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91
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 69 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
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 55 (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.
The complexity of class polynomial computation via floating point approximations. ArXiv preprint
, 601
"... Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest ..."
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Cited by 45 (7 self)
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Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmeticgeometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time “p “p ”” 3 2 O Dlog D  M Dlog D  ⊆ O ` Dlog 6+ε D  ´ ⊆ O ` h 2+ε´ for any ε> 0, where D is the CM discriminant, h is the degree of the class polynomial and M(n) is the time needed to multiply two nbit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials. 1. Motivation and
Computing discrete logarithms in real quadratic congruence function fields of large genus
 Math. Comp
, 1999
"... Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the dif ..."
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Cited by 43 (9 self)
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Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields. 1.
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 35 (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...
Computing Hilbert class polynomials with the Chinese Remainder Theorem
, 2010
"... We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O ..."
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Cited by 31 (3 self)
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We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O(D  1+ɛ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with D  as large as 1013 and h(D) up to 106. We apply these results to construct pairingfriendly elliptic curves of prime order, using the CM method.
Open Problems in Number Theoretic Complexity, II
"... this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new ..."
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Cited by 30 (0 self)
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this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...
MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES
, 2010
"... We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese R ..."
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Cited by 26 (7 self)
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We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(l3 (log l) 3 log log l), and compute Φl mod m using O(l2 (log l) 2 + l2 log m) space. We have used the new algorithm to compute Φl with l over 5000, and Φl mod m with l over 20000. We also consider several modular functions g for which Φ g l is smaller than Φl, allowing us to handle l over 60000.
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 23 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Some results on pseudosquares
 Math. Comp
, 1996
"... Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe ..."
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Cited by 16 (5 self)
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Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp √ e p/2, an inequality that must hold under the extended Riemann Hypothesis. 1.