On the generic complexity of discrete logarithm problem in groups of elliptic curves over finite fields

AbstractIn this paper, the authors analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP) for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field F2N, where N is in [100,600], elliptic curve parameters are identified such that: (i) there should exist a cryptographically interesting elliptic curve E over F2N with these parameters; and (ii) the GHS attack is more efficient for solving the ECDLP in E(F2N) than for solving the ECDLP on any other cryptographically interesting elliptic curve over F2N. The feasibility of the GHS attack on the specific elliptic curves is examined over F2176, F2208, F2272, F2304 and F2368, which are provided as examples in the ANSI X9.62 standard for the elliptic curve signature scheme ECDSA. Finally, several concrete instances are provided of the ECDLP over F2N, N composite, of increasing difficulty; these resist all previously known attacks, but are within reach of the GHS attack.

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The Subset Search Algorithm: A New Attack on the Discrete Logarithm Problem over Finite Fields and Elliptic Curves

Information Technology Journal ◽

10.3923/itj.2006.520.523 ◽

2006 ◽

Vol 5 (3) ◽

pp. 520-523

Author(s):

Kaqish Malek

Keyword(s):

Elliptic Curves ◽

Finite Fields ◽

Search Algorithm ◽

Discrete Logarithm ◽

Discrete Logarithm Problem ◽

Subset Search

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An O(m^2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2^m)

Quantum Information and Computation ◽

10.26421/qic9.7-8-4 ◽

2009 ◽

Vol 9 (7&8) ◽

pp. 610-621

Author(s):

D. Maslov ◽

J. Mathew ◽

D. Cheung ◽

D.K. Pradhan

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Finite Fields ◽

Nearest Neighbor ◽

Discrete Logarithm ◽

Quantum Algorithm ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Discrete Logarithm Problem ◽

Quantum Polynomial

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over $GF(2^m)$. We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation, executable in the Linear Nearest Neighbor (LNN) architecture, is $O(m^2)$, which is an improvement over the previous bound of $O(m^3)$ derived assuming no architectural restrictions.

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PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

Visnyk Universytetu “Ukraina” ◽

10.36994/2707-4110-2019-2-23-26 ◽

2019 ◽

Author(s):

Anna ILYENKO ◽

Sergii ILYENKO ◽

Yana MASUR

Keyword(s):

Information Systems ◽

Elliptic Curve ◽

Elliptic Curves ◽

Finite Fields ◽

Computational Efficiency ◽

Digital Signature ◽

Discrete Logarithm ◽

Algebraic Groups ◽

Schnorr Signature ◽

Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

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