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49
The hidden subgroup problem and eigenvalue estimation on a quantum computer
 Lecture Notes in Computer Science
, 1999
"... Abstract. A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of e ..."
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Cited by 68 (2 self)
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Abstract. A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits. 1
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 53 (3 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.
Circuit Minimization Problem
 In ACM Symposium on Theory of Computing (STOC
, 1999
"... We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surpris ..."
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Cited by 35 (4 self)
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We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NPcomplete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NPcompleteness. 1 Introduction An nvariable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...
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 32 (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...
Linear recurrences with polynomial coefficients and computation of the CartierManin operator on hyperelliptic curves
 In International Conference on Finite Fields and Applications (Toulouse
, 2004
"... Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of h ..."
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Cited by 29 (9 self)
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Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.
Action of modular correspondences around CM points
"... We study the action of modular correspondences in the p adic neighborhood of CM points. We deduce and prove two stable and ecient padic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imagin ..."
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Cited by 28 (0 self)
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We study the action of modular correspondences in the p adic neighborhood of CM points. We deduce and prove two stable and ecient padic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over Fq can be computed in probabilistic time exp((log q) ) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
 Journal of Number Theory
"... Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the se ..."
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Cited by 27 (12 self)
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Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log DE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed. 1.
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 24 (6 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.
An introduction to quantum complexity theory
 Collected Papers on Quantum Computation and Quantum Information Theory
, 2000
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