Abstract:
Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2 with addition and multiplication being effectively computable relative to the computability of their own spaces. Here we emphasize the algorithmic notions of the Clifford algebras. We recall the reduction of Clifford algebras into isomorphic structures also suitable for symbolic manipulation.

Abstract:
Given a real objective function defined over the symmetric group, a direct local-search algorithm is proposed, and its complexity is estimated. In particular for an $n$-dimensional unit vector we are interested in the permutation isometry that acts on this vector by mapping it into a cone of a given angle.

Abstract:
By considering probability distributions over the set of assignments the expected truth values assignment to propositional variables are extended through linear operators, and the expected truth values of the clauses at any given conjunctive form are also extended through linear maps. The probabilistic satisfiability problems are discussed in terms of the introduced linear extensions. The case of multiple truth values is also discussed.

Abstract:
The Clifford algebra over the three-dimensional real linear space includes its linear structure and its exterior algebra, the subspaces spanned by multivectors of the same degree determine a gradation of the Clifford algebra. Through these geometric notions, natural one-to-one and two-to-one homomorphisms from $\mbox{SO}(3)$ into $\mbox{SU}(2)$ are built conventionally, and the set of qubits, is identified with a subgroup of $\mbox{SU}(2)$. These constructions are suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilber spaces. realise a notion of entanglement in the tensor product of symmetry groups.

Abstract:
Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge $\edg{v_1}{v_2}$ is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs.

Abstract:
modular arithmetic with prime moduli has been crucial in present day cryptography. the primes of mersenne, solinas, crandall and the so called ike-modp primes have been widely used in efficient implementations. in this paper we study the density of primes with binary signed representation involving a small number of non-zero ±1-digits, and its repercussion in the generation of those primes.

Abstract:
We develop an approach of key distribution protocol(KDP) proposed recently by T.Aono et al., where the security of KDP is only partly estimated in terms of eavesdropper's key bit errors. Instead we calculate the Shannon's information leaking to a wire tapper and also we apply the privacy amplification procedure from the side of the legal users. A more general mathematical model based on the use of Variable-Directional Antenna (VDA) under the condition of multipath wave propagation is proposed. The new method can effectively be used even in noiseless interception channels providing thus a widened area with respect to practical applications. Statistical characteristics of the VDA are investigated by simulation, allowing to specify the model parameters. We prove that the proposed KDP provides both security and reliability of the shared keys even for very short distances between legal users and eavesdroppers. Antenna diversity is proposed as a mean to enhance the KDP security. In order to provide a better performance evaluation of the KDP, it is investigated the use of error correcting codes.

Abstract:
We develop an approach of key distribution protocol (KDP) proposed recently by T. Aono et al. A more general mathematical model based on the use of Variable-Directional Antenna (VDA) under the condition of multipath wave propagation is proposed. Statistical characteristics of VDA were investigated by simulation, that allows us to specify model parameters. The security of the considered KDP is estimated in terms of Shannon's information leaking to an eavesdropper depending on the mutual locations of the legal users and the eavesdropper. Antenna diversity is proposed as a mean to enhance the KDP security. In order to provide a better agreement of the shared keys it is investigated the use of error-correcting codes.