a commitment scheme. Another important application of secure hashes is verification of message integrity. Determination of whether or not any changes have been made to a message (or a file), for example, can be accomplished by comparing message digests calculated before, and after, transmission (or any other event) (see Tripwire, a system using this property as a defense against malware and malfeasance). A message digest can also serve as a means of reliably identifying a file.

A related application is password verification. Passwords should not be stored in clear text, for obvious reasons, but instead in digest form. In a later chapter, Password handling will be discussed in more detail—in particular, why hashing the password once is inadequate.

A hash function is a key part of message authentication (HMAC).

Most distributed version control systems (DVCSs) use cryptographic hashes.[2]

For both security and performance reasons, most digital signature algorithms specify that only the digest of the message be "signed", not the entire message. The Hash functions can also be used in the generation of pseudo-random bits.

SHA-1, MD5, and RIPEMD-160 are among the most commonly-used message digest algorithms as of 2004. In August 2004, researchers found weaknesses in a number of hash functions, including MD5, SHA-0 and RIPEMD. This has called into question the long-term security of later algorithms which are derived from these hash functions. In particular, SHA-1 (a strengthened version of SHA-0), RIPEMD-128 and RIPEMD-160 (strengthened versions of RIPEMD). Neither SHA-0 nor RIPEMD are widely used since they were replaced by their strengthened versions.

Other common cryptographic hashes include SHA-2 and Tiger.

Later we will discuss the "birthday attack" and other techniques people use for Breaking Hash Algorithms.

Hash speed edit

When using hashes for file verification, people prefer hash functions that run very fast. They want a corrupted file can be detected as soon as possible (and queued for retransmission, quarantined, or etc.). Some popular hash functions in this category are:

  • BLAKE2b
  • SHA-3

In addition, both SHA-256 (SHA-2) and SHA-1 have seen hardware support in some CPU instruction sets.

When using hashes for password verification, people prefer hash functions that take a long time to run. If/when a password verification database (the /etc/passwd file, the /etc/shadow file, etc.) is accidentally leaked, they want to force a brute-force attacker to take a long time to test each guess.[3] Some popular hash functions in this category are:

  • Argon2
  • scrypt
  • bcrypt
  • PBKDF2

We talk more about password hashing in the Cryptography/Secure Passwords section.

Further reading edit

  1. applications of non-cryptographic hash functions are described in Data Structures/Hash Tables and Algorithm Implementation/Hashing.
  2. Eric Sink. "Version Control by Example". Chapter 12: "Git: Cryptographic Hashes".
  3. "Speed Hashing"


Common flaws and weaknesses

Cryptography relies on puzzles. A puzzle that can not be solved without more information than the cryptanalyst has or can feasibly acquire is an unsolvable puzzle for the attacker. If the puzzle can be understood in a way that circumvents the secret information the cryptanalyst doesn't have then the puzzle is breakable. Obviously cryptography relies on an implicit form of security through obscurity where there currently exists no likely ways to understand the puzzle that will break it. The increasing complexity and subtlety of the mathematical puzzles used in cryptography creates a situation where neither cryptographers or cryptanalysts can be sure of all facets of the puzzle and security.

Like any puzzle, cryptography algorithms are based on assumptions - if these assumptions are flawed then the underlying puzzle may be flawed.

Secret knowledge assumption - Certain secret knowledge is not available to unauthorised people. Attacks such as packet sniffing, keylogging and meet in the middle attacks try to breach this assumption.

Secret knowledge masks plaintext - The secret knowledge is applied to the plaintext so that the nature of the message is no longer obvious. In general the secret knowledge hides the message in way so that the secret knowledge is required in order rediscover the message. Attacks such as chosen plaintext, brute force and frequency analysis try to breach this assumption


Secure Passwords

Passwords edit

A serious cryptographic system should not be based on a hidden algorithm, but rather on a hidden password that is hard to guess (see Kerckhoffs's law in the Basic Design Principles section). Passwords today are very important because access to a very large number of portals on the Internet, or even your email account, is restricted to those who can produce the correct password. This usually involves humans in choosing, remembering, and using passwords. All three aspects are commonly weaknesses: humans are notoriously bad at choosing hard-to-break passwords,[1] do not easily remember strong passwords, and are sloppy and too trusting in their use of passwords when they remember them. It is nearly overwhelmingly tempting to base passwords on already known items. As well, we can remember simple (e.g. short), or familiar (e.g. telephone number) pretty well, but stronger passwords are more than most of us can reliably remember; this leads to insecurity as easy methods of password recovery, or even password bypass, are required. These are universally insecure. Finally, humans are too easily prey to phishing fraud scams, to shoulder surfing, to helping out a friend who has forgotten their own password, etc.

