In the field of molecular computing, in particular P systems, synchronization is an essential requirement for composing or sequentially linking together congenial P system activities. We provide an improved deterministic algorithm based on static structures and traditional rules, which runs in 4e + 13 steps, where e is the eccentricity of the initiating cell. Using the same model, extended with...

This paper describes the design of a fast software library for the computation of the optimal ate pairing on a Barreto–Naehrig elliptic curve. Our library is able to compute the optimal ate pairing over a 254-bit prime field Fp, in just 2.33 million of clock cycles on a single core of an Intel Core i7 2.8GHz processor, which implies that the pairing computation takes 0.832msec. We are able to a...

In this paper we introduce a new form of decomposition of graphs, the (P, Q)-decomposition. We first give an optimal algorithm for finding the 1-decomposition of a graph which is a special case of the (P, Q)-decomposition which was first introduced in [21]. We then examine the connections between the 1-decomposition and well known forms of decomposition of graphs, namely, modular and homogeneou...

graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution ι to projective space, such that F is a projective embedding of X(4p)/ι aw...

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution ι to projective space, such that F is a projective embedding of X(4p)/ι aw...

Let F be a totally real field, p ≥ 3 a rational prime unramified in F , and p a place of F over p. Let ρ : Gal(F/F ) → GL2(Fp) be a two-dimensional mod p Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at p is irreducible. We specify a set of weights, determined by the restriction of ρ to inertia at p, which contains all the ...

We propose two hardware architectures for the modular multiplication over a finite field F p using Adapted Modular Number System (AMNS). present their optimized implementations, with low latency.

We will determine the structure of the modular standard modules of association schemes of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than th...

We first propose a modular framework for recursive composition of P systems. This modular approach provides encapsulation and information hiding, facilitating the design of P programs for complex algorithms. Using this framework, we developed a P program that solves the classical version of the Byzantine agreement problem, for N participants connected in a complete graph, according to the well ...