A Generic Approach to Searching for Jacobians

We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio

Computer Architectures for Cryptosystems Based on Hyperelliptic Curves

Security issues play an important role in almost all modern communication and computer networks. As Internet applications continue to grow dramatically, security requirements have to be strengthened. Hyperelliptic curve cryptosystems (HECC) allow for shorter operands at the same level of security than other public-key cryptosystems, such as RSA or Diffie-Hellman. These shorter operands appear promising for many applications. Hyperelliptic curves are a generalization of elliptic curves and they can also be used for building discrete logarithm public-key schemes. A major part of this work is the development of computer architectures for the different algorithms needed for HECC. The architectures are developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients and underlying finite field. In this work we first summarized the theoretical background of hyperelliptic curve cryptosystems. In order to realize the operation addition and doubling on the Jacobian, we developed architectures for the composition and reduction step. These in turn are based on architectures for arithmetic in the underlying field and for arithmetic in the polynomial ring. The architectures are described in VHDL (VHSIC Hardware Description Language) and the code was functionally verified. Some of the arithmetic modules were also synthesized. We provide estimates for the clock cycle count for a group operation in the Jacobian. The system targeted was HECC of genus four over GF(2^41)

Remarks on the zeta function of some diagonal hyperelliptic curves

AbstractWe compute the Jacobi sums and the zeta functions associated to the family of diagonal curves defined over Fp by y2=γx5+δ (with γδ≠0), then we prove that their Jacobians are simple. We discuss about the hardness of the associated discrete logarithm problem with respect to known attacks and we show that our family is suitable for cryptographic purpose

Counting Points on Genus 2 Curves with Real Multiplication

We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field (\F_{q}) of large characteristic from (\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian

The Q-curve construction for endomorphism-accelerated elliptic curves

We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}\_{p^2}$ with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when $p$ is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $\mathbb{F}\_{p^2}$ equipped with efficient endomorphisms for every p \textgreater{} 3, and exhibit examples of twist-secure curves over $\mathbb{F}\_{p^2}$ for the efficient Mersenne prime $p = 2^{127}-1$.Comment: To appear in the Journal of Cryptology. arXiv admin note: text overlap with arXiv:1305.540

Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves

One of the challenges in the designing of pairing-based cryptographic protocols is to construct suitable pairing-friendly curves: Curves which would provide e�cient implementation without compromising the security of the protocols. These curves have small embedding degree and large prime order subgroup. Random curves are likely to have large embedding degree and hence are not practical for implementation of pairing-based protocols. In this thesis we review some mathematical background on elliptic and hyperelliptic curves in relation to the construction of pairing-friendly hyper-elliptic curves. We also present the notion of pairing-friendly curves. Furthermore, we construct new pairing-friendly elliptic curves and Jacobians of genus two hyperelliptic curves which would facilitate an efficient implementation in pairing-based protocols. We aim for curves that have smaller values than ever before reported for di�erent embedding degrees. We also discuss optimisation of computing pairing in Tate pairing and its variants. Here we show how to e�ciently multiply a point in a subgroup de�ned on a twist curve by a large cofactor. Our approach uses the theory of addition chains. We also show a new method for implementation of the computation of the hard part of the �nal exponentiation in the calculation of the Tate pairing and its varian

Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves (Update)

For most of the time since they were proposed, it was widely believed that hyperelliptic curve cryptosystems (HECC) carry a substantial performance penalty compared to elliptic curve cryptosystems (ECC) and are, thus, not too attractive for practical applications. Only quite recently improvements have been made, mainly restricted to curves of genus 2. The work at hand advances the state-of-the-art considerably in several aspects. First, we generalize and improve the closed formulae for the group operation of genus 3 for HEC defined over fields of characteristic two. For certain curves we achieve over 50% complexity improvement compared to the best previously published results. Second, we introduce a new complexity metric for ECC and HECC defined over characteristic two fields which allow performance comparisons of practical relevance. It can be shown that the HECC performance is in the range of the performance of an ECC; for specific parameters HECC can even possess a lower complexity than an ECC at the same security level. Third, we describe the first implementation of a HEC cryptosystem on an embedded (ARM7) processor. Since HEC are particularly attractive for constrained environments, such a case study should be of relevance