Related Concepts
Definition
The ECDLP is a special case of the discrete logarithm problem
Background
Let E be an elliptic curve defined over a finite field \({\mathbb{F}}_{q}\), and let \(P \in E({\mathbb{F}}_{q})\) be a point of order n. Given \(Q \in \langle P\rangle\), the elliptic curve discrete logarithm problem (ECDLP) is to find the integer l, \(0 \leq l \leq n - 1\), such that Q = lP. The ECDLP is a special case of the discrete logarithm problem in which the cyclic group G is represented by the group \(\langle P\rangle\) of points on an elliptic curve. It is of cryptographic interest because its apparent intractability is the basis for the security of elliptic curve cryptography.
Theory
If the order n of the base point P is composite and its factorization is known, then the Pohlig–Hellman algorithm [19] (see the discrete logarithm problem entry) can be used to efficiently reduce the ECDLP in \(\langle P\rangle\)to...
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Recommended Reading
Adleman L, DeMarrais J, Huang M (1994) A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields. Algorithmic Number Theory—ANTS-I, Lecture Notes in Computer Science, vol 877. Springer, Berlin, pp 28–40
Diem C (in press), On the discrete logarithm problem in class groups of curves. Mathematics of Computation, to appear
Diem C (2009) On the discrete logarithm problem in elliptic curves. Preprint
Frey G (2001) Applications of arithmetical geometry to cryptographic constructions. Proceedings of the Fifth International Conference on Finite Fields and Applications, Springer, Berlin, pp 128–161
Frey G, Rück H (1994) A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math Comput 62:865–874
Gallant R, Lambert R, Vanstone S (2000) Improving the parallelized Pollard lambda search on anomalous binary curves. Math Comput 69:1699–1705
Gaudry P (2000) An algorithm for solving the discrete log problem in hyperelliptic curves. Advances in cryptology—EUROCRYPT 2000, Lecture Notes in Computer Science, vol 1807. Springer, Berlin, pp 19–34
Gaudry P (2009), Index calculus for abelian varieties and the elliptic curve discrete logarithm problem. J Symbol Computation 44(12):1690–1702
Gaudry P, Hess F, Smart N (2002) Constructive and destructive facets of Weil descent on elliptic curves. J Cryptol 15:19–46
Hess F (2004) Generalising the GHS attack on the elliptic curve discrete logarithm problem. LMS J Comput Math 7:167–192
Jacobson M, Koblitz N, Silverman J, Stein A, Teske E (2000) Analysis of the xedni calculus attack. Design Codes Cryptogr 20:41–64
Jacobson M, Menezes A, Stein A (2001) Solving elliptic curve discrete logarithm problems using Weil descent. J Ramanujan Math Soc 16:231–260
Maurer M, Menezes A, Teske E (2002) Analysis of the GHS Weil descent attack on the ECDLP over characteristic two finite fields of composite degree. LMS J Comput Math 5:127–174
Menezes A, Okamoto T, Vanstone S (1993) Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans Inf Theory 39:1639–1646
Menezes A, Qu M (2001) Analysis of the Weil descent attack of Gaudry, Hess and Smart. Topics in cryptology—CT-RSA 2001, Lecture Notes in Computer Science, vol 2020. Springer, Berlin, pp 308–318
Menezes A, Teske E (2006) Cryptographic implications of Hess’ generalized GHS attack. Appl Algebr Eng Comm Comput 16:439–460
Menezes A, Teske E, Weng A (2004) Weak fields for ECC. Topics in cryptology–CT-RSA 2004, Lecture Notes in Computer Science, vol 2964. Springer, Berlin, pp 366–386
Miller V (1986) Use of elliptic curves in cryptography. Advances in cryptology—CRYPTO ’85, Lecture Notes in Computer Science, vol 218. Springer, Berlin, pp 417–426
Pohlig S, Hellman M (1978) An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans Inf Theory 24, 106–110
Pollard J (1978) Monte Carlo methods for index computation (mod p). Math Comput 32:918–924
Satoh T, Araki K (1998) Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment Math Universitat Sancti Pauli 47:81–92
Semaev I (1998) Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p. Math Comput 67:353–356
Silverman J (2000) The xedni calculus and the elliptic curve discrete logarithm problem. Design Codes Cryptogr 20:5–40
Silverman J, Suzuki J (1998) Elliptic curve discrete logarithms and the index calculus. Advances in cryptology—ASIACRYPT ’98, Lecture Notes in Computer Science, vol 1514. Springer, Berlin, pp 110–125
Smart N (1999) The discrete logarithm problem on elliptic curves of trace one. J Cryptol 12:193–196
Teske E (1998) Speeding up Pollard’s rho method for computing discrete logarithms. Algorithmic number theory—ANTS-III, Lecture Notes in Computer Science, vol 1423. Springer, Berlin, pp 541–554
van Oorschot P, Wiener W (1998) Parallel collision search with cryptanalytic applications. J Cryptol 12:1–28
Wiener M, Zuccherato R (1999) Faster attacks on elliptic curve cryptosystems. Selected Areas in Cryptography—SAC ’98, Lecture Notes in Computer Science 1556:190–200, Springer, Berlin
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Hankerson, D., Menezes, A. (2011). Elliptic Curve Discrete Logarithm Problem. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_246
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