The ideal class group of hyperelliptic curves can be used in cryptosystems based on the discrete logarithm problem. In this article we present explicit formulae to perform the group operations for genus 2 curves. The formulae are completely general but to achieve the lowest number of operations we treat odd and even characteristic separately. We present 3 different coordinate systems which are suitable for different environments, e. g. on a smart card we should avoid inversions while in software a limited number is acceptable. The presented formulae render genus two hyperelliptic curves very useful in practice. The first system are affine coordinates where each group operation needs one inversion. Then we consider projective coordinates avoiding inversions on the cost of more multiplications and a further coordinate. Finally, we introduce a new system of coordinates and state algorithms showing that doublings are comparably cheap and no inversions are needed. A comparison between the systems concludes the paper.

Abstract. This paper explains the design and implementation of a highsecurity elliptic-curve-Diffie-Hellman function achieving record-setting speeds: e.g., 832457 Pentium III cycles (with several side benefits: free key compression, free key validation, and state-of-the-art timing-attack protection), more than twice as fast as other authors ’ results at the same conjectured security level (with or without the side benefits). 1

Abstract. In most algorithms involving elliptic and hyperelliptic curves, the costliest part consists in computing multiples of ideal classes. This paper investigates how to compute faster doubling over fields of characteristic two. We derive explicit doubling formulae making strong use of the defining equation of the curve. We analyze how many field operations are needed depending on the curve making clear how much generality one loses by the respective choices. Note, that none of the proposed types is known to be weak – one only could be suspicious because of the more special types. Our results allow to choose curves from a large enough variety which have extremely fast doubling needing only half the time of an addition. Combined with a sliding window method this leads to fast computation of scalar multiples. We also speed up the general case.

Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i p 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre’s algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography. 1

Abstract. In this paper, we present several improvements on the best known explicit formulæ for hyperelliptic curves of genus three and four in characteristic two, including the issue of reducing memory requirements. To show the effectiveness of these improvements and to allow a fair comparison of the curves of different genera, we implement all formulæ using a highly optimized software library for arithmetic in binary fields. This library was designed to minimize the impact of a whole series of overheads which have a larger significance as the genus of the curves increases. The current state of the art in attacks against the discrete logarithm problem is taken into account for the choice of the field and group sizes. Performance tests are done on two personal computers with very different architectures. Our results can be shortly summarized as follows: Curves of genus three provide performance similar, or better, to that of curves of genus two, and these two types of curves can perform faster than elliptic curves – indeed on some processors often twice as fast. Curves of genus four attain a performance level comparable to elliptic curves. A large choice of curves is therefore available for the deployment of curve-based cryptography, with curves of genus three and four providing their own advantages as larger cofactors can be allowed for the group order.

Abstract. In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller’s algorithm can be up to g times shorter than for the Tate pairing, with g the genus of the curve, and the pairing is automatically reduced, i.e. no final exponentiation is needed.

At Crypto 2004, van Dijk and Woodruff introduced a new way of using the algebraic tori Tn in cryptography, and obtained an asymptotically optimal n/φ(n) savings in bandwidth and storage for a number of cryptographic applications. However, the computational requirements of compression and decompression in their scheme were impractical, and it was left open to reduce them to a practical level. We give a new method that compresses orders of magnitude faster than the original, while also speeding up the decompression and improving on the compression factor (by a constant term). Further, we give the first efficient implementation that uses T30 , compare its performance to XTR, CEILIDH, and ECC, and present new applications. Our methods achieve better compression than XTR and CEILIDH for the compression of as few as two group elements. This allows us to apply our results to ElGamal encryption with a small message domain to obtain ciphertexts that are 10% smaller than in previous schemes.

Abstract. It has recently been reported that the performance of hyperelliptic curve cryptosystems (HECC) is competitive to that of elliptic curve cryptosystems (ECC). However, it is expected that HECC still can be improved due to their mathematically rich structure. We consider here the application of degenerate divisors of HECC to scalar multiplication. We investigate the operations of the degenerate divisors in the Harley algorithm and the Cantor algorithm of genus 2. The timings of these operations are reported. We then present a novel efficient scalar multiplication method using the degenerate divisors. This method is applicable to cryptosystems with fixed base point, e.g., ElGamal-type encryption, sender of Diffie-Hellman, and DSA. Using a Xeon processor, we found that the double-and-add-always method using the degenerate base point can achieve about a 20 % increase in speed for a 160-bit HECC. However, we mounted an timing attack using the time difference to designate the degenerate divisors. The attack assumes that the secret key is fixed and the base point can be freely chosen by the attacker. Therefore, the attack is applicable to ElGamal-type decryption and single-pass Diffie-Hellman — SSL using a hyperelliptic curve could be vulnerable to the proposed attack. Our experimental results show that one bit of the secret key for a 160-bit HECC can be recovered by calling the decryption oracle 500 times.

Abstract. Although cryptographic software implementation is often performed by expert programmers, the range of performance and security driven options, as well as more mundane software engineering issues, still make it a challenge. The use of domain specific language and compiler techniques to assist in description and optimisation of cryptographic software is an interesting research challenge. Our results, which focus on Elliptic Curve Cryptography (ECC), show that a suitable language allows description of ECC based software in a manner close to the original mathematics; the corresponding compiler allows automatic production of an executable whose performance is competitive with that of a handoptimised implementation. Our work are set within the context of CACE, an ongoing EU funded project on this general topic.

Abstract. We describe an implementation of binary field arithmetic written in the C programming language. Even though the implementation targets 32-bit CPUs, the results can be applied also to CPUs with different granularity. We begin with separate routines for each operand size in words to minimize performance penalties that have a bigger relative impact for shorter operands – such as those used to implement modern curve based cryptography. We then proceed to use techniques specific to operand size in bits for several field sizes. This results in an implementation of field arithmetic where the curve representing field multiplication performance closely resembles the theoretical quadratic bit-complexity that can be expected for small inputs. This has important practical consequences: For instance, it will allow us to compare the performance of the arithmetic on curves of different genera and defined over fields of different sizes without worrying about penalties introduced by field arithmetic and concentrating on the curve arithmetic itself. Moreover, the cost of field inversion is very low, makingthe use of affine coordinates in curve arithmetic more interesting. These applications will be mentioned.