# Information Security and Cryptography Research Group

## Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters

### Ueli Maurer

Journal of Cryptology, vol. 8, no. 3, pp. 123–155, 1995, Preliminary version: [Maurer89].

A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudo-prime of the same size that passes the Miller-Rabin test for only one base. Therefore our algorithm is even faster than presently-used algorithms for generating only pseudo-primes because several Miller-Rabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval.

Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA public-key cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSA-moduli that satisfy these security constraints. Further results described in this paper include an analysis of the optimal upper bound for trial division in the Miller-Rabin test as well as an analysis of the distribution of the number of bits of the smaller prime factor of a random $k$-bit RSA-modulus, given a security bound on the size of the two primes.

Keywords: Public-key cryptography, Prime numbers, Primality proof, Miller-Rabin test, RSA cryptosystem, Number theory.

## BibTeX Citation

@article{Maurer95a,
author       = {Ueli Maurer},
title        = {Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters},
journal      = {Journal of Cryptology},
pages        = {123--155},
number       = {3},
volume       = {8},
year         = {1995},
note         = {Preliminary version: \cite{Maurer89}},
}