In this paper, we give some results towards the conjecture that σ2t+1l-1,2t are the only nonlinear balanced elementary symmetric Boolean functions where t and l are positive integers. At first, a unified and simple proof of some earlier results is shown. Then a property of balanced elementary symmetric Boolean functions is presented. With this property, we prove that the conjecture is true for n=2m+2t-1 where m,t (m>t) are two non-negative integers, which verified the conjecture for a large infinite class of integer n.

In this paper, we investigate the security property of RSA when some middle bits of the private key d are known to an attacker. Using the technique of unravelled linearization, we present a new attack on RSA with known middle bits, which improves a previous result under certain circumstance. Our approach is based on Coppersmith's method for finding small roots of modular polynomial equations.

In this paper, by using the properties of the cyclic Hadamard matrices of order 4t, an infinite class of (4t-1)-variable 2-resilient rotation symmetric Boolean functions is constructed, and the nonlinearity of the constructed functions are also studied. To the best of our knowledge, this is the first class of direct constructions of 2-resilient rotation symmetric Boolean functions. The spirit of this method is different from the known methods depending on the solutions of an equation system proposed by Du Jiao, et al. Several situations are examined, as the direct corollaries, three classes of (4t-1)-variable 2-resilient rotation symmetric Boolean functions are proposed based on the corresponding sequences, such as m sequences, Legendre sequences, and twin primes sequences respectively.

In this paper, we study partial key exposure attacks on RSA where the number of unexposed blocks of the private key is greater than or equal to one. This situation, called generalized framework of partial key exposure attack, was first shown by Sarkar [22] in 2011. Under a certain condition for the values of exposed bits, we present a new attack which needs fewer exposed bits and thus improves the result in [22]. Our work is a generalization of [28], and the approach is based on Coppersmith's method and the technique of unravelled linearization.

At FSE 2014, Grosso et al. proposed LS-designs which are a family of bitslice ciphers aiming at efficient masked implementations against side-channel analysis. They also presented two specific LS-designs, namely the non-involutive cipher Fantomas and the involutive cipher Robin. The designers claimed that the longest impossible differentials of these two ciphers only span 3 rounds. In this paper, for the two ciphers, we construct 4-round impossible differentials which are one round more than the longest impossible differentials found by the designers. Furthermore, with the 4-round impossible differentials, we propose impossible differential attacks on Fantomas and Robin reduced to 6 rounds (out of the full 12/16 rounds). Both of the attacks need 2119 chosen plaintexts and 2101.81 6-round encryptions.

In this paper, we study the problem of a Boolean function can be represented as the sum of two bent functions. This problem was recently presented by N. Tokareva when studying the number of bent functions [27]. Firstly, several classes of functions, such as quadratic Boolean functions, Maiorana-MacFarland bent functions, many partial spread functions etc, are proved to be able to be represented as the sum of two bent functions. Secondly, methods to construct such functions from low dimension ones are also introduced. N. Tokareva's main hypothesis is proved for n≤6. Moreover, two hypotheses which are equivalent to N. Tokareva's main hypothesis are presented. These hypotheses may lead to new ideas or methods to solve this problem. Finally, necessary and sufficient conditions on the problem when the sum of several bent functions is again a bent function are given.

MDS transformation plays an important role in resisting against differential cryptanalysis (DC) and linear cryptanalysis (LC). Recently, M. Sajadieh, et al.[15] designed an efficient recursive diffusion layer with Feistel-like structures. Moreover, they obtained an MDS transformation which is related to a linear function and the inverse is as lightweight as itself. Based on this work, we consider one specific form of linear functions to get the diffusion layer with low XOR gates for the hardware implementation by using temporary registers. We give two criteria to reduce the construction space and obtain six new classes of lightweight MDS transformations. Some of our constructions with one bundle-based LFSRs have as low XOR gates as previous best known results. We expect that these results may supply more choices for the design of MDS transformations in the (lightweight) block cipher algorithm.

(m+k,m)-functions with good cryptographic properties when 1≤k

Because of the algebraic attacks, a high algebraic immunity is now an important criteria for Boolean functions used in stream ciphers. In 2011, X.Y. Zeng et al. proposed three constructions of balanced Boolean functions with maximum algebraic immunity, the constructions are based on univariate polynomial representation of Boolean functions. In this paper, we will improve X.Y. Zeng et al.' constructions to obtain more even-variable Boolean functions with maximum algebraic immunity. It is checked that, our new functions can have as high nonlinearity as X.Y. Zeng et al.' functions.

In this comment, an inequality of algebraic immunity of the sum of two Boolean functions is pointed out to be generally incorrect. Then we present some results on how to impose conditions such that the inequality is true. Finally, complete proofs of two existing results are given.

Constructing degree-optimized resilient Boolean functions with high nonlinearity is a significant study area in Boolean functions. In this letter, we provide a construction of degree-optimized n-variable (n odd and n ≥ 35) resilient Boolean functions, and it is shown that the resultant functions achieve the currently best known nonlinearity.

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, odd-variable balanced RSBFs with maximum algebraic immunity (AI) are investigated. We provide a construction of n-variable (n=2k+1 odd and n ≥ 13) RSBFs with maximum AI and nonlinearity ≥ 2n-1-¥binom{n-1}{k}+2k+2k-2-k, which have nonlinearities significantly higher than the previous nonlinearity of RSBFs with maximum AI.