[Resource Topic] 2006/107: The number field sieve for integers of low weight

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The number field sieve for integers of low weight

Authors: Oliver Schirokauer


We define the weight of an integer N to be the smallest w such that N can be represented as \sum_{i=1}^w \epsilon_i 2^{c_i}, with \epsilon_1,\ldots,\epsilon_w\in\{1,-1\}. Since arithmetic modulo a prime of low weight is particularly efficient, it is tempting to use such primes in cryptographic protocols. In this paper we consider the difficulty of the discrete logarithm problem modulo a prime N of low weight, as well as the difficulty of factoring an integer N of low weight. We describe a
version of the number field sieve which handles both problems. Our analysis leads to the conjecture that, for N\to\infty with w fixed, the worst-case running time of the method is bounded above by
{\rm exp}((c+o(1))(\log\,N)^{1/3}(\log\log\,N)^{2/3}) with
c<((32/9)(2w-3)/(w-1))^{1/3} and below by the same expression with c=(32/9)^{1/3}((\sqrt{2}w-2\sqrt{2}+1)/(w-1))^{2/3}.
It also reveals that on average the method performs significantly better than it does in the worst case. We consider all the examples given in a recent paper of Koblitz and Menezes and demonstrate that in every case but one, our algorithm runs faster than the standard versions of the number field sieve.

ePrint: https://eprint.iacr.org/2006/107

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