Security edit

But passwords must protect access and messages against more than just human attackers. There are many machine-based ways of attacking cryptographic algorithms and cryptosystems, so passwords should also be hard to attack automatically. To prevent one important class of automatic attack, the brute force search, passwords must be difficult for the bad guys to guess. be both long (single character passwords are easily guessed, obviously) and, ideally, random—that is, without pattern of any kind. A long enough password will require so much machine time as to be impractical for an attacker. A password without pattern will offer no shortcut to brute force search. These considerations suggest several properties passwords should possess:

  • sufficient length to preclude brute force search (common recommendations as of 2010 are at least 8 characters, and more when any class of character is not allowed (e.g. if lower case is not permitted, or non alphanumeric characters are not permitted, ..., a longer password is required); more length is required if the password should remain unbreakable into the future (when computers will be faster and brute force searches more effective)
  • no names (pets, friends, relatives, ...), no words findable in any dictionary, no phrases found in any quotation book
  • no personally connected numbers or information (telephone numbers, addresses, birthdays)

Password handling edit

Password handling is simultaneously one of the few Solved Problems of Cryptography, *and* one of the most misunderstood.

Any web server that stores user passwords in plain text in some file or database somewhere, is doing it wrong.[2][3][4][5][6]

Passwords are usually not stored in cleartext, for obvious reasons, but instead in digest form. To authenticate a user, the password presented by the user is salted, hashed, and compared with the stored hash.[7]

PBKDF2 was originally designed to "generating a cryptographic key from a password", but it turns out to be good for generating password digests for safe storage for authentication purposes.[8][9]

In 2013, because only 3 algorithms are available for generating password digests for safe storage for authentication purposes—PBKDF2, bcrypt, and scrypt[6]—In 2013, the Password Hashing Competition (PHC) was announced. [10][11][12][13]

passphrase hashing algorithm edit

 

To do:
mention some historic password hashing algorithms, such as the Purdy polynomial by Wikipedia: George B. Purdy; the traditional DES-based scheme that truncates the user's password to eight characters; and also say a few words about how the Modular Crypt Format Wikipedia: Crypt (C) makes it easier to transition to a new password hashing algorithm.


The main difference between a password hashing algorithm and other cryptographic hash algorithms is that a password hashing algorithm should make it difficult for attackers who have massively parallel GPUs and FPGAs to recover a passphrase—even if the passphrase is relatively weak—from the stored password digest. The most common way of doing this is to design the algorithm so the amount of time it takes for such an attacker to recover a weak passphrase should be much longer than the amount of time it takes for an authorized server, when given the correct passphrase, to verify that it is in fact correct.

The password verification utility passwd uses a secret password and non-secret salt to generate password hash digests using the crypt (C) library, which in turn uses many password hashing algorithms.

Often a single shadow password file stores password hash digests generated by several different hash algorithms. The particular hash algorithm used can be identified by a unique code prefix in the resulting hashtext, following a pseudo-standard called Modular Crypt Format.[14][15][16]

Any available hash algorithm is vastly superior to the shameful[17] practice of simply storing passwords in plain text; or storing "encrypting" passwords that can be quickly decrypted to recover the original password.

Unfortunately, practically all available hash algorithms were not originally designed for password hashing.

While one iterations of SHA-2 and SHA-3 are more than adequate for calculating a file verification digest, a message authentication code, or a digital signature when applied to long files, they are too easy to crack when applied to short passwords even when iterated a dozen times. Practically all available hash algorithms as of 2013 are either

(a) already known to be relatively easy to crack (recover a weak password) using GPU-based brute-force password cracking tools, or (b) are too new or haven't been studied enough to know whether it's easy to crack or not.[18]

For example, systems have been built that can recover a valid password from any Windows XP LM hash or 6-printable-character password in at most 6 minutes,[19][20] and can recover any 8-printable-character password from a NTLM hash in at most 5.5 hours.[19]

For example, systems have been built that can run through a dictionary of possible words and common "leet speak" substitutions[20] in order to recover the password that produced some given MD5 hash at a rate of 180 Ghashes/second,[21] or produced some given DEC crypt() digest at a rate of 73 Mhashes/second.[22]

A technique called "key stretching" makes key search attacks more expensive.[23]

