Keqin Feng

An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity

After the improvement by Courtois and Meier of the algebraic attacks on stream ciphers and the introduction of the related notion of algebraic immunity, several constructions of infinite classes of Boolean functions with optimum algebraic immunity have been proposed. All of them gave functions whose algebraic degrees are high enough for resisting the Berlekamp-Massey attack and the recent Ronjom-Helleseth attack, but whose nonlinearities either achieve the worst possible value (given by Lobanovs bound) or are slightly superior to it. Hence, these functions do not allow resistance to fast correlation attacks. Moreover, they do not behave well with respect to fast algebraic attacks. In this paper, we study an infinite class of functions which achieve an optimum algebraic immunity. We prove that they have an optimum algebraic degree and a much better nonlinearity than all the previously obtained infinite classes of functions. We check that, at least for small values of the number of variables, the functions of this class have in fact a very good nonlinearity and also a good behavior against fast algebraic attacks.

On the Weight Distributions of Two Classes of Cyclic Codes

Let q=p^m where p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from F_q to F_p and zeta_p=e^2pii/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Sigma_xisinF _qchi(alphax^p ^k ⁺¹+betax²) (alpha, betaisinF_q) where chi(x)=zeta_p ^Tr(x) is the canonical additive character of F_q. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C₁ and C₂ over F_pt with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where t is a divisor of d=gcd(m, k), and h₁(x), h₂(x) , and h₃(x) are the minimal polynomials of pi^-1, pi^-2, and pi^-(p ^k ⁺¹⁾ over F_pt, respectively, for a primitive element pi of F_q. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).

Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums Sigma_xisinF(q)zeta_p ^Tr(af(x)+bx)(a,bisinF_q,q=p^m,pges3) for all perfect nonlinear functions f which is a Dembowski-Ostrom polynomial or p = 3,f=^x(3(k)+1)/2 where k is odd and (k,m)=1. As applications, we determine (1) the correlation distribution of the m-sequence {a_lambda=Tr(gamma^lambda)}(lambda=0,1,...) and the sequence {b_lambda=Tr(f(gamma^lambda))}(lambda=0,1,...) over F_p where gamma is a primitive element of F_q and (2) the weight distributions of the linear codes over F_p defined by f.

Constructing Symmetric Boolean Functions With Maximum Algebraic Immunity

Symmetric Boolean functions with even variables 2k and maximum algebraic immunity AI(f) = k have been constructed in Braekens thesis (2006). In this paper, we show more constructions of such Boolean functions including the generalization of a result and prove a conjecture raised in Braekens thesis (2006).

Asymptotic bounds on quantum codes from algebraic geometry codes

We generalize a characterization of p-ary (p is a prime) quantum codes given by Feng and Xing to q-ary (q is a prime power) quantum codes. This characterization makes it possible to convert an asymptotic bound of Stichtenoth and Xing for nonlinear algebraic geometry codes to a quantum asymptotic bound. Besides, we also investigate the asymptotic behavior of quantum codes

A finite Gilbert-Varshamov bound for pure stabilizer quantum codes

A finite Gilbert-Varshamov (GV) bound for pure stabilizer (binary and nonbinary) quantum error correcting codes is presented in analogy to the GV bound for classical codes by using several enumerative results in finite unitary geometry. From this quantum GV bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes.

A Note on Symmetric Boolean Functions With Maximum Algebraic Immunity in Odd Number of Variables

In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.

Asymmetric Quantum Codes: Characterization and Constructions

The stabilizer method for constructing a class of asymmetric quantum codes (AQC), called additive AQC, has been established by Aly et.al. In this paper, we present a new characterization of AQC, which generalizes a result of the symmetric case known previously. As an application of the characterization, we establish a relationship of AQC with classical error-correcting codes and show a few examples of good AQC with specific parameters. By using this relationship, we obtain an asymptotic bound on AQCs from algebraic geometry codes.

Maximal values of generalized algebraic immunity

The notion of algebraic immunity of Boolean functions has been generalized in several ways to vector-valued functions and/or over arbitrary finite fields and reasonable upper bounds for such generalized algebraic immunities has been proved in Armknecht and Krause (Proceedings of ICALP 2006, LNCS, vol. 4052, pp 180–191, 2006), Ars and Faugere (Algebraic immunity of functions over finite fields, INRIA, No report 5532, 2005) and Batten (Canteaut, Viswanathan (eds.) Progress in Cryptology—INDOCRYPT 2004, LNCS, vol. 3348, pp 84–91, 2004). In this paper we show that the upper bounds can be reached as the maximal values of algebraic immunities for most of generalizations by using properties of Reed–Muller codes.

Linear error-block codes

A linear error-block code is a natural generalization of the classical error-correcting code and has applications in experimental design, high-dimensional numerical integration and cryptography. This article formulates the concept of a linear error-block code and derives basic results for this kind of code by direct analogy to the classical case. Some problems for further research are raised.