The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm for Symplectic and Orthogonal Groups

Abstract

In this chapter, we study the MOR cryptosystem with symplectic and orthogonal groups over finite fields of odd characteristics. There are four infinite families of finite classical Chevalley groups. These are special linear groups SL(d, q), orthogonal groups O(d, q), and symplectic groups Sp(d, q). The family O(d, q) splits into two different families of Chevalley groups depending on the parity of d. The MOR cryptosystem over SL(d, q) was studied by the second author. In that case, the hardness of the MOR cryptosystem was found to be equivalent to the discrete logarithm problem in F q d . In this chapter, we show that the MOR cryptosystem over Sp(d, q) has the security of the discrete logarithm problem in F q d . However, it seems likely that the security of the MOR cryptosystem for the family of orthogonal groups is F q d 2 . We also develop an analog of row-column operations in symplectic and orthogonal groups which is of independent interest as an appendix.

Cryptanalysis of a new version of the MOR scheme

“…We suppose that ϕ can be naturally extended to a linear transformation of V =Lin Fq (G) that is a linear subspace generated by G in M(d, F q ). It happens for example, if ϕ is an inner automorphism of G. Note, that the case of inner automorphism ϕ is considered in [1] as the most significant.…”

“…We are to show that the version of MOR in [1] is not entirely accurate. It should be supplemented with an additional assumption.…”

“…In [1], S. Bhunia, A. Mahalanobis, P. Shinde and A. Singh study the ElGamal-type version of the MOR cryptosystem with symplectic and orthogonal groups over finite fields F q of odd characteristics. The MOR cryptosystem over SL(d, F q ) was previously investigated by the second of these authors.…”

“…In that case, the hardness of the MOR cryptosystem was found to be equivalent to the discrete logarithm problem in F q d . It is shown in [1] that the MOR cryptosystem over Sp(d, q) has the security of the discrete logarithm problem in F q d . The MOR cryptosystem also studied in [2] and in [3], and was cryptanalyzed in [4].…”

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We show that an attack based on the linear decomposition method introduced by the author can be efficiently applied to the new version of the MOR scheme proposed in [1]. We draw attention to some inaccuracies in the description of this version.

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