Question

I’m new to cryptography and modular arithmetic. So, I’m sure it’s a silly question, but I can’t help it.

How do I calculate a from
     pow(a,q) = 1 (mod p),
where p and q are known? I don’t get the ‘1 (mod p)’ part, it equals to 1, doesn’t it? If so, than what is ‘mod p‘ about?
Is this the same as
     pow(a,-q) (mod p) = 1?

Answer 1

The (mod p) part refers not to the right hand side, but to the equality sign: it says that modulo p, pow(a,q) and 1 are equal. For instance, ‘modulo 10, 246126 and 7868726 are equal’ (and they are also both equal to 6 modulo 10): two numbers x and y are equal modulo p if they have the same remainder on dividing by p, or equivalently, if p divides x-y.

Since you seem to be coming from a programming perspective, another way of saying it is that pow(a,q)%p=1, where ‘%’ is the ‘remainder’ operator as implemented in several languages (assuming that p>1).

You should read the Wikipedia article on Modular arithmetic, or any elementary number theory book (or even a cryptography book, since it is likely to introduce modular arithmetic).

To answer your other question: there is no general formula for finding such an a (to the best of my knowledge) in general. Assuming that p is prime, and using Fermat’s little theorem to reduce q modulo p-1, and assuming that q divides p-1 (or else no such a exists), you can produce such an a by taking a primitive root of p and raising it to the power (p-1)/q. [And more generally, when p is not prime, you can reduce q modulo φ(p), then assuming it divides φ(p) and you know a primitive root (say r) mod p, you can take r to the power of φ(p)/q, where φ is the totient function — this comes from Euler’s theorem.]