We introduce the definition ofm-to-1 mappings between two finite sets, which unifies and generalizes the definitions of 2-to-1 andn-to-1 mappings in recent literature. We also characterize thesem-to-1 mappings in terms of the generalized local criterion and thus provide three generic constructions ofm-to-1 mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whetherxrh(xs) ism-to-1 on the multiplicative groupF∗qis converted into that whether an associated polynomialxr1h(x)s1ism2-to-1 on the order ℓ subgroupUℓ ofF∗q, wherem2=m/(r,s) and ℓ = (q− 1)/s. Furthermore, them2-to-1 property ofxr1h(x)s1onUℓ is studied in detail in five different cases. In addition, a recursive construction ofm-to-1 mappings fromm-to-1 mappings is proposed.

We propose to study the construction of Boolean bent functions from cyclotomic mappings. By considering Dillon functions, Niho functions and Kasami functions as different branch functions respectively, we obtain three generic constructions from this new perspective. As a result, several infinite classes of bent functions belonging to the${\mathcal {PS}}_{ap}$class, class$\mathcal {H}$and the completed$\mathcal {MM}$class are derived, thereby providing simple representations of known classes of bent functions through cyclotomic mappings. In addition, computer experiments show that examples of bent functions outside these three well-known classes can also be obtained by selecting other branch functions.

In caching system, it is desirable to design a coded caching scheme with the transmission load$R$and subpacketization$F$as small as possible, in order to improve efficiency of transmission in the peak traffic times and to decrease implementation complexity. Yan et al. reformulated the centralized coded caching scheme as designing a corresponding$F\times K$array called placement delivery array (PDA), where$F$is the subpacketization and$K$is the number of users. Motivated by several constructions of PDAs, we introduce a framework for constructing PDAs, where each row is indexed by a row vector of some matrix called row index matrix and each column’s index is labelled by an element of a direct product set. Using this framework, a new scheme is obtained, which can be regarded as a generalization of some previously known schemes. When$K$is equal to${\binom{m}{ t}}q^{t}$for positive integers$m$,$t$with$t < m$and$q\geq 2$, we show that the row index matrix must be an orthogonal array if all the users have the same memory size. Furthermore, the row index matrix must be a covering array if the coded gain is${\binom{m}{ t}}$, which is the maximal coded gain under our framework. Consequently the lower bounds on the transmission load and subpacketization of the schemes are derived under our framework. Finally, using orthogonal arrays as the row index matrix, we obtain two more explicit classes of schemes which have significantly advantages on the subpacketization while the transmission load is equal or close to that of the schemes constructed by Shangguan et al. for the same number of users and memory size.

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.

In this paper, we study a class of binary nonlinear feedback shift register sequences generated by cascaded feedback registers, one is an LFSR and the other one generates a de Bruijn sequence. The cycle structure (in particular, the initial state of each cycle) is determined by solving a system of linear equations. As an application, we can generate de Bruijn sequences of large period algorithmically.

Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree ≤ 6 over finite fields Fq for all q and the inverses of all PPs of degree 7 over F2(n). The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas' theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields.

Coded caching, which could significantly reduce the maximum amount of transmission rate during the peak traffic times in wireless network, has been widely studied recently. Apart from the transmission rate, sub-packetization F reflecting the implementation complexity, is also concerned in coded caching. The grouping method proposed by Shanmugam et al. is wellknown and widely used to reduce the sub-packetization level of the coded caching problem. In this paper, we propose a concatenating construction method for coded caching schemes, which generalizes the grouping method. Moreover, we demonstrate the advantage of our method in reducing the transmission rate over the grouping method. In particular, some new explicit schemes are obtained from previously known schemes. From one of these schemes, we can derive all the results by Tang and Ramamoorthy as special cases. Furthermore, the analysis and comparison of these new schemes are also performed.

In 2007, Carlet and Ding introduced two parameters, denoted by NbF and NBF, quantifying respectively the balancedness of general functions F between finite Abelian groups and the (global) balancedness of their derivatives DaF(x) = F(x + a) - F(x), a ∈ G \ {0} (providing an indicator of the nonlinearity of the functions). These authors studied the properties and cryptographic significance of these two measures. They provided inequalities relating the nonlinearity NL(F) to NBF for S-box and specifically obtained an upper bound on the nonlinearity that unifies Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the Workshop WCC 2009 and in its postproceedings in 2011, a further study of these parameters was made; in particular, the first parameter was applied to the functions F + L, where L is affine, providing more nonlinearity parameters. In 2010, motivated by the study of Costas arrays, two parameters called ambiguity and deficiency were introduced by Panario et al. for permutations over finite Abelian groups to measure the injectivity and surjectivity of the derivatives, respectively. These authors also studied some fundamental properties and cryptographic significance of these two measures. Further studies followed without comparing the second pair of parameters to the first one. In this paper, we observe that ambiguity is the same parameter as NBF up to additive and multiplicative constants (i.e., up to rescaling). We perform the necessary work of comparison and unification of the results on NBF and on ambiguity, which have been obtained in the five papers devoted to these parameters. We generalize some known results to any finite Abelian groups. More importantly, we derive many new results on these parameters.

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

The concepts of ambiguity and deficiency for a bijection on a finite Abelian group were recently introduced. In this paper, we present some further fundamental results on the ambiguity and deficiency of functions; in particular, we note that they are invariant under the well-known Carlet-Charpin-Zinoviev-equivalence, we obtain upper and lower bounds on the ambiguity and deficiency of differentially k-uniform functions, and we give a lower bound on the nonlinearity of functions that achieve the lower bound of ambiguity and deficiency. In addition, we provide an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a Dembowski-Ostrom (DO) polynomial, and using this technique, we find exact values for known cases of DO permutations with few terms. We also derive exact values for the ambiguities and deficiencies of DO permutations obtained from trace functions. The key relationship between the above polynomials is that they all have linearized difference map.

The concepts of ambiguity and deficiency for a given bijection on a finite Abelian group were recently introduced [13]. In this work we investigate the ambiguity and deficiency of some well-known polynomials which satisfy Dn(x+y, xy) = xn+ynfor every x, y ϵ Fqand n ϵ N, as well as linearized polynomials and Dembowski-Ostrom polynomials (DO polynomials). For some specific values of n (related to q) these polynomials generate permutations on Fq. We derive explicitly the ambiguity and deficiency of some of them. Numerical results on the ambiguity and deficiency of the others are also provided. Some of these polynomials are almost perfect nonlinear (APN) functions.

We introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite Abelian groups of the same size. Then, we study the optimum lower bounds of these measures for permutations of an Abelian group. A construction of permutations, by modifying some permutation functions over finite fields, is given. Their ambiguity and deficiency is investigated; most of these functions are APN permutations. We show that, when they are not optimal, the Möbius function in the multiplicative group of \BBFqis closer to being optimal in ambiguity than the inverse function in the additive group of \BBFq. We note that the inverse function over \BBF28is used in AES. Finally, we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the APN function employed in the SAFER cryptosystem.

In this paper, we present a new method for multiplying polynomials in Chebyshev form. Our approach has two steps. First, the well-known Karatsuba's algorithm is applied to polynomials constructed by using Chebyshev coefficients. Then, from the obtained result, extra arithmetic operations are used to write the final result in Chebyshev form. The proposed algorithm has a quadratic computational complexity. We also compare our method to other approaches.