V. M uller, A. Stein, and C. Thiel. Computing discrete logarithms in real quadratic congruence function fields of large genus. Math. Comp., 68(226):807--822, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....hyperelliptic curve is an elliptic curve. Jacobian elements can be compactly represented by a pair of polynomials of degree at most g over F q , and eciently added using Cantor s algorithm [6] When g is large, there is a subexponential algorithm due to Adleman, DeMarrais and Huang [1] see also [20, 8]) for the discrete logarithm problem in J (F q ) Moreover, when g is small and 5, Gaudry s algorithm [10] is faster than Pollard s rho algorithm. If g = 2 or g = 3, and n is the largest prime divisor of #J C (F q ) the best algorithm known takes O ( p n) steps, i.e. the algorithm takes ....

V. Muller, A. Stein and C. Thiel, Computing discrete logarithms in real quadratic congruence function elds of large genus, Math. Comp. 68 (1999), 807-822.

....to fail in practice. 14. Hyperelliptic Curves. Hyperelliptic curves are a family of algebraic curves of arbitrary genus that includes elliptic curves. Hence, an elliptic curve can be viewed as a hyperelliptic curve of genus 1. Adleman, DeMarrais and Huang [1] see also Stein, Muller and Thiel [92]) presented a subexponential time algorithm for the discrete logarithm problem in the jacobian of a large genus hyperelliptic curve over a finite field. However, in the case of elliptic curves, the algorithm is worse than naive exhaustive search. 15. Equivalence to Other Discrete Logarithm ....

A. Stein, V. Muller and C. Thiel, "Computing discrete logarithms in real quadratic congruence function fields of large genus", Mathematics of Computation, 68 (1999), 807-822.

....as factor base It is to be expected that (for xed eld F q ) this method becomes more and more e ective with increasing g. And indeed Adleman, DeMarrais and Huang ( ADH] proposed an index calculus algorithm for hyperelliptic curves with subexponential complexity in g. This was made precise in [MST] and [E] and nally settled by a smoothnessness result in [ES] Proposition 2.5. E] For g= log(q) t the discrete logarithm in the divisor class group of a hyperelliptic curve of genus g de ned over F q can be computed with complexity bounded by L q g [ 5 p 6 ( 1 3 2t ) 1=2 ( 3 ....

V. Muller, A. Stein and A. Thiel, Computing discrete logarithms in real quadratic congruence function elds, Math. Comp., 68 (226) (1999), 807{ 822.

....function fields for which (A) holds: Lemma 4.3 Let d = 2 log(4g Gamma 2) log q : Then the set F consisting of the classes of nonprincipal prime ideals whose absolute norm is at most q d form a generating system of C. Furthermore, #F 4dq d . Proof: Follows from Corollary 1 of [11] and Theorem 5.4.3 and Lemma 6.2.3 of [16] 2 Corollary 4.4 If q is bounded by a polynomial in g, then assumption (A) holds for K. Proof: Let F and d be as in the previous Lemma. Since d (2 log(4g Gamma 2) log q) 1, we have q d (4g Gamma 2) 2 q which is polynomially bounded in g. ....

V. M uller, A. Stein & C. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus. To appear in Math. Comp.

....complexity O(q (2g Gamma1) 5 ) see Theorem 2.2.33, p. 78, of [37] If log q 2g 1, i.e. q is again small compared to g, then discrete logarithms, including the regulator, can be computed probabilisticly in subexponential time exp(c p log q g log log q g ) where c 0 is a constant ([26], Theorem 6.3.2, p. 203, of [37] The algorithm does not appear feasible in practice; nevertheless, to be safe, one might again wish to choose q to be large relative to g. The computations in [34] show that the elliptic case g = 1 performed best computationally; for a 50 digit prime q, a call ....

V. M uller, A. Stein & C. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus. To appear in Math. Comp.

....2g 1 [c] yielding an overall expected runtime of O(L q 2g 1 [c] with c = 3(l 1) 4 l) The analysis in [3] assumes l 7:376 for the linear algebra, yielding c 2:313: The MST algorithm. The second index calculus algorithm for solving the HCDLP was proposed by M uller, Stein, and Thiel [53]. Their algorithm, which works for any odd characteristic nite eld, solves the discrete logarithm problem in the infrastructure of a real quadratic function eld. For elliptic curves, it was shown in [76] odd characteristic) and [87] even characteristic) that the ECDLP is polynomially ....

....is derived, so the smoothness assumption used to analyze the ADH algorithm is not required. The second of these results is described in detail by Enge and Stein [20] Finally, the analysis of the algorithm is not only rigorous, the complexity is better than the complexity of the ADH method. In [53], it is shown that that if 2g 2 log q, then with n 2 O(L q 2g 2 [ the expected number of trials before nding a relation is O(L q 2g 2 [ 4 ] The linear algebra step was assumed to take time O(L q 2g 2[5 ] and the relation generation stage takes time O(L q 2g 2[2 4 ] as m 2 O(L ....

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V. Muller, A. Stein and C. Thiel, \Computing discrete logarithms in real quadratic congruence function elds of large genus", Mathematics of Computation, 68 (1999), 807-822.

....O(exp( c o(1) p log n log log n) The algorithm does not assume that the group order #JC (k) is known, necessitating an expensive Smith Normal Form computation on a sparse integer matrix. Index calculus algorithms with rigorously proved running times were presented by M uller, Stein and Thiel [31] and Enge [7] Their algorithms have an expected running time of L q 2g 1[1:44] and are superior, both in theory and in practice, to the ADH algorithm. Gaudry [16] building on earlier work of Adleman, DeMarrais and Huang [1] and Hafner and McCurley [18] presented an algorithm speci cally ....

V. Muller, A. Stein and C. Thiel, \Computing discrete logarithms in real quadratic congruence function elds of large genus", Mathematics of Computation, 68 (1999), 807-822. 16

....= 4, is equivalent to the DLP for elliptic curves. So far, the only known algorithm for solving the DLP for elliptic curves is exponential (except for the supersingular case) For real quadratic congruence function fields of large genus, the DLP turned out to be of subexponential complexity (see [6]) The main results are stated in Theorem 12, Theorem 13, and also in Theorem 6, Theorem 7. Mostly, we derive the formulae from Adams and Razar [2] by combining their theorems with theorems about the arithmetic in elliptic congruence function fields (see [15] or [14] We explicitly give the ....

M ULLER, V., STEIN A., THIEL, C., Computing Discrete Logarithms in Real Quadratic Congruence Function Fields of Large Genus. Submitted Manuscript.

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V. M uller, A. Stein, and C. Thiel. Computing discrete logarithms in real quadratic congruence function fields of large genus. Math. Comp., 68(226):807--822, 1999.

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V. Muller, A. Stein, and C. Thiel. Computing discrete logarithms in real quadratic congruence function elds of large genus. Math. Comp., 68(226):807-822, 1999.

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V. Muller, A. Stein, Ch. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus, preprint,

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