International Association
for Cryptologic Research

Strategies and their evaluations play important roles in speeding up the computation of large smooth-degree isogenies. The concept of optimal strategies for such computation was introduced by De Feo et al., and virtually all implementations of isogeny-based protocols have adopted this approach, which is provably optimal for single-core platforms. In spite of its inherent sequential nature, several recent works have studied ways of speeding up this isogeny computation by exploiting the rich parallelism available in vectorized and multi-core platforms. One obstacle to taking full advantage of this parallelism, however, is that De Feo et al.’s strategies are not necessarily optimal in multi-core environments. To illustrate how the speed of vectorized and parallel isogeny computation can be improved at the strategylevel, we present two novel software implementations that utilize a state-of-the-art evaluation technique, called precedence-constrained scheduling (PCS), presented by Phalakarn et al., with our proposed strategies crafted for these environments. Our first implementation relies only on the parallelism provided by multi-core processors. The second implementation targets multi-core processors supporting the latest generation of the Intel’s Advanced Vector eXtensions (AVX) technology, commonly known as AVX-512IFMA instructions. To better handle the computational concurrency associated with PCS, we equip both implementations with extensive synchronization techniques. Our first implementation outperforms the implementation of Cervantes-Vázquez et al. by yielding up to 14.36% reduction in the execution time, when targeting platforms with two- to four-core processors. Our second implementation, equipped with four cores, achieves up to 34.05% reduction in the execution time compared to the single-core implementation of Cheng et al. of CHES 2022.

Hashing arbitrary values to points on an elliptic curve is a required step in many cryptographic constructions, and a number of techniques have been proposed to do so over the years. One of the first ones was due to Shallue and van de Woestijne (ANTS-VII), and it had the interesting property of applying to essentially all elliptic curves over finite fields. It did not, however, have the desirable property of being *indifferentiable from a random oracle* when composed with a random oracle to the base field. Various approaches have since been considered to overcome this limitation, starting with the foundational work of Brier et al. (CRYPTO 2011). For example, if f: F_q→E(F_q) is the Shallue--van de Woestijne (SW) map and H, H' are *two* independent random oracles, we now know that m↦f(H(m))+f(H'(m)) is indifferentiable from a random oracle. Unfortunately, this approach has the drawback of being twice as expensive to compute than the straightforward, but not indifferentiable, m↦f(H(m)). Most other solutions so far have had the same issue: they are at least as costly as two base field exponentiations, whereas plain encoding maps like f cost only one exponentiation. Recently, Koshelev (DCC 2022) provided the first construction of indifferentiable hashing at the cost of one exponentiation, but only for a very specific class of curves (some of those with j-invariant 0), and using techniques that are unlikely to apply more broadly. In this work, we revisit this long-standing open problem, and observe that the SW map actually fits in a one-parameter family (f_u)_{u∈F_q} of encodings, such that for independent random oracles H, H', F: m↦f_{H'(m)}(H(m)) is indifferentiable. Moreover, on a very large class of curves (essentially those that are either of odd order or of order divisible by 4), the one-parameter family admits a rational parametrization, which lets us compute F at almost the same cost as small f, and finally achieve indifferentiable hashing to most curves with a single exponentiation. Our new approach also yields an improved variant of the Elligator Squared technique of Tibouchi (FC 2014) that represents points of arbitrary elliptic curves as close-to-uniform random strings.

In this work, we retake an old idea that Koblitz presented in his landmark paper (Koblitz, in: Proceedings of CRYPTO 1991. LNCS, vol 576, Springer, Berlin, pp 279–287, 1991 ), where he suggested the possibility of defining anomalous elliptic curves over the base field $${\mathbb {F}}_4$$ F 4 . 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 $${\mathbb {F}}_4$$ F 4 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 $${\mathbb {F}}_{4^{m}},$$ F 4 m , with m a prime number, into a ring of higher order defined by an almost irreducible trinomial. We also suggest 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.