A Note on Cross Correlation Distribution of Ternary m-Sequences

In this note, we prove a conjecture proposed by Tao Zhang, Shuxing Li, Tao Feng and Gennian Ge, IEEE Transaction on Information Theory, vol. 60, no. 5, May 2014. This conjecture is about the cross correlation distribution of ternary $m$-sequences.Comment: After posting our manuscript on arxiv, we recived an email from Yongbo Xia. He told us that they also got the same result as in our manuscript. And their paper was accepted by 2014 SETA on June 14th. And the information of their paper is"Y. Xia, T. Helleseth and G. Wu, A note on cross-correlation distribution between a ternary m-sequence and its decimated sequence, to appear in SETA2014

New Classes of Permutation Binomials and Permutation Trinomials over Finite Fields

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and trinomials attract people's interest due to their simple algebraic form and additional extraordinary properties. In this paper, several new classes of permutation binomials and permutation trinomials are constructed. Some of these permutation polynomials are generalizations of known ones.Comment: 18 pages. Submitted to a journal on Aug. 15t

New Constructions of Permutation Polynomials of the Form xrh(xq−1)x^rh\left(x^{q-1}\right) over Fq2\mathbb{F}_{q^2}

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing the corresponding fractional polynomials permute a smaller set $\mu_{q+1}$, where $\mu_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. Motivated by these results, we characterize the permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$ such that $h(x)\in\mathbb{F}_q[x]$ is arbitrary and $q$ is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing permutations over a small subset $S$ of a proper subfield $\mathbb{F}_{q}$, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic.Comment: 29 pages. An early version of this paper was presented at Fq13 in Naples, Ital

A New Method to Compute the 2-adic Complexity of Binary Sequences

In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are uniformly determined. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum. Moreover, 2-adic complexities of two classes of optimal autocorrelation sequences with period $N\equiv1\mod4$, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated. Besides, this method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.Comment: 16 page

New Permutation Trinomials Constructed from Fractional Polynomials

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures

More Constructions of Differentially 4-uniform Permutations on \gf_{2^{2k}}

Differentially 4-uniform permutations on \gf_{2^{2k}} with high nonlinearity are often chosen as Substitution boxes in both block and stream ciphers. Recently, Qu et al. introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations \cite{QTTL}. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially 4-uniform permutations over \gf_{2^{2k}}. Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially 4-uniform permutations over \gf_{2^{2k}} grows exponentially when $k$ increases, which gives a positive answer to an open problem proposed in \cite{QTTL}

On two conjectures about the intersection distribution

Recently, S. Li and A. Pott\cite{LP} proposed a new concept of intersection distribution concerning the interaction between the graph \{(x,f(x))~|~x\in\F_{q}\} of $f$ and the lines in the classical affine plane $AG(2,q)$. Later, G. Kyureghyan, et al.\cite{KLP} proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over \F_{q} with $q$ both odd and even. They also proposed several conjectures in \cite{KLP}. In this paper, we completely solve two conjectures in \cite{KLP}. Namely, we prove two classes of power functions having intersection distribution: $v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}$. We mainly make use of the multivariate method and QM-equivalence on $2$-to-$1$ mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations

Cryptographically Strong Permutations from the Butterfly Structure

In this paper, we present infinite families of permutations of $\mathbb{F}_{2^{2n}}$ with high nonlinearity and boomerang uniformity $4$ from generalized butterfly structures. Both open and closed butterfly structures are considered. It appears, according to experiment results, that open butterflies do not produce permutation with boomerang uniformity $4$. For the closed butterflies, we propose the condition on coefficients $\alpha, \beta \in \mathbb{F}_{2^n}$ such that the functions $$V_i := (R_i(x,y), R_i(y,x))$$ with $R_i(x,y)=(x+\alpha y)^{2^i+1}+\beta y^{2^i+1}$ are permutations of $\mathbb{F}_{2^n}^2$ with boomerang uniformity $4$, where $n\geq 1$ is an odd integer and $\gcd(i, n)=1$. The main result in this paper consists of two major parts: the permutation property of $V_i$ is investigated in terms of the univariate form, and the boomerang uniformity is examined in terms of the original bivariate form. In addition, experiment results for $n=3, 5$ indicates that the proposed condition seems to cover all coefficients $\alpha, \beta \in \mathbb{F}_{2^n}$ that produce permutations $V_i$ with boomerang uniformity $4$. However, the experiment result shows that the quadratic permutation $V_i$ seems to be affine equivalent to the Gold function. Therefore, unluckily, we may not to obtain new permutations with boomerang uniformity $4$ from the butterfly structure

Further Study of Planar Functions in Characteristic Two

Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In 2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar functions over \F_{2^n}. Very recently, D. Bartoli and M. Timpanella \cite{Bartoli} characterized the condition on coefficients $a,b$ such that the function f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x] is a planar function over \F_{2^{3m}} by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in \cite{Q}, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over \F_{q^k}, where $q=2^m$ with $m$ sufficiently large (see Theorem \ref{main}). The first and last classes of them are over \F_{q^2} and \F_{q^4} respectively, while the other two classes are over \F_{q^3}. One class over \F_{q^3} is an extension of $f_{a,b}(x)$ investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In addition, although the planar binomial over \F_{q^2} of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in \cite{Q}

Finding compositional inverses of permutations from the AGW criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.Comment: 24 pages. Revision submitte