Some Proposed Problems on Permutation Polynomials over Finite Fields

Long period pseudo-random sequence plays an important role in modern information processing systems. Base on residue number system (RNS) and permutation polynomials over finite fields, a pseudorandom sequence generation scheme is proposed in this paper. It extends several short period random sequences to a long period pseudo-random sequence by using RNS. The short period random sequences are generated parallel by the iterations of permutation polynomials over finite fields. Due to the small dynamic range of each iterative calculation, the bit width in hardware implementation is reduced. As a result, we can use full look-up table (LUT) architecture to achieve high-speed sequence output. The methods to find proper permutation polynomials to generate long period sequences and the optimization algorithm of Chinese remainder theorem (CRT) mapping are also proposed in this paper. The period of generated pseudorandom sequence can exceed 2100 easily based on common used field programmable gate array (FPGA) chips. Meanwhile, this scheme has extensive freedom in choosing permutation polynomials. For example, 10905 permutation polynomials meet the long period requirement over the finite field F q with q ≢ 1(mod 3) and q ⩽ 503. The hardware implementation architecture is simple and multiplier free. Using Xilinx XC7020 FPGA chip, we implement a sequence generator with the period over 250, which only costs 20 18kb-BRAMs (block RAM) and a small amount of logics. And the speed can reach 449.236 Mbps. The National Institute of Standards and Technology (NIST) test results show that the sequence has good random properties.

Two cryptosystems using polynomials on finite fields are introduced; a permutation cipher based on permutation polynomials and a flexible secret sharing scheme. Polynomials are called permutation polynomials if they induce bijective functions. Metric and algebraic properties of permutation polynomials over a finite fields are investigated, especially properties associated with orbits and permutation cycles. Flexible secret sharing schemes have features that they can change a secret, enroll/disenroll members, increase/decrease threshold, without changing shares. A simple scheme using polynomials over finite fields is presented.

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f ( x ) f(x) and g ( x ) g(x) are integral polynomials of degree n ≥ 2 n \geq 2 and p is a prime exceeding ( n 2 − 3 n + 4 ) 2 {({n^2} - 3n + 4)^2} for which f and g are both permutation polynomials of the finite field F p {F_p} , then their difference h = f − g h = f - g cannot be such that h ( x ) = c x h(x) = cx for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in F p {F_p} and t is the degree of h, then t ≥ 3 n / 5 t \geq 3n/5 and, provided t ≤ n − 3 t \leq n - 3 , t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

The study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G. L. Mullen (1991), as well as the potential applications of these permutation polynomials (Dillon 1974, Khachatrian and Kyureghyan, Discrete Appl. Math. 216, 622–626 2017, Lidl 1985, Lidl and Muller 1984, Rivest et al., ACM Commun. Comput. Algebra. 1978, 120–126 1976, Schwenk and Huber, Electron. Lett. 34, 759–760 1998). It is well known that every permutation polynomial over a finite field $\mathbb {F}_{q}$ can be reduced to a permutation polynomial of the form xrh(xs) with s∣(q − 1) and $h(x) \in \mathbb {F}_{q}[x]$ (Akbary et al., Finite Fields Appl. 15(2), 195–206 2009, Wang, Finite Fields Appl. 22, 57–69 2013). Recently, several explicit classes of permutation polynomials of the form xrh(xs) over ${\mathbb F}_{q}$ have been constructed. However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications. In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form xrh(xs) over ${\mathbb F}_{q}$ , where s∣(q − 1) and $\gcd (r,q-1)= 1$ . The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over ${\mathbb F}_{q}$ into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over $\mathbb {F}_{q}$ and the other is the polynomial xrh(x)s over a particular subgroup of $\mathbb {F}_{q}^{*}$ with order (q − 1)/s. This is a multiplicative analog of Tuxanidy and Wang (Finite Fields Appl. 28, 244–281 2014), Wu and Liu (Finite Fields Appl. 24, 136–147 2013). We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases. As consequences, many explicit compositional inverses of permutation polynomials given in Zieve (Proc. Am. Math. Soc. 137, 2209–2216 2009), Zieve (arXiv: 1310.0776 , 2013), Zieve (arXiv: 1312.1325v3 , 2013) are obtained using this method.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

A link between two classes of permutation polynomials

Known results on permutation polynomials with coefficients in a finite field are extended to algebras of the form Lv=K[X]/(p(X)v)), where K is a finite field, p(X) belongs to K[X] and is an irreducible polynomial, v = 1,2..., and to the algebra of power series L[[Z]]. Finally analogues of Dickson polynomials in these algebras are studied.

The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2

The topic of permutation polynomials over finite fields is important both in theory and in engineering application. Let p be an odd prime number and q a power of p. A necessary and sufficient condition for a family of polynomials over F q to be permutation polynomials is presented. Moreover, the number of permutation polynomials of this type is obtained. The polynomials discussed in this note are generalizations and improvement of those discussed by Xi-Wang Cao(2009).

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field $${\mathbb {F}}_{q}$$.

New classes of permutation binomials and permutation trinomials over finite fields

Finite (Galois) field arithmetic finds applications in cryptography, error correction codes, signal processing, etc. Multiplication usually lies at the core of all Galois field computations and is a high-complexity operation. This paper addresses the problem of formal verification of hardware implementations of modulo-multipliers over Galois fields of the type F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k, using a computer-algebra/algebraic-geometry based approach. The multiplier circuit is modeled as a polynomial system in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k[x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ,⋯,x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> ] and the verification test is formulated as a Nullstellensatz proof over the finite field. A Grobner basis engine is used as the underlying computational framework. The efficiency of Grobner basis computations depends heavily upon the variable (and term) ordering used to represent and manipulate the polynomials. We present a variable (and term) ordering heuristic that significantly improves the efficiency of Grobner basis engines. Using our approach, we can verify the correctness of up to 96-bit multipliers, whereas contemporary BDDs/SAT/SMT-solver based methods are infeasible.

Ten years ago, Ko et al. described a Diffie-Hellman like protocol based on decomposition with respect to a non-commutative semigroup law. Instantiation with braid groups had first been considered, however intense research on braid groups revealed vulnerabilities of the group structure and of the braid based DH problem itself. Recently, Boucher et al. proposed a similar scheme based on a particular non-commutative multiplication of polynomials over a finite field. These so called skew polynomials have a well-studied theory and have many applications in mathematics and coding theory, however they are quite unknown in a cryptographic application. In this paper, we show that the Diffie-Hellman problem based on skew polynomials is susceptible to a very efficient attack. This attack is in fact general in nature, it uses the availability of a one-sided notion of gcd and exact division. Given such tools, one can shift the Diffie-Hellman problem to a linear algebra type problem.

Two types of permutation polynomials with special forms

Let p be prime, q = pm, and q − 1 = 7s. We completely describe the permutation behavior of the binomial P(x) = xr(1 + xes) (1 ≤ e ≤ 6) over a finite field in terms of the sequence {an} defined by the recurrence relation an = an−1 + 2an−2 − an−3 (n ≥ 3) with initial values a0 = 3, a1 = 1, and a2 = 5.