## NORMAL ELLIPTIC BASES AND TORUS-BASED CRYPTOGRAPHY (909)

### BibTeX

@MISC{909normalelliptic,

author = {},

title = {NORMAL ELLIPTIC BASES AND TORUS-BASED CRYPTOGRAPHY},

year = {909}

}

### OpenURL

### Abstract

Abstract. We consider representations of algebraic tori Tn(Fq) over finite fields. We make use of normal elliptic bases to show that, for infinitely many squarefree integers n and infinitely many values of q, we can encode m torus elements, to a small fixed overhead and to m ϕ(n)-tuples of Fq elements, in quasi-linear time in log q. This improves upon previously known algorithms, which all have a quasi-quadratic complexity. As a result, the cost of the encoding phase is now negligible in Diffie-Hellman cryptographic schemes. 1.

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Citation Context ...ve groups defined by finite fields F × qn are of first importance in numerous applications, especially in discrete-log based public key cryptography. In this field, Diffie and Hellman’s seminal paper =-=[DH76]-=- opened the way to their use in numerous cryptographic standards in the eighties. It turns out that elliptic curves are often prefered today, since there exist subexponential algorithms to solve the d... |

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Citation Context ...f order Φn(q), where Φn denotes the n-th cyclotomic polynomial (the minimal polynomial over Q of e 2iπ n ), has reattracted attention since the publication of Lenstra and Verheul’s xtr scheme in 2000 =-=[LV00]-=-. Lenstra and Verheul noticed that in the very particular case n = 6, working in the F × q6-subgroup of order Φ6(q) = q2 − q + 1 can be done with a F × q2 arithmetic, whereas the best way to break the... |

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(Show Context)
Citation Context ... that can be computed very efficiently, that is with log 1+o(1) q bit operations too. To this purpose, we start from the interpretation of xtr-subgroups as algebraic tori, due to Rubin and Silverberg =-=[RS03]-=-, and the explicit encoding proposed by van Dĳk and Woodruff [DW04]. Algebraic tori over Fq are algebraic groups defined over Fq that are isomorphic to some (Gm) d over Fq, where Gm denotes the multip... |

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(Show Context)
Citation Context ...it operations too. To this purpose, we start from the interpretation of xtr-subgroups as algebraic tori, due to Rubin and Silverberg [RS03], and the explicit encoding proposed by van Dĳk and Woodruff =-=[DW04]-=-. Algebraic tori over Fq are algebraic groups defined over Fq that are isomorphic to some (Gm) d over Fq, where Gm denotes the multiplicative group and d is the dimension of the Date: September 1, 200... |

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(Show Context)
Citation Context ...es(Φe, Φf ) = 1 and it is widely known that this is equivalent to the condition f ̸= e pi with p prime and i � 1 (see [Dun09] for a proof). This is a corollary of the following formula due to Apostol =-=[Apo70]-=-, for f > e > 1, ∏ ϕ(f) µ(e/d) Res(Φf , Φe) = p ϕ(pi ) . (3.2) d | e f p prime, (f,d) =pi There remains to check that when f = e p i , there exist integers q such that Eq. (3.1) is satisfied. Since n ... |

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Citation Context ...plexities. We can now state more precisely the complexity of Algorithm 1. We first construct an irreducible polynomial P (X) of degree n over Fq, which can be done in n 2+o(1) log 2+o(1) q operations =-=[PR98]-=-. Let α = X mod P (X). Then (1, α, . . . , α n−1 ) is6 CLÉMENT DUNAND AND REYNALD LERCIER an Fq-basis of Fqn. Additions, subtractions and comparisons require O(n log q) elementary operations. Multipl... |

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(Show Context)
Citation Context ... succeed in speeding up the algorithm with the help of a new representation of field extensions. Couveignes and Lercier recently constructed a new family of normal bases, called normal elliptic bases =-=[CL09]-=-. They allow to perform low cost arithmetic in Fqn and in the context of tori this yields encodings with a log q smaller computational cost. In order to reach this complexity, we need inputs q and n s... |

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(Show Context)
Citation Context ...e(q), Φf (q)) = 1 . (3.1) The right hand side condition is always satisfied when Res(Φe, Φf ) = 1 and it is widely known that this is equivalent to the condition f ̸= e pi with p prime and i � 1 (see =-=[Dun09]-=- for a proof). This is a corollary of the following formula due to Apostol [Apo70], for f > e > 1, ∏ ϕ(f) µ(e/d) Res(Φf , Φe) = p ϕ(pi ) . (3.2) d | e f p prime, (f,d) =pi There remains to check that ... |

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1 |
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(Show Context)
Citation Context ...icular values of n (for instance, 2, 3 or 6 with luc [SL93], xtr[LV00] or ceilidh[RS03]), the rationality or stable rationality of such structures for every n has been a concern for several years now =-=[Vos91]-=-. A nice workaround proposed by van Dĳk and Woodruff [DW04] consists in adding to the torus Tn(Fq) some well chosen finite fields and mapping the whole set into another product of finite fields, θ : T... |