Permutation polynomials on matrices

The universal binary neurons are a central point of this Chapter. The mathematical models of the universal binary neuron over the field of the complex numbers, the residue class ring and the finite field are considered. A notion of the P-realizable Boolean function, which may be considered as a generalization of a notion of the threshold Boolean function, is introduced. It is shown that the implementation with a single neuron the input/output mapping described by non-threshold Boolean functions is possible, if weights are complex, and activation function of the neuron is a function of the argument of the weighted sum (similar to multi-valued neuron). It is also possible to define an activation function on the residue class ring and the finite field in such a way that the implementation of non-threshold Boolean functions will be possible on the single neuron with weights from these sets. P-realization of multiple-valued function over finite algebras and residue class ring in particular is also considered. The general features of P-realizable Boolean functions are considered, also as the synthesis of universal binary neuron.

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

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.

Diestelkamp, Hartke and Kenney recently studied local permutation polynomials over finite field $F_s$ and proved that $2s-4$ is the sharp bound of their degrees when $s \\gt 3$. In this paper, we focus on local permutation polynomials over residue class rings. We prove two specific formulas computing the number and the sharp bound of degrees of local permutation polynomials modulo $p^n$ respectively.

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.

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.

Letbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

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

Two types of permutation polynomials with special forms