Further reading edit

  1. Burr, Dodson, Polk. "Electronic Authentication Guideline: Recommendations of the National Institute of Standards and Technology" section A.2.2 "Min Entropy Estimates": "Experience suggests that a significant share of users will choose passwords that are very easily guessed ("password" may be the most commonly selected password, where it is allowed)." [1]
  2. http://plaintextoffenders.com/
  3. http://crypto.stackexchange.com/tags/passwords/info
  4. http://security.stackexchange.com/questions/17979/is-sending-password-to-user-email-secure
  5. http://security.stackexchange.com/tags/passwords/info
  6. a b Adam Bard. "3 Wrong Ways to Store a Password: And 5 code samples doing it right". 2013.
  7. "How to securely hash passwords?"
  8. "Does NIST really recommend PBKDF2 for password hashing?"
  9. "Is it safe to use PBKDF2 for hashing?"
  10. http://crypto.stackexchange.com/questions/12795/why-do-i-need-to-add-the-original-salt-to-each-hash-iteration-of-a-password
  11. "Increase the security of an already stored password hash". quote: "a new open competition for password hashing algorithms has been launched, using the model of the previous AES, eSTREAM and SHA-3 competitions. Submissions are due for the end of January 2014."
  12. "Password Hashing Competition"
  13. "Are there more modern password hashing methods than bcrypt and scrypt?"
  14. Simson Garfinkel, Alan Schwartz, Gene Spafford. "Practical Unix & Internet Security". 2003. section "4.3.2.3 crypt16( ), DES Extended, and Modular Crypt Format". "The Modular Crypt Format (MCF) specifies an extensible scheme for formatting encrypted passwords. MCF is one of the most popular formats for encrypted passwords"
  15. "Modular Crypt Format: or, a side note about a standard that isn’t".
  16. "Binary Modular Crypt Format"
  17. "Plain Text Offenders"
  18. Dennis Fisher. "Cryptographers aim to find new password hashing algorithm". 2013.
  19. a b Paul Roberts. "Update: New 25 GPU Monster Devours Passwords In Seconds". 2012.
  20. a b Dan Goodin "Anatomy of a hack: even your 'complicated' password is easy to crack". 2013.
  21. Jeremi M. Gosney. "Password Cracking HPC". 2012.
  22. "John the Ripper benchmarks".
  23. Arnold Reinhold. "HEKS: A Family of Key Stretching Algorithms". 1999.


S-box

In cryptography, an S-Box (Substitution-box) is a basic component of symmetric key algorithms which performs substitution. In block ciphers, they are typically used to obscure the relationship between the key and the ciphertext — Claude Shannon's property of confusion.[1] In many cases, the S-Boxes are carefully chosen to resist cryptanalysis.

In general, an S-Box takes some number of input bits, m, and transforms them into some number of output bits, n: an m×n S-Box can be implemented as a lookup table with   words of n bits each. Fixed tables are normally used, as in the Data Encryption Standard (DES), but in some ciphers the tables are generated dynamically from the key; e.g. the Blowfish and the Twofish encryption algorithms. Bruce Schneier describes IDEA's modular multiplication step as a key-dependent S-Box.

Given a 6-bit input, the 4-bit output is found by selecting the row using the outer two bits (the first and last bits), and the column using the inner four bits. For example, an input "011011" has outer bits "01" and inner bits "1101"; the corresponding output would be "1001".

The 8 S-Boxes of DES were the subject of intense study for many years out of a concern that a backdoor — a vulnerability known only to its designers — might have been planted in the cipher. The S-Box design criteria were eventually published[2] after the public rediscovery of differential cryptanalysis, showing that they had been carefully tuned to increase resistance against this specific attack. Other research had already indicated that even small modifications to an S-Box could significantly weaken DES.

There has been a great deal of research into the design of good S-Boxes, and much more is understood about their use in block ciphers than when DES was released.

References edit

  1. Chandrasekaran, J. et al. (2011). "A Chaos Based Approach for Improving Non Linearity in the S-Box Design of Symmetric Key Cryptosystems". Advances in Networks and Communications: First International Conference on Computer Science and Information Technology, CCSIT 2011, Bangalore, India, January 2-4, 2011. Proceedings, Part 2. Springer. p. 516. ISBN 978-3-642-17877-1. {{cite book}}: Explicit use of et al. in: |authors= (help); Unknown parameter |editors= ignored (|editor= suggested) (help)CS1 maint: uses authors parameter (link)
  2. Coppersmith, Don (1994). "The Data Encryption Standard (DES) and its strength against attacks" (PDF). IBM Journal of Research and Development. 38 (3): 243–250. doi:10.1147/rd.383.0243. Retrieved 2007-02-20.{{cite journal}}: CS1 maint: ref duplicates default (link)


Basic Design Principles

Good ciphers often attempt to have the following traits.

Kerckhoffs's principle edit

Kerckhoffs's principle, also called Kerckhoffs's law:

A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.

