Smoothness and factoring polynomials over finite fields
Information Processing Letters
Counting the integers factorable via cyclotomic methods
Journal of Algorithms
Cryptography: Theory and Practice
Cryptography: Theory and Practice
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes
CRYPTO '96 Proceedings of the 16th Annual International Cryptology Conference on Advances in Cryptology
Discrete Logarithms: The Effectiveness of the Index Calculus Method
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
Generating EIGamal signatures without knowing the secret key
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
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We obtain rigorous upper bounds on the number of primes p 驴 x for which p-1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.