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We investigate the security of Diffie-Hellman key exchange as used in popular Internet protocols and find it to be less secure than widely believed. First, we present Logjam, a novel flaw in TLS that lets a man-in-the-middle downgrade connections to "...

The security of the digital signature algorithm (DSA) and Diffie-Hellman key exchange is based on the difficulty of the discrete logarithm problems (DLP) over prime field GF(p), and thus it is important to evaluate the difficulty of the DLP over GF(p) ...

This paper reports on the factorization of the 768-bit number RSA-768 by the number field sieve factoring method and discusses some implications for RSA.

This paper shows experimental results of the linear algebra step in the number field sieve on parallel environment with implementation techniques. We developed an efficient algorithm that shares the sum of vectors in each node, and the network structure ...

Recently, Shirokauer's algorithm to solve the discrete logarithm problem modulo a prime p has been modified by Matyukhin, yielding an algorithm with running time $L_{p}[\frac{1}{3},1.09018...]$, which is, at the present time, the best known estimate of ...

Motivated by the goal of factoring large integers using the Number Field Sieve, several special-purpose hardware designs have been recently proposed for solving large sparse systems of linear equations over finite fields using Wiedemann's algorithm. ...

We describe a hardware device for supporting the sieving step in integer factoring algorithms like the quadratic sieve or the number field sieve. In analogy to Bernstein's proposal for speeding up the linear algebra step, we rely on a mesh of very ...

In [1], Bernstein proposed a circuit-based implementation of the matrix step of the number field sieve factorization algorithm. These circuits offer an asymptotic cost reduction under the measure "construction cost run time". We evaluate the cost of ...

A set A of primes p involving numbers such as ab^t + c, where |a|, |b|, |c| = O(1) and t → ∞, is defined. An algorithm for computing discrete logs in the finite field of order p with p ∈ A is suggested. Its heuristic expected running time is L_p[1/3; (32/...

Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.

The first practical public key cryptosystem to be published, the Diffie–Hellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security ...