In the words of Claude Shannon, "The enemy knows the system." (Shannon's maxim).


 

To do:
Say something about "security through obscurity" here. Perhaps also something about Kerckhoff's other 5 principles.


Diffusion edit

Having good diffusion means that making a small change in the plain text should ideally cause as much as possible of cipher text to have a fifty percent possibility of change.

For example a Caesar cipher has almost no diffusion while a block cypher may contain lots of it.

Confusion edit

For good confusion the relationship between the cypher text and the plain text should be as complex as possible.

Further reading edit


Open Algorithms

Creating a good cryptographic algorithm that will stand against all that the best cryptanalysis can throw at it, is hard. Very hard. So, this is why most people design algorithms by first designing the basic system, then refining it, and finally letting it lose for all to see.

Why, do this? Surely, if you let everyone see your code that turns a plain bit of text into garbled rubbish, then they will be able to reverse it! This assumption is unfortunately wrong. Now the algorithms that have been/ are being made are so strong, that just reversing the algorithm is not effective when trying to crack it. And when you let people look at your algorithm, they may spot a security flaw that nobody else could see. We talk more about this counter-intuitive idea in another chapter, Basic Design Principles#Kerckhoffs's principle.

AES, one of the newest and strongest (2010) algorithms in the world, was created by a team of two people, and was put forward into a sort of competition, where only the best algorithm would be examined and put forward to be selected for the title of the Advanced Encryption Standard. There were about 35 entrants, and although all of them appeared strong at first, it soon became clear that some of these apparently strong algorithms were in fact, very weak!

AES is a good example of open algorithms.


Mathematical Background

See Talk page for other suggestions.

Introduction edit

Modern public-key (asymmetric) cryptography is based upon a branch of mathematics known as number theory, which is concerned solely with the solution of equations that yield only integer results. These type of equations are known as diophantine equations, named after the Greek mathematician Diophantos of Alexandria (ca. 200 CE) from his book Arithmetica that addresses problems requiring such integral solutions.

One of the oldest diophantine problems is known as the Pythagorean problem, which gives the length of one side of a right triangle when supplied with the lengths of the other two side, according to the equation

 

where   is the length of the hypotenuse. While two sides may be known to be integral values, the resultant third side may well be irrational. The solution to the Pythagorean problem is not beyond the scope, but is beyond the purpose of this chapter. Therefore, example integral solutions (known as Pythagorean triplets) will simply be presented here. It is left as an exercise for the reader to find additional solutions, either by brute-force or derivation.

Pythagorean Triplets
     
3 4 5
5 12 13
7 24 25
8 15 17

Prime Numbers edit

Description edit

Asymmetric key algorithms rely heavily on the use of prime numbers, usually exceedingly long primes, for their operation. By definition, prime numbers are divisible only by themselves and 1. In other words, letting the symbol | denote divisibility (i.e. -   means "  divides into  "), a prime number strictly adheres to the following mathematical definition

  |   Where   or   only

The Fundamental Theorem of Arithmetic states that all integers can be decomposed into a unique prime factorization. Any integer greater than 1 is considered either prime or composite. A composite number is composed of more than one prime factor

  |   where ultimately  

in which   is a unique prime number and   is the exponent.

Numerical Examples edit

543,312 = 24   32   50   73   111
553,696 = 25   30   50   70   113   131

As can be seen, according to this systematic decomposition, each factorization is unique.

In order to deterministically verify whether an integer   is prime or composite, only the primes   need be examined. This type of systematic, thorough examination is known as a brute-force approach. Primes and composites are noteworthy in the study of cryptography since, in general, a public key is a composite number which is the product of two or more primes. One (or more) of these primes may constitute the private key.

There are several types and categories of prime numbers, three of which are of importance to cryptography and will be discussed here briefly.

Fermat Primes edit

Fermat numbers take the following form

 

If Fn is prime, then it is called a Fermat prime.

Numerical Examples edit

 
 
 
 
 
 

The only Fermat numbers known to be prime are  . Moreover, the primality of all Fermat numbers was disproven by Euler, who showed that  .

Mersenne Primes edit

Mersenne primes - another type of formulaic prime generation - follow the form

 

where   is a prime number. The [8] Wolfram Alpha engine reports Mersenne Primes, an example input request being "4th Mersenne Prime".

Numerical Examples edit

The first four Mersenne primes are as follows

 
 
 
 

Numbers of the form Mp = 2p without the primality requirement are called Mersenne numbers. Not all Mersenne numbers are prime, e.g. M11 = 211−1 = 2047 = 23 · 89.

