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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 33 (5 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
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 29 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
The security of Hidden Field Equations (HFE
 In The Cryptographer’s Track at RSA Conference 2001, volume 2020 of Lecture Notes in Computer Science
, 2001
"... Abstract. We consider the basic version of the asymmetric cryptosystem HFE from Eurocrypt 96. We propose a notion of nontrivial equations as a tentative to account for a large class of attacks on oneway functions. We found equations that give experimental evidence that basic HFE can be broken in e ..."
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Cited by 28 (2 self)
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Abstract. We consider the basic version of the asymmetric cryptosystem HFE from Eurocrypt 96. We propose a notion of nontrivial equations as a tentative to account for a large class of attacks on oneway functions. We found equations that give experimental evidence that basic HFE can be broken in expected polynomial time for any constant degree d. It has been independently proven by Shamir and Kipnis [Crypto’99]. We designed and implemented a series of new advanced attacks that are much more efficient that the ShamirKipnis attack. They are practical for HFE degree d ≤ 24 and realistic up to d = 128. The 80bit, 500$ Patarin’s 1st challenge on HFE can be broken in about 2 62. Our attack is subexponential and requires n 3 2 log d computations. The original ShamirKipnis attack was in at least n log2 d. We show how to improve the ShamirKipnis attack, by using a better method of solving the involved algebraical problem MinRank. It becomes then in n 3 log d+O(1). All attacks fail for modified versions of HFE: HFE − (Asiacrypt’98), HFEv (Eurocrypt’99), Quartz (RSA’2000) and even for Flash (RSA’2000).
Gauß Periods in Finite Fields
"... In this survey, we review two recent applications of a venerable tool: Gauß periods. In Section 2, we describe Gauß' original construction, and how it can be used to generate normal bases in extensions of finite fields. Section 3 contains the first application: finding elements of exponentially larg ..."
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Cited by 1 (0 self)
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In this survey, we review two recent applications of a venerable tool: Gauß periods. In Section 2, we describe Gauß' original construction, and how it can be used to generate normal bases in extensions of finite fields. Section 3 contains the first application: finding elements of exponentially large order in certain finite fields. This can be viewed as a step towards solving the famous open problem of finding efficiently a primitive element in a given finite field. A pleasant feature is that the prime factorization of the order of the multiplicative group is not required. In Section 4 we give another example of the method, yielding a different kind of bound: among the q shifts + a of an element of an extension of F q , where a runs through F q , at most one has "small" order. The second application, in Section 5, deals with efficient exponentiation...
On the Complexity of Certain Algebraic and Number Theoretic Problems
"... It is certified that the work contained in the thesis entitled “On the Complexity ..."
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It is certified that the work contained in the thesis entitled “On the Complexity
Fast Algorithms for Towers of Finite Fields and Isogenies Algorithmes Rapides pour les Tours de Corps Finis et les Isogénies
, 2011
"... Thèse de doctorat ..."