## Cryptology ePrint Archive: Report 2016/961

A kilobit hidden SNFS discrete logarithm computation

Joshua Fried and Pierrick Gaudry and Nadia Heninger and Emmanuel Thomé

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 $\mathbb{F}_p^*$, yet detecting that $p$ has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of 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.

Category / Keywords: public-key cryptography / Discrete logarithm, Special number field sieve, Trapdoor

Original Publication (with minor differences): IACR-EUROCRYPT-2017

Date: received 4 Oct 2016, last revised 18 Jul 2017

Contact author: nadiah at cis upenn edu

Available format(s): PDF | BibTeX Citation

Note: Final version as published in proceedings.

Short URL: ia.cr/2016/961

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