Shapeless quasigroups derived by Feistel orthomorphisms

Abstract

Abstract.Shapeless quasigroups are needed for cryptography purposes. In this paper, we construct shapeless quasigroups by the diagonal method from orthomorphisms over abelian groups. We use generalizations of Feistel networks as orthomorphisms. We introduce parameters into several types of Extended Feistel networks and Generalized Feistel-non linear feedback shift registers and, by suitable choice of the parameter values, different shapeless quasigroup can be used in every application.

Quasigroup Representation of Some Feistel and Generalized Feistel Ciphers

“…Now we are going to modify slightly these definitions in order to be suitable for our next purposes. All of the proofs given in [20] are almost immediately applicable for the modified versions as well. As an illustration, we will give only the proof of Theorem 2.…”

“…In [20] we have defined parameterized versions of the Feistel network, the type-1 Extended Feistel network, and the Generalized Feistel-Non Linear Feedback Shift Register (GF-NLFSR), and we have proved that if a bijection f is used for their creation, then they are orthomorphisms of abelian groups. Now we are going to modify slightly these definitions in order to be suitable for our next purposes.…”

“…We go one step further. In this paper we examine several block ciphers that use Feistel networks or their generalizations, and we rewrite their rounds by quasigroup operations, using ideas from [20]. Special attention is given to the block ciphers Misty1, Camellia, Four-Cell, Four-Cell + and SMS4.…”

Regular complete permutation polynomials over quadratic extension fields

“…Niederreiter and Robinson later gave a detailed study of CPPs over finite fields [29]. CPPs have widely applications in the design of nonlinear dynamic substitution device [23,24], the Lay-Massey scheme [37], the block cipher SMS4 [6], the stream cipher Loiss [7], the design of Hash functions [34,36], quasigroups [18,21,22], and the constructions of some cryptographically strong functions [26,35,40]. The two most important concepts related to a permutation are the existence of fixed points and the specification of its cycle structure.…”