Coprimes (Relatively Prime Numbers) edit

Two numbers are said to be coprime if the largest integer that divides evenly into both of them is 1. Mathematically, this is written

 

where   is the greatest common divisor. Two rules can be derived from the above definition

If   |   and  , then   |  
If   with  , then both   and   are squares, i.e. -  ,  

The Prime Number Theorem edit

The Prime Number Theorem estimates the probability that any integer, chosen randomly will be prime. The estimate is given below, with   defined as the number of primes  

 

  is asymptotic to  , that is to say  . What this means is that generally, a randomly chosen number is prime with the approximate probability  .

The Euclidean Algorithm edit

Introduction edit

The Euclidean Algorithm is used to discover the greatest common divisor of two integers. In cryptography, it is most often used to determine if two integers are coprime, i.e. -  .

In order to find   where   efficiently when working with very large numbers, as with cryptosystems, a method exists to do so. The Euclidean algorithm operates as follows - First, divide   by  , writing the quotient  , and the remainder  . Note this can be written in equation form as  . Next perform the same operation using   in  's place:  . Continue with this pattern until the final remainder is zero. Numerical examples and a formal algorithm follow which should make this inherent pattern clear.

Mathematical Description edit

 
 
 
 
 
 
 
 

When  , stop with  .

Numerical Examples edit

Example 1 - To find gcd(17,043,12,660)

17,043 = 1   12,660 + 4383
12,660 = 2   4,383 + 3894
 4,383 = 1   3,894 + 489
 3,894 = 7   489 + 471
   489 = 1   471 + 18
   471 = 26   18 + 3
    18 = 6   3 + 0

gcd (17,043,12,660) = 3

Example 2 - To find gcd(2,008,1,963)

2,008 = 1   1,963 + 45
1,963 = 43   45 + 28
   45 = 1   28 + 17
   28 = 1   17 + 11
   17 = 1   11 + 6
   11 = 1   6 + 5
    6 = 1   5 + 1
    5 = 5   1 + 0

gcd (2,008,1963) = 1 Note: the two number are coprime.

Algorithmic Representation edit

Euclidean Algorithm(a,b)
Input:     Two integers a and b such that a > b
Output:    An integer r = gcd(a,b)
  1.   Set a0 = a, r1 = r
  2.   r = a0 mod r1
  3.   While(r1 mod r   0) do:
  4.      a0 = r1
  5.      r1 = r
  6.      r = a0 mod r1
  7.   Output r and halt

The Extended Euclidean Algorithm edit

In order to solve the type of equations represented by Bézout's identity, as shown below

 

where  ,  ,  , and   are integers, it is often useful to use the extended Euclidean algorithm. Equations of the form above occur in public key encryption algorithms such as RSA (Rivest-Shamir-Adleman) in the form   where  . There are two methods in which to implement the extended Euclidean algorithm; the iterative method and the recursive method.

As an example, we shall solve an RSA key generation problem with e = 216 + 1, p = 3,217, q = 1,279. Thus, 62,537d + 51,456w = 1.

Methods edit

The Iterative Method edit

This method computes expressions of the form   for the remainder in each step   of the Euclidean algorithm. Each modulus can be written in terms of the previous two remainders and their whole quotient as follows:

 

By substitution, this gives:

 
 

The first two values are the initial arguments to the algorithm:

 
 

The expression for the last non-zero remainder gives the desired results since this method computes every remainder in terms of a and b, as desired.

Example edit
Step Quotient Remainder Substitute Combine terms
1 4,110,048 = a 4,110,048 = 1a + 0b
2 65,537 = b 65,537 = 0a + 1b
3 62 46,754 = 4,110,048 - 65,537   62 46,754 = (1a + 0b) - (0a + 1b)   62 46,754 = 1a - 62b
4 1 18,783 = 65,537 - 46,754   1 18,783 = (0a + 1b) - (1a - 62b)   1 18,783 = -1a + 63b
5 2 9,188 = 46,754 - 18,783   2 9,188 = (1a - 62b) - (-1a + 62b)   2 9,188 = 3a - 188b
6 2 407 = 18,783 - 9,188   2 407 = (-1a + 63b) - (3a - 188b)   2 407 = -7a + 439b
7 22 234 = 9,188 - 407   22 234 = (3a - 188b) - (-7a + 439b)   22 234 = 157a - 9,846b
8 1 173 = 407 - 234   1 173 = (-7a + 439b) - (157a - 9,846b)   1 173 = -164a + 10,285b
9 1 61 = 234 - 173   1 61 = (157a - 9,846b) - (-164a + 10,285b)   1 61 = 321a + 20,131b
10 2 51 = 173 - 61   2 51 = (-164a + 10,285b) - (321a +20,131b)   2 51 = -806a + 50,547b
11 1 10 = 61 - 51   1 61 = (321a +20,131b) - (-806a + 50,547b)   1 10 = 1,127a - 70,678b
12 5 1 = 51 -10   5 1 = (-806a + 50,547b) - (1,127a - 70,678b)   5 1 = -6,441a + 403,937b
13 10 0 End of algorithm

