Blind signcryption scheme based on hyper elliptic curves cryptosystem

Abstract

In modern cryptography, the Discrete Logarithm Problem (DLP) plays a vital role, but a classical computer cannot efficiently solve it. Nowadays, the Hyper Elliptic Curve Discrete Logarithm Problem (HECDLP) of Genus two (G₂) widely used in industry and also a research field of hot interest. This paper presents a novel blind signcryption scheme based on Hyper Elliptic Curves having properties of confidentiality, integrity, authenticity, non-repudiation, unforgeability, un-traceability, and message and original sender unlink-ability. The efficiency of our scheme is 76.85% and 80.95% in terms of communication cost w.r.t minor and major operations respectively. Also, the efficiency of our scheme 33.33% and 50% in terms of computation cost w.r.t minor and major operations. Our proposed scheme more suitable for emerging resource constraints environment like mobile commerce and digital democracy.

Appendix

In 1976 Deffie-Hellman proposed a concept about a new direction in cryptography [16], in which the author presented that two parties can agree on a common secret key across a public channel to interchange his/her information. Deffie-Hellman key agreement protocols are based on a finite field (finite Abelian group). The idea behind Deffie-Hellman key exchange protocol is shown in Fig. 5.

Fig. 5

Deffie-Hellman key exchange protocols

In 2010 Micheal J. et al., [19] says about the scheme [16], for efficient arithmetic the group must be sufficiently large. Two parties (Alice and Bob) agree on an imaginary hyper elliptic curve and a public random class in its Jacobian (known as reduced element/divisor D). Alice generates a random secret key a and sends the reduced element Da in the class of aD to Bob. Similarly, Bob sends the reduced element Db in the class of bD to Alice, where b is his secret key. The common key is the reduced element Dab in the class of abD = a(bD) = b(aD).

1) DLP

The underlying discrete logarithm problem reads as follows: given D and the reduced element in the class of nD, find n. This problem has undergone considerable study and is widely believed to be intractable for HEC of small genus (g). The genus (g) of HEC over F_q is defined to be an absolutely irreducible, non-singular curve of the form shown in Eq. (1). it having two main models one Imaginary model and the second one is a Real model.

Imaginary model

If f is monic, deg(f) = 2g + 1, and deg(h) ≤ g if q is even.

Real model

If q is odd, then f is monic and deg(f) = 2g + 2. If q is even, then h is monic, deg(h) = g + 1, deg(f) ≤ 2g + 2, and the coefficient of x^2g+ 2 ∈ f is of the form s² + s for some s ∈ F_q.

2) Divisors of a hyper elliptic curve

Let C be a HEC equation(1) of genus g over a finite field F_q, and S denote the set of infinite points on C, $S = {\infty }$ if C is imaginary, and $S = {\infty ,\overline {\infty }}$ if C is real. A divisor D on C is a formal finite sum of (finite and infinite) points p on C; we can write $D = {\sum }_{p} n_{p}p$ where n(p) ∈ Z and n(p) = 0 for all but finitely many p. The support of D, denoted by supp(D), is the set of points p on C for which n(p)6 = 0, and the degree of D is $deg(D)= {\sum }_{p} n_{p}(p)\in Z$. A principal divisor has the form $D = {\sum }_{p} v_{p}(\alpha )p$ for some α ∈ F_q(C), where v(p)(α) is the order of vanishing of α at p; write D = div(α). Every principal divisor has degree zero.

3) Finite and infinite Divisors of a hyper elliptic curve

A divisor D is finite if $supp(D)\bigcap S = \oslash$ and infinite if $supp(D) \subseteq S$. For any divisor $D ={\sum }_{p} n_{p}p$, let $D_{s} ={\sum }_{p\notin s} n_{p}p$ denote its finite portion. If D has degree zero, then there is a unique representation

$$ D= \begin{cases} D_{s}-deg(D_{s})\infty ~if~ C~ is~ imaginary,\\ D_{s}-deg(D_{s})\infty +n_{\overline{\infty}}(\overline{\infty}-\infty)~\\ if~ C ~is~ real. \end{cases} $$

(20)

The hyper elliptic involution on C can be linearly extended P to send D_s to $\overline {D}_{s} ={\sum }_{p\notin s} n_{p}p$, and hence D to the divisor

$$ \overline{D}= \begin{cases} \overline{D}_{s}-deg(D_{s})\infty~if~ C~ is~ imaginary, \overline{D}_{s}-deg(D_{s})\infty +n_{\overline{\infty}}(\overline{\infty}-\infty)~if\\~ C~is~ real. \end{cases} $$

(21)

From Eqs. (20 & 21) , it is clear that $D +\overline {D}$ is a principal divisor.