Speeding up the Arithmetic on Koblitz Curves of Genus Two

Abstract

Koblitz, Solinas, and others investigated a family of elliptic curves which admit faster cryptosystem computations. In this paper, we generalize their ideas to hyperelliptic curves of genus 2. We consider the following two hyperelliptic curves Cα : v 2 + uv = u 5 + α u 2 + 1 defined over F2 with α = 0, 1, and show how to speed up the arithmetic in the Jacobian JC α (F2n) by making use of the Frobenius automorphism. With two precomputations, we are able to obtain a speed-up by a factor of 5.5 compared to the generic double-and-add-method in the Jacobian. If we allow 6 precomputations, we are even able to speed up by a factor of 7.

Speeding Up Point Multiplication on Hyperelliptic Curves with Efficiently-Computable Endomorphisms

“…As Koblitz curves were generalized to hyperelliptic Koblitz curves for faster point multiplication by Günter, et al [10] we extend the recent work of Gallant, et al [8] to hyperelliptic curves. So the extended method for speeding point multiplication applies to a much larger family of hyperelliptic curves over finite fields that have efficiently-computable endomorphisms.…”

“…Along this idea, Meier and Staffelbach [15], Müller [18], Smart [24], and Solinas [26,27] have thoroughly investigated elliptic curves defined over small finite fields. In addition, the idea of Koblitz curves was generalized to hyperelliptic curves of genus 2 by Günter, Lange and Stein [10]. We also refer the reader to [14] for a detailed investigation on hyperelliptic Koblitz curves of small genus defined over small base fields.…”

Genus Two Hyperelliptic Curve Coprocessor

“…We are interested in carrying the implementation forward and completing the ASIC level design and eventual fabrication of the chip. Additionally, we are extending the implementation to use the τ -adic method for point multiplication, as described in [8] and more extensively in [13]. We are also examining implementations of a genus three curve over F 2 61 , on a 64-bit system.…”

On the distribution of the coefficients of normal forms for Frobenius expansions

“…These ideas have been generalised by several authors to other settings, see, for example, [9,11,18,20]. It should also be noted that, unlike expansions to integral rational bases, very little is known about Frobenius expansions of minimal weight, even in the case q = 2; see [2,3,10].…”