A proof of maximum contention-free property of interleavers for Turbo codes using permutation polynomials over integer rings

It is well known that interleavers play a critical role in Turbo coding/decoding schemes, and contention-free interleaver design has become a serious problem in the paraUelization of Turbo decoding, which is indispensable to meet the demands for high throughput and low latency in next generation mobile communication systems. This paper unveils the fact that interleavers based on permutation polynomials modulo N are contention-free for every window size W, a factor of the intedeaver length N, which, also called maximum contention-free interleavers.

Some classes of complete permutation polynomials over Fq

By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Niho type are presented. Second, for any odd prime p, we give a sufficient and necessary condition for a-1xdto be a complete permutation polynomial over Fp4 k, where d =(p4k-1)/(pk-1)+ 1 and a ∈ F*p4k. Finally, we present a class of complete permutation multinomials, which is a generalization of recent work.

Permutation Polynomials over Finite Fields

§1.IntroductionLet Fqdenote the finite field with q=pmelements,wherep is a prime.A polynomial f(x)inFqis called a permutation polynomial if f(x)=a has a solution in Fqfor every a in Fq.Many studieshave been made to develop properties of permutation polynomials.For a survey of the work on thissubject prior to 1920 we refer to Dickson.During this period it was Dickson himself who

On Permutation Polynomials in Two Variables over ■/p^2■

We give a good characterization of the permutation polynomials in two variables over Z/p2Z.

New Permutation Reversed Dickson Polynomials over Finite Fields

Let p be an odd prime,and n,k be nonnegative integers.Let Dn,k(1,x)be the reversed Dickson polynomial of the(k+1)-th kind.In this paper,by using Hermite's criterion,we study the permutational properties of the reversed Dickson polynomials Dn,k(1,x)over finite fields in the case of n=mp* with 0<m<p-1.In particular,we provide some precise characterizations for Dn,k(1,x)being permutation polynomials over finite fields with characteristic p when n=2p^(s),or n=3p^(s),or n=4p^(s).

Zeros of Complete Symmetric Polynomials over Finite Fields

Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author improves Wan-Zhang's bound from three aspects.The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial.And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.

The Linear Span of a Class of Optimal Frequency Hopping Sequences

For the anti-jamming purpose,frequency hopping sequences are required to have a large linear span. In this paper,we firstly give the linear span of a class of optimal frequency hopping sequences. The results show that the linear span is very small compared with their periods. To improve the linear span,we transform these optimal frequency hopping sequences into new optimal frequency hopping sequences with large linear span by using a general type of permutation polynomials over a finite field. Furthermore,we give the exact values of the linear span of the transformed optimal frequency hopping sequences.

On the Lower Bound of the Divisibility of Exponential Sums in Binomial Case

In this article, we analyze the lower bound of the divisibility of families of exponential sums for binomials over prime field. An upper bound is given for the lower bound, and, it is related to permutation polynomials.

On Dickson Polynomials and Difference Sets

In 1998, Maschietti constructed several cyclic difference sets from monomial hyperovals. R. Evans, H.D.L. Holloman, C. Krattnthaler and Qing Xiang gave an algebraic proof of the two autocorrelation property of the related binary sequence. In this paper, we show that hyperovals are very closely related to two-to-one maps, and then we proceed to generalize Maschietti＇s result.

Constructing Permutation Binomials from Permutations of Subfields

Permutation polynomials is a hot topic in finite fields,they have many applications in different areas.Permutation binomials and trinomials over finite fields were studied recently.In thispaper,by using a powerful lemma given by Zieve and some degree 5 and 6 permutation polynomials over Fq,we construct somepermutation binomials over Fqm.

On the Differential Uniformity and Nonlinearity of a Class of Permutation Quadrinomials Over F_(2 2 m)

Permutation polynomials with low differential uniformity and high nonlinearity are preferred in cryptographic systems. In 2018, Tu, Zeng and Helleseth constructed a new class of permutation quadrinomials over the finite field F_(2 2m) for an odd integer m. In this paper, we aim to investigate the differential uniformity and nonlinearity of this class of permutation polynomials so as to find 4-uniform permutation polynomials with high nonlinearity.

Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form

This paper is concerned with so-called index d generalized cyclotomic mappings of a finite field F_(q), which are functions F_(q)→F_(q) that agree with a suitable monomial function x↦axr on each coset of the index d subgroup of F_(q)^(*). We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index d generalized cyclotomic permutations of F_(q) and pertain to cycle structures, the classification of (q−1)-cycles and involutions, as well as inversion.

On the Number Counting of Polynomial Functions

Polynomial functions （in particular, permutation polynomials） play an important role in the design of modern cryptosystem. In this note the problem of counting the number of polynomial functions over finite commutative rings is discussed. Let A be a general finite commutative local ring. Under a certain condition, the counting formula of the number of polynomial functions over A is obtained. Before this paper, some results over special finite commutative rings were obtained by many authors.

LTE turbo decoder design

This paper presents a turbo decoder supporting all 188 block sizes in 3GPP long term evolution （LTE） standard which can be employed in the LTE micro-eNodB system. The design allows 1, 2, 4, 8 or 16 soft-in/softout （SISO） decoders to concurrently process each block size, and the number of iterations can be adjusted. By adding a register in core structure add-compare-select-add, this article proposes an improved SISO algorithm and interleaving design, calculated forward state matrix and backward state matrix alternately, and the branch transition probability can be used in the Turbo decode process directly just after one clock delay. The structure enables a decoder processing radix-2 algorithm with high speed, instead of radix-4 as the conventional decoder. Moreover, the paper details an interleaver/de-interleaver, which is combined by two operational steps. One is column address mapping and the other is intra-row permutation. Decoder realizes interleaving by loading data from memories whose address is generated by column mapping and then lets data passing through inter-row permutation. For de-interleaving, the system can adopt reverse operation.

A new construction for involutions over finite fields

In 2020,Niu et al.[Cryptogr.Commun.,2020,12(2):165–185]studied the fixed points of involutions over the finite field with q-elements.This paper further discusses the relationship between the fixed points set and the non-fixed points set of two involutions f_(1)(x)and f_(2)(x)over the finite field F_(q),and then obtains a necessary and sufficient condition for that the composite function f_(1)■f_(2)(x)is also an involution over F_(q).In particular,a special class of involutions over some finite fields is determined completely.