Exponential sums over points of elliptic curves

Ahmadi, Omran; Shparlinski, Igor E.

doi:10.1016/j.jnt.2014.01.016

Cited by 5 publications

(5 citation statements)

References 31 publications

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“…In this section, we introduce some definitions and results on exponential sums over finite fields and over elliptic curves (see [1,18,22]).…”

Section: Exponential Sumsmentioning

confidence: 99%

“…Then we manipulate the sums, separate some terms (a = 0) which gives 1 p k with the rest. So for a ∈ F p…”

Section: Randomness Extraction In F P Nmentioning

confidence: 99%

“…If e > 1 and k > 1 are two integers such as k ≤ (l 1 + l 2 ) − 2e − n + 2 then, F k is a ((U G1 , U G2 ), 1 2 e )-deterministic extractor .…”

Section: Randomness Extraction In F P Nmentioning

confidence: 99%

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Two-sources randomness extractors in finite fields and in elliptic curves

Tchapgnouo

Ciss²,

Sow³

et al. 2017

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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We propose two-sources randomness extractors over finite fields and on elliptic curves that can extract from two sources of information without consideration of other assumptions that the starting algorithmic assumptions with a competitive level of security. These functions have several applications. We propose here a description of a version of a Diffie-Hellman key exchange protocol and key extraction. Nous proposons des extracteurs d'aléas 2-sources sur les corps finis et sur les courbes elliptiques capables d'extraire à partir de plusieurs sources d'informations sans considération d'autres hypothèses que les hypothèses algorithmiques de départ avec un niveau de sécurité compétitif. Ces fonctions possèdent plusieurs applications. Nous proposons ici une version du protocole d'échange de clé Diffie-Hellman incluant la phase d'extraction.

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“…In this section, we introduce some definitions and results on exponential sums over finite fields and over elliptic curves (see [1,18,22]).…”

Section: Exponential Sumsmentioning

confidence: 99%

“…Then we manipulate the sums, separate some terms (a = 0) which gives 1 p k with the rest. So for a ∈ F p…”

Section: Randomness Extraction In F P Nmentioning

confidence: 99%

See 1 more Smart Citation

Two-sources randomness extractors in finite fields and in elliptic curves

Tchapgnouo

Ciss²,

Sow³

et al. 2017

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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“…Then given two sets A, B ⊆ Z * T , we consider the following two pair of sets Ahmadi & Shparlinski [2,3] have shown that the method of Garaev [96] and some bounds of bilinear exponential sums over points of elliptic curves, imply that at least one of the sets (3.14) and at least one of the sets (3.15) is "large". Similar arguments are used in [179] to show that for two sets P, Q ⊆ E(F q ) at least one of the sets {x(P ) + x(Q): P ∈ P, Q ∈ Q}, and {x(P ⊕ Q): P ∈ P , Q ∈ Q} , is also "large".…”

Section: Elliptic Curve Analoguesmentioning

confidence: 99%

Additive Combinatorics over Finite Fields: New Results and Applications

Shparlinski¹

2013

Finite Fields and Their Applications

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“…For given two points P ∈ P and Q ∈ Q, the first extractor outputs the k-least significant bits of the abscissa of the point P ⊕ Q. We show that the extracted bits are indistinguishable from a random bit-string of length k. In fact, we use bilinear exponential sums, recently proposed by Ahmadi and Shparlinski [1] to bound the the statistical distance.…”

Section: Introductionmentioning

confidence: 98%

Two-Source Randomness Extractors for Elliptic Curves for Authenticated Key Exchange

Ciss

Sow

2017

Codes, Cryptology and Information Security

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This paper studies the task of two-sources randomness extractors for elliptic curves defined over a finite field K, where K can be a prime or a binary field. In fact, we introduce new constructions of functions over elliptic curves which take in input two random points from two different subgroups. In other words, for a given elliptic curve E defined over a finite field Fq and two random points P ∈ P and Q ∈ Q, where P and Q are two subgroups of E(Fq), our function extracts the least significant bits of the abscissa of the point P ⊕ Q when q is a large prime, and the k-first Fp coefficients of the abscissa of the point P ⊕ Q when q = p n , where p is a prime greater than 5. We show that the extracted bits are close to uniform. Our construction extends some interesting randomness extractors for elliptic curves, namely those defined in [7] and [9,10], when P = Q. The proposed constructions can be used in any cryptographic schemes which require extraction of random bits from two sources over elliptic curves, namely in key exchange protocol , design of strong pseudo-random number generators, etc.

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Exponential sums over points of elliptic curves

Abstract: We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.

Cited by 5 publications

References 31 publications

Two-sources randomness extractors in finite fields and in elliptic curves

Two-sources randomness extractors in finite fields and in elliptic curves

Additive Combinatorics over Finite Fields: New Results and Applications

Two-Source Randomness Extractors for Elliptic Curves for Authenticated Key Exchange

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