Putting the equation in its original form   yields  , it is shown that   and  . During the process of key generation for RSA encryption, the value for w is discarded, and d is retained as the value of the private key In this case

d = 0x629e1 = 01100010100111100001

The Recursive Method edit

This is a direct method for solving Diophantine equations of the form  . Using this method, the dividend and the divisor are reduced over a series of steps. At the last step, a trivial value is substituted into the equation, and is then worked backward until the solution is obtained.

Example edit

Using the previous RSA vales of   and  

Euclidean Expansion Collect Terms Substitute Retrograde Substitution Solve For dx
4,110,048 w0 + 65,537d0 = 1
(62   65,537 + 46,754) w0 + 65,537d0 = 1
65,537 (62w0 + d0) + 46,754w0 = 1 w1 = 62w0 + d0 4,595 = (62)(-6441) + d0 d0 = 403,937
65,537 w1 + 46,754d1 = 1 d1 = w0 w1 = -6,441
(1   46,754 + 18,783) w1 + 46,754d1 = 1
46,754 (w1 + d1) + 18,783w1 = 1 w2 = w1 + d1 -1,846 = 4,595 + d1 d1 = -6,441
46,754 w2 + 18,783d2 = 1 d2 = w1
(2   18,783 + 9,188) w2 + 18,783d2 = 1
18,783 (2w2 + d2) + 9,188w2 = 1 w3 = 2w2 + d2 903 = (2)(-1,846) + d2 d2 = 4,595
18,783 w3 + 9,188d3 = 1 d3 = w2
(2   9,188 + 407) w3 + 9,188d3 = 1
9,188 (2w3 + d3) + 407w3 = 1 w4 = 2w3 + d3 -40 = (2)(903) + d3 d3 = -1846
9,188 w4 + 407d4 = 1 d4 = w3
(22   407 + 234) w4 + 407d4 = 1
407 (22w4 + d4) + 234w4 = 1 w5 = 22w4 +d4 23 = (22)(-40) + d4 d4 = 903
407 w5 + 234d5 = 1 d5 = w4
(1   234 + 173) w5 + 234d5 = 1
234 (w5 + d5) + 173w5 = 1 w6 = w5 +d5 -17 = 23 + d5 d5 = -40
234 w6 + 173d6 = 1 d6 = w5
(1   173 + 61) w6 + 173d6 = 1
173 (w6 + d6) + 61w6 = 1 w7 = w6 +d6 6 = -17 + d6 d6 = 23
173 w7 + 61d7 = 1 d7 = w6
(2   61 + 51) w7 + 61d7 = 1
61 (2w7 + d7) + 51w7 = 1 w8 = 2w7 +d7 -5 = (2)(6) + d7 d7 = -17
61 w8 + 51d8 = 1 d8 = w7
(1   51 + 10) w8 + 51d8 = 1
51 (w8 + d8) + 10w8 = 1 w9 = w8 +d8 1 = -5 + d8 d8 = 6
51 w9 + 10d9 = 1 d9 = w8
(5   10 + 1) w9 + 10d9 = 1
10 (5w9 + d9) + 1w9 = 1 w10 = 5w9 +d9 0 = (5)(1) + d9 d9 = -5
10 w10 + 1d10 = 1 d10 = w9
(1   10 + 0) w10 + 1d10 = 1
1 (10w10 + d10) + 0w10 = 1 w11 = 10w10 +d10 1 = (10)(0) + d10 d10 = 1
1 w11 + 0d11 = 1 d11 = w10 w11 = 1, d11 = 0

Euler's Totient Function edit

Significant in cryptography, the totient function (sometimes known as the phi function) is defined as the number of nonnegative integers   less than   that are coprime to  . Mathematically, this is represented as

 

Which immediately suggests that for any prime  

 

The totient function for any exponentiated prime is calculated as follows

 
 
 

The Euler totient function is also multiplicative

 

where  

Finite Fields and Generators edit

A field is simply a set   which contains numerical elements that are subject to the familiar addition and multiplication operations. Several different types of fields exist; for example,  , the field of real numbers, and  , the field of rational numbers, or  , the field of complex numbers. A generic field is usually denoted  .

