Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms compared with the linear block codes. The objective of this letter is to present a family of p-ary cyclic codes with length $rac{p^m-1}{p-1}$ and dimension $rac{p^m-1}{p-1}-2m$, where p is an arbitrary odd prime and m is a positive integer with gcd(p-1,m)=1. The minimal distance d of the proposed cyclic codes are shown to be 4≤d≤5 which is at least almost optimal with respect to some upper bounds on the linear code.

Semi-bent functions have very high nonlinearity and hence they have many applications in symmetric-key cryptography, binary sequence design for communications, and combinatorics. In this paper, we focus on studying the additive autocorrelation of semi-bent functions. We provide a lower bound on the maximum additive autocorrelation absolute value of semi-bent functions with three-level additive autocorrelation. Semi-bent functions with three-level additive autocorrelation achieving this bound with equality are said to have perfect three-level additive autocorrelation. We present two classes of balanced semi-bent functions with optimal algebraic degree and perfect three-level additive autocorrelation.

Dynamic linear feedback shift registers (DLFSRs) are a scheme to transfer from one LFSR to another. In cryptography each LFSR included in a DLFSR should generate maximal-length sequences, and the number of switches transferring LFSRs should be small for efficient performance. This corresponding addresses on searching such conditioned DLFSRs. An efficient probabilistic algorithm is given to find such DLFSRs with two or four switches, and it is proved to succeed with nonnegligible probability.

In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.

Let f be a Boolean function in n variables. The Möbius transform and its converse of f can describe the transformation behaviors between the truth table of f and the coefficients of the monomials in the algebraic normal form representation of f. In this letter, we develop the Möbius transform and its converse into a more generalized form, which also includes the known result given by Reed in 1954. We hope that our new result can be used in the design of decoding schemes for linear codes and the cryptanalysis for symmetric cryptography. We also apply our new result to verify the basic idea of the cube attack in a very simple way, in which the cube attack is a powerful technique on the cryptanalysis for symmetric cryptography.

In recent years, algebraic attacks and fast algebraic attacks have received a lot of attention in the cryptographic community. There are three Boolean functions achieving optimal algebraic immunity based on primitive element of F2n. The support of Boolean functions in [1]-[3] have the same parameter s, which makes us have a large number of Boolean functions with good properties. However, we prove that the Boolean functions are affine equivalence when s takes different values.

Boolean functions used in the filter model of stream ciphers should have balancedness, large nonlinearity, optimal algebraic immunity and high algebraic degree. Besides, one more criterion called strict avalanche criterion (SAC) can be also considered. During the last fifteen years, much work has been done to construct balanced Boolean functions with optimal algebraic immunity. However, none of them has the SAC property. In this paper, we first present a construction of balanced Boolean functions with SAC property by a slight modification of a known method for constructing Boolean functions with SAC property and consider the cryptographic properties of the constructed functions. Then we propose an infinite class of balanced functions with optimal algebraic immunity and SAC property in odd number of variables. This is the first time that such kind of functions have been constructed. The algebraic degree and nonlinearity of the functions in this class are also determined.

To resist algebraic and fast algebraic attacks, Boolean functions used in stream ciphers should have optimal algebraic immunity and good fast algebraic immunity. One challenge of cryptographic Boolean functions is to determine their ability to resist fast algebraic attacks, which can be measured by their fast algebraic immunities. In this letter, we determine the exact values of fast algebraic immunity of the majority function of 2m and 2m+1 variables. This is the first time that the exact values of the fast algebraic immunity of an infinite class of symmetric Boolean functions with optimal algebraic immunity are determined.

Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.