Evolution of Hyperelliptic Curve Cryptosystems

Due to short operand size, Hyperelliptic Curve Cryptosystem (HECC) of genus 3 is well suited for all kinds of embedded processor architectures, where resources such as storage, time or power are constrained. In the implementation of HECC, a significant step is the selection of secure hyperelliptic curves on which the Jacobian is constructed and speed up the scalar multiplications in the Jacobians of hyperelliptic curves. In this paper, we have explored various possible attacks to the discrete logarithm in the Jacobian of a Hyperelliptic Curve (HEC) that are to be considered to establish a secure HEC, analysed addition and doubling of divisor which are the prime steps of scalar multiplication and then proposed certain improvements in the existing explicit formula that will result in a performance gain for HECC of genus 3.

[1]  Y. Wang,et al.  Inversion-free arithmetic on genus 3 hyperelliptic curves and its implementations , 2005, International Conference on Information Technology: Coding and Computing (ITCC'05) - Volume II.

[2]  Neal Koblitz,et al.  Hyperelliptic cryptosystems , 1989, Journal of Cryptology.

[3]  Kazumaro Aoki,et al.  Improvements of Addition Algorithm on Genus 3 Hyperelliptic Curves and Their Implementation , 2005, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[4]  Jeffrey Shallit,et al.  Algorithmic Number Theory , 1996, Lecture Notes in Computer Science.

[5]  D. Cantor.  Computing in the Jacobian of a hyperelliptic curve , 1987 .

[6]  Benjamin A. Smith.  Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves , 2008, EUROCRYPT.

[7]  Nigel P. Smart,et al.  Advances in Cryptology - EUROCRYPT 2008, 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Istanbul, Turkey, April 13-17, 2008. Proceedings , 2008, EUROCRYPT.

[8]  Christof Paar,et al.  Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves , 2003, CHES.

[9]  Guang Gong,et al.  Efficient explicit formulae for genus 3 hyperelliptic curve cryptosystems over binary fields , 2007, IET Inf. Secur..

[10]  R. Zuccherato,et al.  An elementary introduction to hyperelliptic curves , 1996 .

[11]  Hans-Georg Rück,et al.  On the discrete logarithm in the divisor class group of curves , 1999, Math. Comput..

[12]  Leonard M. Adleman,et al.  A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields , 1994, ANTS.

[13]  G. Frey,et al.  A remark concerning m -divisibility and the discrete logarithm in the divisor class group of curves , 1994 .