Finite Fields edit

Cryptography utilizes primarily finite fields, nearly exclusively composed of integers. The most notable exception to this are the Gaussian numbers of the form   which are complex numbers with integer real and imaginary parts. Finite fields are defined as follows

  The set of integers modulo  
  The set of integers modulo a prime  

Since cryptography is concerned with the solution of diophantine equations, the finite fields utilized are primarily integer based, and are denoted by the symbol for the field of integers,  .

A finite field   contains exactly   elements, of which there are   nonzero elements. An extension of   is the multiplicative group of  , written  , and consisting of the following elements

  such that  

in other words,   contains the elements coprime to  

Finite fields form an abelian group with respect to multiplication, defined by the following properties

  The product of two nonzero elements is nonzero  
  The associative law holds  
  The commutative law holds  
  There is an identity element  
  Any nonzero element has an inverse  

A subscript following the symbol for the field represents the set of integers modulo  , and these integers run from   to   as represented by the example below

 

The multiplicative order of   is represented   and consists of all elements   such that  . An example for   is given below

 

If   is prime, the set   consists of all integers   such that  . For example

Composite n Prime p
   
   

Generators edit

Every finite field has a generator. A generator   is capable of generating all of the elements in the set   by exponentiating the generator  . Assuming   is a generator of  , then   contains the elements   for the range  . If   has a generator, then   is said to be cyclic.

The total number of generators is given by

 

Examples edit

For   (Prime)

 
 

Total number of generators   generators

Let  , then  ,   is a generator

Since   is a generator, check if  
 , and  ,  , therefore,   is not a generator
 , and  ,  , therefore,   is not a generator

Let  , then  ,   is a generator
Let  , then  ,   is a generator
Let  , then  ,   is a generator

There are a total of   generators,   as predicted by the formula  
For   (Composite)

 
 

Total number of generators   generators

Let  , then  ,   is a generator
Let  , then  ,   is a generator

There are a total of   generators   as predicted by the formula  

Congruences edit

Description edit

Number theory contains an algebraic system of its own called the theory of congruences. The mathematical notion of congruences was introduced by Karl Friedrich Gauss in Disquisitiones (1801).

Definition edit

If   and   are two integers, and their difference is evenly divisible by  , this can be written with the notation

 

This is expressed by the notation for a congruence

 

where the divisor   is called the modulus of congruence.   can equivalently be written as

 

where   is an integer.

Note in the examples that for all cases in which  , it is shown that  . with this in mind, note that

  Represents that   is an even number.

  Represents that   is an odd number.

Examples edit

   
   
   
   

Properties of Congruences edit

All congruences (with fixed  ) have the following properties in common

 
  if and only if  
If   and   then  
Given   there exists a unique   such that  

These properties represent an equivalence class, meaning that any integer is congruent modulo   to one specific integer in the finite field  .

Congruences as Remainders edit

If the modulus of an integer  , then for every integer  

 

which can be understood to mean   is the remainder of   divided by  , or as a congruence

 

Two numbers that are incongruent modulo   must have different remainders. Therefore, it can be seen that any congruence   holds if and only if   and   are integers which have the same remainder when divided by  .

Example edit

  is equivalent to
  implies
  is the remainder of   divided by  

The Algebra of Congruences edit

Suppose for this section we have two congruences,   and  . These congruences can be added or subtracted in the following manner

 
 

If these two congruences are multiplied together, the following congruence is obtained

 

or the special case where  

 

Note: The above does not mean that there exists a division operation for congruences. The only possibility for simplifying the above is if and only if   and   are coprime. Mathematically, this is represented as

  implies that   if and only if  

The set of equivalence classes defined above form a commutative ring, meaning the residue classes can be added, subtracted and multiplied, and that the operations are associative, commutative and have additive inverses.

Reducing Modulo m edit

Often, it is necessary to perform an operation on a congruence   where  , when what is desired is a new integer   such that   with the resultant   being the least nonnegative residue modulo m of the congruence. Reducing a congruence modulo   is based on the properties of congruences and is often required during exponentiation of a congruence.

