Algorithmic number theory
Asymptotic semismoothness probabilities
Mathematics of Computation
Approximating the number of integers free of large prime factors
Mathematics of Computation
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Elementary Numerical Analysis: An Algorithmic Approach
Elementary Numerical Analysis: An Algorithmic Approach
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
A Fast Algorithm for Appoximately Counting Smooth Numbers
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Factorization of a 768-bit RSA modulus
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Fast bounds on the distribution of smooth numbers
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
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An integer n is (y,z)-semismooth if n=pm where m is an integer with all prime divisors ≥ y and p is 1 or a prime ≥ z. Large quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the number field sieve, that incorporate the so-called large prime variant. Thus, it is useful for factoring practitioners to be able to estimate the value of Ψ(x,y,z), the number of (y,z)-semismooth integers up to x, so that they can better set algorithm parameters and minimize running times, which could be weeks or months on a cluster supercomputer. In this paper, we explore several algorithms to approximate Ψ(x,y,z) using a generalization of Buchstab's identity with numeric integration.