Efficient Explicit Formulae for Genus 2 Hyperelliptic Curves over Prime Fields and Their Implementations

Abstract

Abstract. We analyze all the cases and propose the corresponding explicit formulae for computing 2D1 + D2 in one step from given divisor classes D1 and D2 on genus 2 hyperelliptic curves defined over prime fields. Compared with naive method, the improved formula can save two field multiplications and one field squaring each time when the arithmetic is performed in the most frequent case. Furthermore, we present a variant which trades one field inversion for fourteen field multiplications and two field squarings by using Montgomery's trick to combine the two inversions. Experimental results show that our algorithms can save up to 13% of the time to perform a scalar multiplication on a general genus 2 hyperelliptic curve over a prime field, when compared with the best known general methods.

Comparative Study of Hyperelliptic Curve Cryptosystem over Prime Field and Its Survey

“…Fan and Gong [12] also proved that HECC provides greater efficiency than either integer factorization systems or discrete logarithm systems, in terms of computational overheads, key sizes, and bandwidth. Deng Jian-zhi [13] illustrate that design of HECC based Digital signature which will solve the problem by checking the integrity of the file and signature ID, and it especially fit for the internet operation which need identity validate.…”

Flexible Prime-Field Genus 2 Hyperelliptic Curve Cryptography Processor with Low Power Consumption and Uniform Power Draw

“…One line of research has focused on finding efficient explicit formulae for calculating the group operations on hyperelliptic curves and also improving the efficiency of calculations by reducing the number of operations. Examples include works presented in [16]- [19] which have covered both prime-field and binary-field curves. These works have used different coordinate systems (both affine and projective) to improve the efficiency of their formulae.…”

“…Since the special cases of the reduced divisors rarely appear in practice, it is reasonable for a hardware implementation to only cover the general case [13]. The same approach was also taken in other implementations of HECC [19].…”

Jacobian Coordinates on Genus 2 Curves

“…To compute [2]P 1 + P 2 , we schedule the higher level operations in the form (P 1 + P 2 )+ P 1 (see [15] and [30] for the same high level scheduling). This means that mDBLADD can be computed using an mADD operation before a zwADD operation.…”