Random primes in arithmetic progressions

TL;DR

This paper presents a simple method for generating large random primes with specific subgroup properties, useful for cryptographic applications, though practical efficiency may be limited by large constants.

Contribution

Introduces an asymptotically efficient algorithm for generating primes with large prime-order subgroups and primitive roots, enhancing cryptographic prime generation techniques.

Findings

Method efficiently generates primes with desired subgroup properties

Algorithm produces large prime p and primitive p'th roots of unity

Practical use may be limited by large constants

Abstract

We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of unity. The methods here are efficient asymptotically, but due to large constants may not be very useful in practical settings.