J. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, Designs, Codes and Cryptography, 20 (2000), pp. 5-40.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....curves. 1. Exhausted search 2. Baby step Giant step 3. Pollard s rho 4. Pohlig Hellman 5. Index calculus 6. Special Curves Upto now nobody was able to find a good factor base in order to apply the Index calculus. In 1998 Silverman suggested a way to apply the Index calculus (cf. [22]) His attack failed (cf. 15] The list of special curves becomes longer and longer: supersingular curves (trace 0 curves over F p ) Menezes, Okamoto, Vanstone anomalous curves (trace 1 curves over F p ) Frey, R uck, Semaev, Satoh, Araki, Smart trace 2 curves over F p (based on the ....

Joseph H. Silverman, The Xedni Calculus and the Elliptic Curve Discrete Logarithm Problem, Designs, Codes & Cryptography, (2000).

....the curve E to a curve E of suciently large rank over Q, then in actually lifting points from E to rational points of reasonably bounded height on E . A careful analysis by Silverman and Suzuki in [10] provides strong theoretical and numerical evidence in support of Miller s arguments. Silverman [8] proposed an alternative approach, dubbed the xedni calculus , for attacking the ECDLP. The xedni idea turns the index calculus on its head by rst lifting a bounded number (nine) of points to Q then nding a lift E=Q of E to t the lifted points. This approach circumvents the diculty of ....

....on an elliptic curve over F p to rational points of canonical height bounded subexponential in log p on an elliptic curve over Q implies a method for constructing elliptic curves over Q of arbitrarily large rank. On the other hand, bounding the number of lifted points, as the xedni algorithm in [8], results in asymptotically negligible probability of success in solving the ECDLP. Our analysis depends on a conjecture of Lang [4] that the canonical height of any nonzero rational point on an elliptic curve E over Q is bounded from below by c log j (E)j where c is a universal constant ....

[Article contains additional citation context not shown here]

J.H. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, preprint.

....for curves of large rank it requires some delicate consideration of precision in order to be sure of the result. The method here, by contrast, involves only discrete computations: finding roots of cubics and evaluating quadratic characters modulo primes. It was also described by Silverman in [6], attributed there to Brumer and myself. In fact, Brumer described the method to me in 1996; it was apparently used by him and Kramer in verifying the examples in [2] though the method is not explicitly mentioned there; so the method goes back to 1975 at least. We give it here as it is closely ....

J. H. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, Design, Codes, and Cryptography 10 (2000), 5--40. 11

....given convincing arguments for why the most natural way in which the index calculus algorithms can be applied to the ECDLP is most likely to fail. 13. Xedni Calculus Attacks. A very interesting line of attack on the ECDLP, called the xedni calculus attack was recently proposed by J. Silverman [83]. One intriguing aspect of the xedni calculus is that it can be adapted to solve both the ordinary discrete logarithm and the integer factorization problems. However, it was subsequently shown by a team of researchers including J. Silverman (see Jacobson et al. 36] that the attack is virtually ....

J. Silverman, "The xedni calculus and the elliptic curve discrete logarithm problem ", Designs, Codes and Cryptography, to appear, 2000. Also available at http://www.cacr.math.uwaterloo.ca

....for curves of large rank it requires some delicate consideration of precision in order to be sure of the result. The method here, by contrast, involves only discrete computations: finding roots of cubics and evaluating quadratic characters modulo primes. It was also described by Silverman in [5], attributed there to Brumer and myself. The method is folklore, and certainly not original indeed, it was mentioned to us by Brumer in 1996 but we give it here as it is closely related to and leads to our second section where we apply similar ideas to 2 Selmer groups. We illustrate the method ....

J. H. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, Design, Codes, and Cryptography 10 (2000), 5--40. 10

....was the rst to suggest using hyperelliptic curves for the same goal. The security of many cryptosystems based on curves relies on the di culty of the discrete log problem. In some special cases, it was shown that this problem was rather easy [22] 10] 39] 34] Apart from new lifting ideas [38, 13, 6] that remain to be tested, it seems that the discrete log on elliptic curves still resists. On ordinary curves, the only known attack is a parallelized version of Pollard s rho method [44] The Certicom challenge 1 records the state of the art in the eld. Among the curves suggested for ....

J. H. Silverman. The XEDNI calculus and the elliptic curve discrete logarithm problem. Preprint, August 1998.

....to the order of vanishing of L(E; s) at s = 1. Mestre s method is to force the first several terms in the formal infinite product for L(E; 1) to be as small as possible, whereas Silverman wants them to be as large as possible. 3. 2 The Algorithm We now describe the steps in the xedni algorithm [33]. Step 1. Choose an integer r with 2 r 9 (most likely 4 r 6) and integers L 0 7 and L 1 100. Set M = Y l prime; L 0 lL 1 l: Also, decide whether you will be working with elliptic curves in general cubic form or in Weierstrass form. In the first case, for any r tuple of projective ....

.... that is needed in order to ensure that one can find coefficients for an elliptic curve over Q that both passes through the lifted points and reduces modulo the primes l and p to the curves E(F l ) for ljM) and E(F p ) that we already have (see Step 7 below) This is proved in Appendix B of [33]. Here we shall motivate the rank r condition for the B matrix by giving an example in a simpler setting. Suppose that r = 2, and we re working with straight lines in the projective plane, rather than elliptic curves, so that the B matrix is just X 1 Y 1 Z 1 X 2 Y 2 Z 2 . Let l = 3. ....

[Article contains additional citation context not shown here]

J. H. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, preprint.

No context found.

J. Silverman, The xedni calculus and the elliptic curve discrete logarithm problem, Designs, Codes and Cryptography, 20 (2000), pp. 5-40.

No context found.

J. Silverman. The Xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr., 20:5--40, 2000.

No context found.

J. H. Silverman, The Xedni calculus and the elliptic curve discrete logarithm problem, to appear Design,Code and Cryptography.

No context found.

J. H. Silverman, The Xedni calculus and the elliptic curve discrete logarithm problem, Design, Code and Cryptography, 20 (2000), 5-40.

No context found.

Silverman, J. H.: The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 20 (2000), no. 1, 5--40.

No context found.

Silverman, Joseph H., The xedni calculus and the elliptic curve discrete logarithm problem, Des. Codes Cryptogr. 20 (2000), 5--40..

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC