594 research outputs found

    Divisibility, Smoothness and Cryptographic Applications

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    This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role

    A deterministic version of Pollard's p-1 algorithm

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    In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's pāˆ’1p-1 algorithm, which finds in random polynomial time the prime divisors pp of an integer nn such that pāˆ’1p-1 is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the kk-th cyclotomic method of factoring (kā‰„2k\ge 2) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function Ļ•\phi. We point out some explicit sets of integers nn that are completely factorable in deterministic polynomial time given Ļ•(n)\phi(n). These sets consist, roughly speaking, of products of primes pp satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of pāˆ’1p-1. Finally, we prove that O(lnā”n)O(\ln n) oracle queries for values of Ļ•\phi are sufficient to completely factor any integer nn in less than expā”((1+o(1))(lnā”n)1/3(lnā”lnā”n)2/3)\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr) deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

    On the average number of divisors of reducible quadratic polynomials

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    We give an asymptotic formula for the divisor sum āˆ‘c<nā‰¤NĻ„((nāˆ’b)(nāˆ’c))\sum_{c<n\leq N}\tau\left((n-b)(n-c)\right) for integers b<cb<c of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a full square. We also provide effective upper bounds of the average divisor sum for some of the reducible quadratic polynomials considered before, with the same main term as in the asymptotic formula.Comment: 16 page

    Computational number theory at CWI in 1970--1994

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    Finite Fields: Theory and Applications

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    Finite ļ¬elds are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of ļ¬nite ļ¬eld techniques in cryptography, error correcting codes, and random number generation

    On the Quadratic Sieve

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    Factoring large integers has long been a subject that has interested mathematicians. And although this interest has been recently increased because of the large usage of cryptography, the thought of factoring integers that are hundreds of digits in length has always been appealing. However it was not until the 1980's that this even seemed fathomable; in fact in 1970 it was extremely difficult to factor a 20-digit number. Then in 1990 the Quadratic Sieve factored a record 116-digit number. While the Quadratic Sieve is not the most recent development in factoring, it is more efficient for factoring numbers below 100-digits than the Number Field Sieve. This paper will discuss the methodology behind the Quadratic Sieve, beginning in its roots in Fermat and Kraitchik's factoring methods. Furthermore our objective is to fully describe the Quadratic Sieve with the goal that the reader could implement a reproduction of the sieve for small numbers
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