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Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
Computing A Square Root For The Number Field Sieve
 The Development of the Number Field Sieve, volume 1554 of Lecture Notes in Mathematics
, 1993
"... . The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume,pp. 1140). A heuristic analysis indicates that this method is asymptotically faster than any other existing one. It has had spectacular successes in factoring numbers of ..."
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Cited by 12 (0 self)
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. The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume,pp. 1140). A heuristic analysis indicates that this method is asymptotically faster than any other existing one. It has had spectacular successes in factoring numbers of a special form. New technical difficulties arise when the method is adapted for general numbers (this volume, pp. 4889). Among these is the need for computing the square root of a huge algebraic integer given as a product of hundreds of thousands of small ones. We present a method for computing such a square root that avoids excessively large numbers. It works only if the degree of the number field that is used is odd. The method is based on a careful use of the Chinese remainder theorem. 1. Introduction We begin by recalling the basic scheme of the number field sieve, cf. [7]. Let n be a positive integer that is not a power of a prime number. In order to factor n, we first find m...