International Association
for Cryptologic Research

This paper accelerates FPGA computations of discrete logarithms on elliptic curves over binary fields. As an illustration, this paper reports successful completion of an attack against the SECG standard curve {\tt sect113r2}, a binary elliptic curve that was not removed from the standard until 2010 and was not disabled in OpenSSL until June 2015. This is a new size record for ECDL computations, using a prime order very slightly larger than the previous record holder. More importantly, this paper uses FPGAs much more efficiently, saving a factor close to $3/2$ in the size of each high-speed ECDL core and allowing 3 cores to be squeezed into a low-cost Spartan-6 FPGA. The paper also covers much larger curves over 127-bit fields.