Modular Addition modulo 2t is one of the most applicable operators in symmetric cryptography Therefore, investigating the properties of this operator has a significant role in design and analysis of symmetric ciphers. Algebraic properties of this operator have been studied for two operands in [1]. In this contribution, to obtain more accurate results in this area, we generalize some of the algebraic properties of this operator for r ³ 2 operands. More precisely, we consider the algebraic degree of the component Boolean functions of Modular Addition as a vectorial Boolean function and determine the number of terms and variables in these Boolean functions. After some theoretical analysis in special cases, we propose an efficient algorithm for finding the degree of these Boolean functions general case. Using this algorithm, the algebraic degree of the component Boolean functions for Modular Addition modulo 232, with three up to eight operands is calculated.

A cryptographic hash function maps an arbitrary length input to a fixed length output. These functions are used in many cryptographic applications such as digital signatures. They must be secure against collision, preimage and 2-preimage attacks. Rotational cryptanalysis is an approach to the analysis of ARX ciphers. The Hash functions Shabal and Cubehash, which are two candidates of the second round of the SHA-3 competition, have an ARX structure. They have been analyzed with respect to rotational cryptanalysis by Tabatabaei et al. In this paper we consider their analysis and present some observations. Our observations show that the results of Tabatabaei et al. ’ s cryptanalysis are not accurate. Then we present some new results about rotational cryptanalysis of Shabal and Cubehash. Thereafter we present some new results and show that rotational cryptanalysis is effective on a smaller number of rounds on Shabal and Cubehash Hash functions.

Hash functions have a very important role in network and telecommunication security. These functions play an important role in hashing a message which are widely used in cryptographic applications such as digital signatures, random number generator algorithms, authentication protocols, and so on. Rotational cryptanalysis is a relatively new attack that is part of a generic attack on hash functions and is effective on algorithms that have an ARX structure. In this paper, for the first time, we apply a rotational cryptanalysis and with the given assumption of the markov chain for the Modular Additions sequence employed in two algorithms Shabal and CubeHash, which are second-round candidates for the SHA-3 competition that use the ARX property in their structure. With the implementation of rotational cryptanalysis we arrived at the complexity of 2-3393. 58 for the entire 16+3-rounds Shabal algorithm and the complexity of 2-57. 6 for the en-tire 16-round CubeHash algorithm. According to the obtained results, it can be seen that due to the large number of Modular Additions with the given assumption of markov chain, the Shabal algorithm exhibits greater resistance to rotational cryptanalysis, compared to the CubeHash algorithm and is less likely to succeed.

Complementary pairs of symmetric $2$-designs are equivalent to coherent configurations of type $(2, 2; 2)$.D. G. Higman studied these coherent configurations and adjacency algebras of coherent configurations over a field of characteristic zero. These are always semisimple.  We investigate these algebras over fields of any characteristic prime and the structures.

Recently, it has been obtained a lot of results on hyperconvex space (see [1, 2, 3, 5]). In this paper we develop some of those results for Modular hyperconvex spaces.