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International Association for Cryptologic Research (IACR)
Abstract
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 p looks random, and p−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 Fp∗, yet detecting that p has this trapdoor
seems out of reach. Twenty-five years ago, there was considerable
controversy around the possibility of backdoored 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
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