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Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of permutation binomials is still unknown. In this paper, we give a classification of permutation binomials of the form $x^i+ax$ over $\mathbb{F}_{2^n}$, where $n\leq 8$ by characterizing three new classes of permutation binomials. In particular one of them has relatively large index $\frac{q^2+q+1}{3}$ over $\mathbb{F}_{q^3}$.

This paper presents a new method to reduce the optimization of a pseudo-Boolean function to QUBO problem which can be solved by quantum annealer. The new method has two aspects, one is coefficient optimization and the other is variable optimization. The former is an improvement on the existing algorithm in a special case. The latter is realized by means of the maximal independent point set in graph theory. We apply this new method in integer factorization on quantum annealers and achieve the largest integer factorization (4137131) with 93 variables, the range of coefficients is [-1024,1024] which is much smaller than the previous results. We also focus on the quantum attacks on block ciphers and present an efficient method with smaller coefficients to transform Boolean equation systems into QUBO problems.

Quantum computers in practice today require strict memory constraints, where 2-qubit operations can only be performed between the qubits closest to each other in a graph structure. So a quantum circuit must undergo a transformation to the graph before it can be implemented. In this paper, we study the optimization of the CNOT circuits on some noisy intermediate-scale quantum(NISQ) devices. Compared with previous works, we decompose it into two sub-problems: optimization with a given initial qubit distribution and optimization without limitations of initial qubit distribution. We find that most of the previous researches focused on the first sub-problem, and ignored the influence of different distribution of qubits in the same topology structure on the optimization results. In this paper, We take both sub-problems into account and give some new optimization algorithms. In short, our method is divided into two steps: matrix optimization and routing optimization. We implement matrix optimization with the algorithm in [XZL+20] and put forward a new heuristic algorithm with MILP method which can solve the second step. We implement our algorithm on IBM20 and some other NISQ devices, the results are better than most other methods in our experiment.

In a deduction system with some propositions and some known relations among these propositions, people usually care about the minimum of propositions by which all other propositions can be deduced according to these known relations. Here we call it a minimizing deduction system. Its common solution is the guess and determine method. In this paper we propose a method of solving the minimizing deduction system based on MILP. Firstly, we introduce the conceptions of state variable, path variable and state copy, which enable us to characterize all rules by inequalities. Then we reduce the deduction problem to a MILP problem and solve it by the Gurobi optimizer. As its applications, we analyze the security of two stream ciphers SNOW2.0 and Enocoro-128v2 in resistance to guess and determine attacks. For SNOW 2.0, it is surprising that it takes less than 0.1s to get the best solution of 9 known variables in a personal Macbook Air(Early 2015, Double Intel Core i5 1.6GHZ, 4GB DDR3). For Enocoro-128v2, we get the best solution of 18 known variables within 3 minutes. What's more, we propose two improvements to reduce the number of variables and inequalities which significantly decrease the scale of the MILP problem., Comment: 24 pages

In this paper, we construct two classes of permutation polynomials over $\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. A complete characterization of their compositional inverses is also given. These permutation polynomials can be generated recursively. As a consequence, we can generate recursively permutation polynomials with arbitrary number of terms. More importantly, the conditions of these polynomials being permutations are very easy to characterize. For wide applications in practice, several classes of permutation binomials and trinomials are given. With the help of a computer, we find that the number of permutation polynomials of these types is very large.

In 2007, Carlet and Ding introduced two parameters, denoted by $Nb_F$ and $NB_F$, quantifying respectively the balancedness of general functions $F$ between finite Abelian groups and the (global) balancedness of their derivatives $D_a F(x)=F(x+a)-F(x)$, $a\in G\setminus\{0\}$ (providing an indicator of the nonlinearity of the functions). These authors studied the properties and cryptographic significance of these two measures. They provided for S-boxes inequalities relating the nonlinearity $\mathcal{NL}(F)$ to $NB_F$, and obtained in particular an upper bound on the nonlinearity which 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 \emph{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 that the second pair of parameters be compared to the first one. In the present paper, we observe that ambiguity is the same parameter as $NB_F$, up to additive and multiplicative constants (i.e. up to rescaling). We make the necessary work of comparison and unification of the results on $NB_F$, respectively on ambiguity, which have been obtained in the five papers devoted to these parameters. We generalize some known results to any Abelian groups and we more importantly derive many new results on these parameters.

In this paper, we construct some new classes of complete permutation monomials with exponent $d=\frac{q^n-1}{q-1}$ using AGW criterion (a special case). This proves two recent conjectures in [Wuetal2] and extends some of these recent results to more general $n$'s.

Recently, a new structure called butterfly introduced by Perrin et at. is attractive for that it has very good cryptographic properties: the differential uniformity is at most equal to 4 and algebraic degree is also very high when exponent $e=3$. It is conjecture that the nonlinearity is also optimal for every odd $k$, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent $e=2^i+1$ have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd $k$, which completely solve the open problem. Finally, we study the butter structures with trivial coefficient and show these butterflies have also optimal nonlinearity. Furthermore, we show that the closed butterflies with trivial coefficient are bijective as well, which also can be used to serve as a cryptographic primitive., Comment: 20 pages