Algorithm edit

Input: Integers   and   from   with  
Output: Integer   such that  

1. Let  
2.      
3.      
4. Output  

Example edit

Given  

 
 
  

Note that   is the least nonnegative residue modulo  

Exponentiation edit

Assume you begin with  . Upon multiplying this congruence by itself the result is  . Generalizing this result and assuming   is a positive integer

 

Example edit

 
 
 

This simplifies to

  implies  
  implies  

Repeated Squaring Method edit

Sometimes it is useful to know the least nonnegative residue modulo   of a number which has been exponentiated as  . In order to find this number, we may use the repeated squaring method which works as follows:

1. Begin with  
2. Square   and   so that  
3. Reduce   modulo   to obtain  
4. Continue with steps 2 and 3 until   is obtained.
   Note that   is the integer where   would be just larger than the exponent desired
5. Add the successive exponents until you arrive at the desired exponent
6. Multiply all  's associated with the  's of the selected powers
7. Reduce the resulting   for the desired result

Example edit

To find  :

 
 
 
 
 
 
 
 

Adding exponents:

 

Multiplying least nonnegative residues associated with these exponents:

 
 

Therefore: 

 

Inverse of a Congruence edit

Description edit

While finding the correct symmetric or asymmetric keys is required to encrypt a plaintext message, calculating the inverse of these keys is essential to successfully decrypt the resultant ciphertext. This can be seen in cryptosystems Ranging from a simple affine transformation

 

Where

 

To RSA public key encryption, where one of the deciphering (private) keys is

 

Definition edit

For the elements   where  , there exists   such that  . Thus,   is said to be the inverse of  , denoted   where   is the   power of the integer   for which  .

Example edit
Find  

This is equivalent to saying  

First use the Euclidean algorithm to verify  .
Next use the Extended Euclidean algorithm to discover the value of  .
In this case, the value is  .

Therefore,  

It is easily verified that  

Fermat's Little Theorem edit

Definition edit

Where   is defined as prime, any integer will satisfy the following relation:

 

Example edit

When   and  

 
 
 
 
  implies that  

Conditions and Corollaries edit

An additional condition states that if   is not divisible by  , the following equation holds

 

Fermat's Little Theorem also has a corollary, which states that if   is not divisible by   and   then

 

Euler's Generalization edit

If  , then  

Chinese Remainder Theorem edit

If one wants to solve a system of congruences with different moduli, it is possible to do so as follows:

 
 
 
 

A simultaneous solution   exists if and only if   with  , and any two solutions are congruent to one another modulo  .

The steps for finding the simultaneous solution using the Chinese Remainder theorem are as follows:

1. Compute  
2. Compute   for each of the different  's
3. Find the inverse   of   for each   using the Extended Euclidean algorithm
4. Multiply out   for each  
5. Sum all  
6. Compute   to obtain the least nonnegative residue

Example edit

Given:

 
 
 
 

 

 
 
 
 

Using the Extended Euclidean algorithm:

 
 
 
 

 

 

 

Quadratic Residues edit

If   is prime and  , examining the nonzero elements of  , it is sometimes important to know which of these are squares. If for some  , there exists a square such that  . Then all squares for   can be calculated by   where  .   is a quadratic residue modulo   if there exists an   such that  . If no such   exists, then   is a quadratic non-residue modulo  .  is a quadratic residue modulo a prime   if and only if  .

Example edit

For the finite field  , to find the squares  , proceed as follows:

 

The values above are quadratic residues. The remaining (in this example) 9 values are known as quadratic nonresidues. the complete listing is given below.

 
Quadratic residues:  
Quadratic nonresidues:  

Legendre Symbol edit

The Legendre symbol denotes whether or not   is a quadratic residue modulo the prime   and is only defined for primes   and integers  . The Legendre of   with respect to   is represented by the symbol  . Note that this does not mean   divided by  .   has one of three values:  .

 

Jacobi Symbol edit

The Jacobi symbol applies to all odd numbers   where  , then:

 

If   is prime, then the Jacobi symbol equals the Legendre symbol (which is the basis for the Solovay-Strassen primality test).

Primality Testing edit

Description edit

In cryptography, using an algorithm to quickly and efficiently test whether a given number is prime is extremely important to the success of the cryptosystem. Several methods of primality testing exist (Fermat or Solovay-Strassen methods, for example), but the algorithm to be used for discussion in this section will be the Miller-Rabin (or Rabin-Miller) primality test. In its current form, the Miller-Rabin test is an unconditional probabilistic (Monte Carlo) algorithm. It will be shown how to convert Miller-Rabin into a deterministic (Las Vegas) algorithm.

Pseudoprimes edit

Remember that if   is prime and  , Fermat's Little Theorem states:

 

However, there are cases where   can meet the above conditions and be nonprime. These classes of numbers are known as pseudoprimes.

  is a pseudoprime to the base  , with   if and only if the least positive power of   that is congruent to   evenly divides  .

If Fermat's Little Theorem holds for any   that is an odd composite integer, then   is referred to as a pseudoprime. This forms the basis of primality testing. By testing different  's, we can probabilistically become more certain of the primality of the number in question.

The following three conditions apply to odd composite integers:

I. If the least positive power of   which is congruent to   and divides   which is the order of   in  , then