High School Mathematics Extensions/Further Modular Arithmetic/Project/Finding the Square Root

High School Mathematics Extensions

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Supplementary Chapters — Primes and Modular Arithmetic — Logic

Mathematical Proofs — Set Theory and Infinite Processes — Counting and Generating Functions — Discrete Probability

Matrices — Further Modular Arithmetic — Mathematical Programming — Markov Chains

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.. to be expanded There is actually a simple way to determine whether a is square

Let g be a generator of G where G is the multiplicative group mod p. Since all the squares form a group therefore, if a is a square, then

$a^{\frac {p-1}{2}}\equiv 1$

and if a is not a square then

$a^{\frac {p-1}{2}}\equiv -1$

we shall use these facts in the next section. .. to be expanded

*Finding the square root*

We aim to describe a way to find a square root in mod m. Let's start with the simplest case, where p is prime. In fact, for square root finding, the simplest case also happens to be the hardest.

If p ≡ 3 (mod 4) then it is easy to find a square root. Just note that if a has a square root then

$(a^{\frac {p+1}{4}})^{2}\equiv a^{\frac {p+1}{2}}\equiv a^{\frac {p-1}{2}}a\equiv a$

So let us consider primes equivalent to 1 mod 4. Suppose we can find the square root of a mod p, and let

$x_{0}^{2}\equiv a{\pmod {p}}$

With the above information, we can find the square of a mod p². We let

$x=x_{0}+x_{1}p\!$

we want x² ≡ a (mod p²), so

$x^{2}=x_{0}^{2}+2x_{0}x_{1}p+x_{1}^{2}p^{2}\!$

$x^{2}=a+kp+2x_{0}x_{1}p{\pmod {p^{2}}}\!$

for some k as $x_{0}^{2}\equiv a{\pmod {p}}$ so $x_{0}^{2}=a+kp$, we see that

$x^{2}=a+p(k+2x_{0}x_{1}){\pmod {p^{2}}}\!$

so if we need to find $x_{1}$ such that $k+2x_{0}x_{1}\equiv 0{\pmod {p}}$, we simply need to make $x_{1}$ the subject

$x_{1}\equiv -k(2x_{0})^{-1}{\pmod {p}}$.

..generalisation ..example

..method for finding a sqr root mod p