Generalized Mersenne Numbers in Pairing-Based Cryptography
01 Jan 2006-
TL;DR: The author’s home country, the United States, and some of the characters from the film adaptation are fictitious.
Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter
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01 Jun 1986
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TL;DR: Some novel methods to compute the index of any integer relative to a given primitive root of a prime p, and how a very simple factorization method results, in which a prime factor p of a number can be found in only 0(pW2) operations.
Abstract: We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our flrst method avoids the use of stored tables and apparently requires O(p 1/2) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O(w/2) operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of application, including the f1actorization of integers. 1. A Rho Method for Index Computation. The concept of a random mapping of a finite set is used by Knuth [1, pp. 7-8] to explain the behavior of a type of random number generator. A sequence obtained by iterating such a function in a set of p elements is 'rho-shaped' with a tail and cycle which are random variables with expectation close to (1) /(irp/8) 0.6267 N/p, (as shown first in [2], [3]). Recently [4], we proposed that this theory be applied to recurrence relations such as (2) xi l x ? 1 (mod p), and showed how a very simple factorization method results, in which a prime factor p of a number can be found in only 0(pW2) operations. The method has been further discussed by Guy [5] and Devitt [6], who have found it suitable for use in programmable calculators. We now suggest that the same theory can be applied to sequences such as xo= 1, i1 qx1 0< x1 < jp 13~~~ (3)~SXi 2? 2i3 ()Xi+ 1 xi' (mod p) for 3' p
846 citations
03 Jan 1991
TL;DR: The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logariths problem in the multiplicative group of an extension of the underlying finite field, thus providing a probabilistic subexponential time algorithm for the former problem.
Abstract: Abstruct- Elliptic cuwe cryptosystems have the potential to provide relatively small block size, high-security public key schemes that can be efficiently implemented. As with other known public key schemes, such as RSA and discrete exponentiation in a finite field, some care must be exercised when selecting the parameters involved, in this case the elliptic curve and the underlying field. Specific classes of cuwes that give little or no advantage over previously known schemes are discussed. The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field. For the class of supersingular elliptic curves, the reduction takes probabilistic polynomial time, thus providing a probabilistic subexponential time algorithm for the former problem. Index Tem- Discrete logarithms, elliptic curves, public key CryPtOSraPhY.
824 citations
Book•
29 May 2012
TL;DR: In this paper, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number, and present over 100 explicit algorithms cast in detailed pseudocode.
Abstract: Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number. Over 100 explicit algorithms cast in detailed pseudocode are included in the book. Applications and theoretical digressions serve to illuminate, justify, and underscore the practical power of these algorithms. The 2nd edition adds new material on primality and algorithms and updates all the numerical records, such as the largest prime, etc. It has been revised throughout.
784 citations
TL;DR: In this paper, the key size for symmetric cryptosystems, RSA, and discrete logarithm-based crypto-systems over finite fields and groups of elliptic curves over prime fields is discussed.
Abstract: In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems.
769 citations
Journal Article•
TL;DR: Recommendations for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosSystems both over finite fields and over groups of elliptic curves over prime fields are offered.
Abstract: In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems.
637 citations