Cyclic Decomposition of Permutations of Finite Fields Obtained Using Monomials

Abstract

Abstract. In this paper we study permutations of finite fields F q that decompose as products of cycles of the same length, and are obtained using monomials. We give the necessary and sufficient conditions on the exponent i to obtain such permutations. We also present formulas for counting the number of this type of permutations. An application to the construction of encoders for turbo codes is also discussed.

Cycle structure of permutation functions over finite fields and their applications

“…Our approach leads us to use self-inverse permutation functions which have cycles of the same length j = 2, or otherwise fixed points. Such permutation monomials are obtained using the following theorem from [24] for j = 2. We recall that j = ord s (n), if j is the smallest integer with the property n j ≡ 1 (mod s).…”

“…The study of permutation monomials x n with a cycle of length j has been treated in [1]. Permutation monomials x n with all cycles of the same length are characterized in [24]. The cycle structure of Dickson permutation polynomials D n (x, a) where a ∈ {0, ±1} has been studied in [17].…”

Rédei permutations with cycles of the same length

“…The results in [20,24,23] reveal a strong connection between Rédei and monomial functions. In fact, the description of Rédei permutations with cycles of the same length given in [24,23] is similar to that given for monomial permutations in [21]. However, it is not clear how to produce permutations with all nontrivial cycles of a given length j.…”

“…This type of function is of interest in the construction of interleavers for turbo codes [21,24,23]. Characterizations of permutations with cycles of the same length have been given for monomial permutations [21], Dickson polynomials [22], Rédei and Möbius functions [24,23], and linear maps [19]. The results in [20,24,23] reveal a strong connection between Rédei and monomial functions.…”

“…In this paper we study Rédei permutations that decompose into cycles of the same length. This type of function is of interest in the construction of interleavers for turbo codes [21,24,23]. Characterizations of permutations with cycles of the same length have been given for monomial permutations [21], Dickson polynomials [22], Rédei and Möbius functions [24,23], and linear maps [19].…”