1,869 research outputs found

    A kilobit hidden SNFS discrete logarithm computation

    Get PDF
    We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime pp looks random, and p1p--1 has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in F_p\mathbb{F}\_p^* , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

    A deterministic version of Pollard's p-1 algorithm

    Full text link
    In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's p1p-1 algorithm, which finds in random polynomial time the prime divisors pp of an integer nn such that p1p-1 is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the kk-th cyclotomic method of factoring (k2k\ge 2) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function ϕ\phi. We point out some explicit sets of integers nn that are completely factorable in deterministic polynomial time given ϕ(n)\phi(n). These sets consist, roughly speaking, of products of primes pp satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of p1p-1. Finally, we prove that O(lnn)O(\ln n) oracle queries for values of ϕ\phi are sufficient to completely factor any integer nn in less than exp((1+o(1))(lnn)1/3(lnlnn)2/3)\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr) deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

    Discrete logarithms in curves over finite fields

    Get PDF
    A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

    Maps between curves and arithmetic obstructions

    Get PDF
    Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.Comment: 8 page

    Computing cardinalities of Q-curve reductions over finite fields

    Get PDF
    We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of Drew Sutherlan
    corecore