Koblitz curves over quadratic fields

In this work, we retake an old idea that Koblitz presented in his landmark paper, where he suggested the possibility of defining anomalous elliptic curves over the base field F4. We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves. We also introduce two ordinary Koblitz-like elliptic curves defined over F4 that are equipped with efficient endomorphisms. To the best of our knowledge these endomorphisms have not been reported before. In order to achieve a fast reduction procedure, we adopted a redundant trinomial strategy that embeds elements of the field F4^m, with m a prime number, into a ring of higher order defined by an almost irreducible trinomial. We also present a number of techniques that allow us to take full advantage of the native vector instructions of high-end microprocessors. Our software library achieves the fastest timings reported for the computation of the timing-protected scalar multiplication on Koblitz curves, and competitive timings with respect to the speed records established recently in the computation of the scalar multiplication over binary and prime fields

Faster discrete logarithms on FPGAs

This paper accelerates FPGA computations of discrete logarithms on elliptic curves over binary fields. As a toy example, this paper successfully attacks the SECG standard curve sect113r2, a binary elliptic curve that was not removed from the SECG standard until 2010 and was not disabled in OpenSSL until June 2015. This is a new size record for completed 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. This paper squeezes 3 cores into a low-cost Spartan-6 FPGA and many more cores into larger FPGAs. The paper also benchmarks many smaller-size attacks to demonstrate reliability of the estimates, and covers a much larger curve over a 127-bit field to demonstrate scalability