Im Kontext von Anwendungen nach dem Signaturgesetz und bei Authentifizierungssystemen wird haufig gefordert, das die verwendeten kryptographischen Schlussel beziehungsweise Schlusselparameter einzigartig (dublettenfrei) und zufallig oder zumindest pseudozufallig sind. In dieser Arbeit wird aufbauend auf zwei bestehenden Verfahren eine Vorgehensweise erlautert, die es ermoglicht, das jeder Systembenutzer seine Schlusselparameter (Primzahlen) effizient und dublettenfrei in seiner lokalen Umgebung erzeugen kann. Trotz der Einzigartigkeit der erzeugten Primzahlen bestehen diese zu einem sehr hohen Anteil aus zufallig gewahlten Bits. Zudem handelt es sich bei den Primzahlen um sogenannte „starke“Primzahlen im Sinne der Resistenz gegen die (p-l)-Faktorisierungsmethode.

Bemerkungen zur Erzeugung dublettenfreier Primzahlen

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 pseudoprime of the same size that passes the Miller-Rabin test for only one base. Therefore our algorithm is even faster than algorithms presently used for generating only pseudoprimes 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.

Fast generation of prime numbers and secure public-key cryptographic parameters

Since the discovery of the RSA encryption scheme, primality domain has gained much interest. For the generation of keys for this code, two prime numbers are used. Amongst the different methods to deal with this problem, we are here interested in generation of certified prime numbers and we present a new method less costly in terms of computation in regard of the other methods of generation for a given size of prime numbers.

https://hal.inria.fr/inria-00074032/document

A new method for the generation of strong prime numbers

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A new method for the generation of strong prime numbers