Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

Efficient (3,3)-isogenies on fast Kummer surfaces

We give an alternative derivation of (N,N)-isogenies between fast Kummer surfaces which complements existing works based on the theory of theta functions. We use this framework to produce explicit formulae for the case of N = 3, and show that the resulting algorithms are more efficient than all prior (3,3)-isogeny algorithms

Kummer for Genus One over Prime Order Fields

This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by Gaudry and Lubicz in 2009 had suggested the use of the associated Kummer line to speed up scalar multiplication. In the present work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necessary security conditions such that the associated Kummer line has small parameters and a base point with small coordinates. It turns out that the ladder step on the Kummer line supports parallelism and can be implemented very efficiently in constant time using the single-instruction multiple-data (SIMD) operations available in modern processors. For the 128-bit security level, this work presents three Kummer lines denoted as $K_1:={\sf KL2519(81,20)}$, $K_2:={\sf KL25519(82,77)}$ and $K_3:={\sf KL2663(260,139)}$ over the three primes $2^{251}-9$, $2^{255}-19$ and $2^{266}-3$ respectively. Implementations of scalar multiplications for all three Kummer lines using Intel intrinsics have been done and the code is publicly available. Timing results on the Skylake and the Haswell processors of Intel indicate that both fixed base and variable base scalar multiplications for $K_1$ and $K_2$ are faster than those achieved by {\sf Sandy2x}, which is a highly optimised SIMD implementation in assembly of the well known {\sf Curve25519}; for example, on Skylake, variable base scalar multiplication on $K_1$ is faster than {\sf Curve25519} by about 30\%. On Skylake, both fixed base and variable base scalar multiplication for $K_3$ are faster than {\sf Sandy2x}; whereas on Haswell, fixed base scalar multiplication for $K_3$ is faster than {\sf Sandy2x} while variable base scalar multiplication for both $K_3$ and {\sf Sandy2x} take roughly the same time. In fact, on Skylake, $K_3$ is both faster and also offers about 5 bits of higher security compared to {\sf Curve25519}. In practical terms, the particular Kummer lines that are introduced in this work are serious candidates for deployment and standardisation. We further illustrate the usefulness of the proposed Kummer lines by instantiating the quotient Digital Signature Algorithm (qDSA) on all the three Kummer lines

Fast, uniform scalar multiplication for genus 2 Jacobians with fast Kummers

International audienceWe give one-and two-dimensional scalar multiplication algorithms for Jacobians of genus 2 curves that operate by projecting to Kummer surfaces, where we can exploit faster and more uniform pseudo-multiplication, before recovering the proper "signed" output back on the Jacobian. This extends the work of López and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. The technique is especially interesting in genus 2, because Kummer surfaces can outperform comparable elliptic curve systems