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In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications. [ABSTRACT FROM AUTHOR]

In 2013, Tang, Carlet, and Tang [IEEE TIT 59(1): 653–664, 2013] presented two classes of Boolean functions. The functions in the first class are unbalanced and the functions in the second one are balanced. Both of those two classes of functions have high nonlinearity, high algebraic degree, optimal algebraic immunity, and high fast algebraic immunity. However, they are not 1-resilient which represents a drawback for their use as filter functions in stream ciphers. In this paper, we first propose a large family of 1-resilient Boolean functions having high lower bound on nonlinearity, optimal algebraic immunity, and optimal algebraic degree, that is, meeting the Siegenthaler bound. Most notably, we can mathematically prove that every function in $n$ variables belonging to this family has fast algebraic immunity no less than $n-6$ , which is the first time that an infinite family of 1-resilient functions with provably high fast algebraic immunity has been invented. Furthermore, we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1-resilient functions with (potentially) optimal algebraic immunity and potentially high fast algebraic immunity. [ABSTRACT FROM AUTHOR]

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ . [ABSTRACT FROM PUBLISHER]

In this paper, for balanced binary sequence pairs (BBSPs for short) of period $N\!\equiv\! 1(\bmod\, 4) $, the optimality of their cross-correlation is determined. Next, a lower bound on the maximal out-of-phase autocorrelation magnitude is established for BBSPs of period $N\!\equiv\! 1(\bmod\, 4) $ with optimal cross-correlation. Furthermore, BBSPs with optimal cross-correlation whose maximal out-of-phase autocorrelation magnitude achieves the new lower bound are constructed based on the cyclotomy of order 4. These proposed sequences have a periodic total squared correlation value meeting the Ganapathy-Pados-Karystinos bound, and are therefore an option for use in asynchronous CDMA systems. [ABSTRACT FROM PUBLISHER]

Because of the recent algebraic attacks, optimal algebraic immunity is now an absolutely necessary (but not sufficient) property for Boolean functions used in stream ciphers. In this paper, we firstly determine the concrete coefficients in the linear expression of the column vectors with respect to a given basis of the generator matrix of Reed-Muller code, which is an important tool for constructing Boolean functions with optimal algebraic immunity. Secondly, as applications of the determined coefficients, we provide simpler and direct proofs for two known constructions. Further, we construct new Boolean functions on odd variables with optimal algebraic immunity based on the generator matrix of Reed-Muller code. Most notably, the new constructed functions possess the highest nonlinearity among all the constructions based on the generator matrix of Reed-Muller code, although which is not as good as the nonlinearity of Carlet-Feng function. Besides, the ability of the new constructed functions to resist fast algebraic attacks is also checked for the variable $$n=11,13$$ and 15. [ABSTRACT FROM AUTHOR]

In this letter, two lower bounds on the maximum cross-correlation magnitude of balanced quaternary sequence pairs with (almost) optimal autocorrelation are presented. The maximum cross-correlation magnitude of a balanced quaternary sequence pair of period N with the maximum out-of-phase autocorrelation magnitude \sqrt{5}, where N=4f+1$ is a prime and f$ is an odd integer, is shown to be \sqrt{N}, achieving one of the new lower bounds. [ABSTRACT FROM PUBLISHER]

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, based on the knowledge of compositions of an integer, we present two new kinds of construction of rotation symmetric Boolean functions having optimal algebraic immunity on either odd variables or even variables. Our new functions are of much better nonlinearity than all the existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, the algebraic degree of our rotation symmetric Boolean functions are also high enough. [ABSTRACT FROM AUTHOR]

Identity-based signcryption is a primitive that combines the functions of identity-based encryption and identity-based signature. In this paper, we first attack two of the existing identity-based signcryption schemes which are claimed to be provably secure without random oracles. Then we construct a new identity-based signcryption scheme and proves its security without random oracles. [ABSTRACT FROM AUTHOR]

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander–Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided. [ABSTRACT FROM AUTHOR]

Boolean functions with high nonlinearity and good autocorrelation properties play an important role in the design of block ciphers and stream ciphers. In this paper, we give a method to construct balanced Boolean functions of n variables, where n ≥ 10 is an even integer, satisfying strict avalanche criterion (SAC), and with high algebraic degree. Compared with the known balanced Boolean functions with SAC property, the constructed functions possess the highest nonlinearity and the best global avalanche characteristics property. [ABSTRACT FROM AUTHOR]

In 2008, Cusick conjectured that certain elementary symmetric Boolean functions of the form \sigma2^{t+1l-1, 2^{t}} are the only nonlinear balanced ones, where t, l are any positive integers, and \sigman,d=\bigoplus1\leq i1 < \cdots < id\leq nxi1xi2\cdots xid for positive integers n, 1\leq d\leq n. In this paper, by analyzing the weight of \sigman,2^{t} and \sigman, d, we prove that wt(\sigman,d) < 2^{n-1} holds in most cases, and so does the conjecture. According to the remainder modulo 4, we also consider the weight of \sigman, d from two aspects: n\equiv 3(mod\;4) and n\not\equiv 3(mod\;4). In particular, our results not only cover the most known results, but also contain some new cases. Thus, we can reduce the conjecture to few remaining cases. We do not fully solve the conjecture, but we also consider the weight of \sigman, 2^{t+2^{s}} and also give some experimental results on it. [ABSTRACT FROM AUTHOR]

Inspired by the previous work of Tu and Deng, we propose two infinite classes of Boolean functions of 2k variables where k\geq 2. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity (as shown by a lower bound that we prove; as a byproduct we also prove a better lower bound on the nonlinearity of the Carlet–Feng function). Thanks to a combinatorial fact, first conjectured by the authors and later proved by Cohen and Flori, we are able to show that they both possess optimal algebraic immunity. It is also checked that, at least for numbers of variables n\leq 16, functions in both classes have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, both of them possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables (n\leq 38). Moreover, these values are very good. The infinite class of functions we propose in Construction 2 presents, among all currently known constructions, the best provable tradeoff between all the important cryptographic criteria. [ABSTRACT FROM AUTHOR]

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N, i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of x^N-1. As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences. [ABSTRACT FROM AUTHOR]

Sequences with optimal autocorrelation property are needed in certain communication systems and cryptography. In this paper, a construction of balanced quaternary sequences with period N \equiv 2 (mod\ 4) and optimal autocorrelation value and a construction of almost balanced binary sequences with period N \equiv 0 (mod\ 4) and optimal autocorrelation value are presented. Both constructions are a generalization of earlier ones. [ABSTRACT FROM AUTHOR]