arXiv subjects

Kangquan Li

Publications and source records attributed to Kangquan Li.

26 records · Page 2Linked to original sources

arXiv2019

Further study of $2$-to-$1$ mappings over $\mathbb{F}_{2^n}$

$2$-to-$1$ mappings over finite fields play an important role in symmetric cryptography, in particular in the constructions of APN functions, bent functions, semi-bent functions and so on. Very recently, Mesnager and Qu \cite{MQ2019} provided a systematic study of $2$-to-$1$ mappings over finite fields. In particular, they determined all $2$-to-$1$ mappings of degree at most 4 over any finite fields. In addition, another research direction is to consider $2$-to-$1$ polynomials with few terms. Some results about $2$-to-$1$ monomials and binomials have been obtained in \cite{MQ2019}. Motivated by their work, in this present paper, we push further the study of $2$-to-$1$ mappings, particularly, over finite fields with characteristic $2$ (binary case being the most interesting for applications). Firstly, we completely determine $2$-to-$1$ polynomials with degree $5$ over $\mathbb{F}_{2^n}$ using the well known Hasse-Weil bound. Besides, we consider $2$-to-$1$ mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of $2$-to-$1$ trinomials, which explain all the examples of $2$-to-$1$ trinomials of the form $x^k+βx^{\ell} + αx\in\mathbb{F}_{2^n}[x]$ over $\mathbb{F}_{2^n}$ with $n\le 7$, and derive twelve classes of $2$-to-$1$ quadrinomials with trivial coefficients over $\mathbb{F}_{2^n}$.

cs.IT↗

arXiv2019

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition about $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x) = x^{3\cdot 2^m} + a_1 x^{2^{m+1}+1}+ a_2 x^{2^m+2} + a_3 x^3$ is a class of permutation polynomials over $\gf_{2^{n}}$ with $n=2m$ and $m$ odd. In this present paper, we prove that the sufficient condition is also necessary.

cs.IT↗

arXiv2019

On the Differential-Linear Connectivity Table of Vectorial Boolean Functions

Vectorial Boolean functions are crucial building-blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria for vectorial Boolean functions, such as differential uniformity and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers $E_0$ and $E_1$ involved in differential-linear attacks. This new notion leads to significant improvements of differential-linear attacks on several ciphers. This paper presents a theoretical characterization of the DLCT of vectorial Boolean functions and also investigates this new criterion for some families of functions with specific forms. More precisely, we firstly reveal the connection between the DLCT and the autocorrelation of vectorial Boolean functions, we characterize properties of the DLCT by means of the Walsh transform of the function and of its differential distribution table, and we present generic bounds on the highest magnitude occurring in the DLCT of vectorial Boolean functions, which coincides (up to a factor~$2$) with the well-established notion of absolute indicator. Next, we investigate the invariance property of the DLCT of vectorial Boolean functions under the affine, extended-affine, and Carlet-Charpin-Zinoviev (CCZ) equivalence and exhaust the DLCT spectra of optimal $4$-bit S-boxes under affine equivalence. Furthermore, we study the DLCT of APN, plateaued and AB functions and establish its connection with other cryptographic criteria. Finally, we investigate the DLCT and the absolute indicator of some specific polynomials with optimal or low differential uniformity, including monomials, cubic functions, quadratic functions and inverses of quadratic permutations.

cs.IT↗

arXiv2019

On the Differential Linear Connectivity Table of Vectorial Boolean Functions

Vectorial Boolean functions are crucial building blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria of vectorial Boolean functions,such as differential distribution table and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the Differential-Linear Connectivity Table (DLCT).This paper is a follow-up work, which presents further theoretical characterization of the DLCT of vectorial Boolean functions and also investigates this new criterion of functions with certain forms. In this paper we introduce a generalized concept of the additive autocorrelation, which is extended from Boolean functions to the vectorial Boolean functions, and use it as a main tool to investigate the DLCT property of vectorial Boolean functions. Firstly, by establishing a connection between the DLCT and the additive autocorrelation, we characterize properties of DLCT by means of the Walsh transform and the differential distribution table, and present generic lower bounds on the differential-linear uniformity (DLU) of vectorial Boolean functions. Furthermore, we investigate the DLCT property of monomials, APN, plateaued and AB functions. Our study reveals that the DLCT of these special functions are closely related to other cryptographic criteria. Next, we prove that the DLU of vectorial Boolean functions is invariant underthe EA equivalence but not invariant under the CCZ equivalence, and that the DLCT spectrum is only invariant under affine equivalence. In addition, under affine equivalence, we exhaust the DLCT spectra and DLU of optimal S-boxes with $4$ bit by Magma. Finally, we investigate the DLCT spectra and DLU of some polynomials over $F_{2^n}$, including the inverse, Gold, Bracken-Leander power functions and all quadratic polynomials.

cs.IT↗

arXiv2019

New Results about the Boomerang Uniformity of Permutation Polynomials

In EUROCRYPT 2018, Cid et al. \cite{BCT2018} introduced a new concept on the cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut \cite{BC2018}. Aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment in this paper. Firstly, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition from \cite{BCT2018}. Secondly, thanks to Carlet's idea \cite{Carlet2018}, we give a characterization of functions $f$ from $\mathbb{F}_{2}^n$ to itself with boomerang uniformity $δ_{f}$ by means of the Walsh transform. Thirdly, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of $4$-uniform BCT permutation polynomials over $\mathbb{F}_{2^n}$, which is the first binomial.

cs.CR↗

arXiv2017

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

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 $μ_{q+1}$, where $μ_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. 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.

cs.IT↗

arXiv2016

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.

cs.IT↗

arXiv2015

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.

cs.IT↗