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Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 42 (4 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
A polynomialtime theory of blackbox groups I
, 1998
"... We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic o ..."
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Cited by 40 (6 self)
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We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomialtime randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “blackbox groups ” under minimal assumptions. In a blackbox group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”
Discrete Logarithms: the Effectiveness of the Index Calculus Method
, 1996
"... . In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the func ..."
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Cited by 24 (1 self)
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. In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the function field sieve. We also provide a sketch of the some of the cryptographic schemes whose security depends on the intractibility of the discrete logarithm problem. 1 Introduction Let G be a cyclic group generated by an element t. The discrete logarithm problem in G is to compute for any b 2 G the least nonnegative integer e such that t e = b. In this case, we write log t b = e. Our purpose, in this paper, is to survey recent work on the discrete logarithm problem. Our approach is twofold. On the one hand, we consider the problem from a purely theoretical perspective. Indeed, the algorithms that have been developed to solve it not only explore the fundamental nature of one of the basic s...
Discrete Logarithms and Smooth Polynomials, in Finite
 Contemporary Mathematics, Volume 168, American Mathematical Society
, 1994
"... ..."
An analytic approach to smooth polynomials over finite fields
 in Algorithmic Number Theory: Third Intern. Symp., ANTSIII
, 1998
"... Abstract. We consider the largest degrees that occur in the decomposition of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discr ..."
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Cited by 13 (2 self)
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Abstract. We consider the largest degrees that occur in the decomposition of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discrete logarithm problem. In addition, we characterize the distribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treatments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method. 1
The role of smooth numbers in number theoretic algorithms
 In International Congress of Mathematicians
, 1994
"... A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin pr ..."
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Cited by 9 (0 self)
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A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin properties of smooth numbers
The index calculus method using nonsmooth polynomials
 Mathematics of Computation
, 2001
"... Abstract. We study a generalized version of the index calculus method for the discrete logarithm problem in Fq, whenq = p n, p is a small prime and n →∞. The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses ..."
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Cited by 6 (2 self)
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Abstract. We study a generalized version of the index calculus method for the discrete logarithm problem in Fq, whenq = p n, p is a small prime and n →∞. The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database. 1.
Discrete Logarithms in Finite Fields
, 1996
"... Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discr ..."
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Cited by 1 (0 self)
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Given a finite field F q of order q, and g a primitive element of F q , the discrete logarithm base g of an arbitrary, nonzero y 2 F q is that integer x, 0 x q \Gamma 2, such that g x = y in F q . The security of many realworld cryptographic schemes depends on the difficulty of computing discrete logarithms in large finite fields. This thesis is a survey of the discrete logarithm problem in finite fields, including: some cryptographic applications (password authentication, the DiffieHellman key exchange, and the ElGamal publickey cryptosystem and digital signature scheme); Niederreiter's proof of explicit formulas for the discrete logarithm; and algorithms for computing discrete logarithms (especially Shank's algorithm, Pollard's aemethod, the PohligHellman algorithm, Coppersmith's algorithm in fields of order 2 n , and the Gaussian integers method for fields of prime order).
SEQUENCES OF CONSECUTIVE SMOOTH POLYNOMIALS
"... (Communicated by WenChing Winnie Li) Abstract. Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n. 1. ..."
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(Communicated by WenChing Winnie Li) Abstract. Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n. 1.