High School Mathematics Extensions/Further Modular Arithmetic/Problem Set

Content HSME Further Modular Arithmetic Multiplicative Group and Discrete Log Problem Set Project Exercises Solutions Problem Set Solutions Definition Sheet Full Version PDF Version

1. Suppose in mod m arithmetic we know xy and

${\displaystyle y^{2}\equiv x^{2}{\pmod {m}}\!}$

find at least 2 divisors of m.

2. Derive the formula for the Carmichael function, λ(m) = smallest number such that aλ(m) ≡ 1 (mod m).

3. Let p be prime such that p = 2s + 1 for some positive integer s. Show that if g is not a square in mod p, i.e. there's no h such that h2g, then g is a generator mod p. That is gq ≠ 1 for all q < p - 1.