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It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

Commutative and non commutative Grobner–Shirshov bases were first studied over fields and after extended to some particular rings. In theses works, the monomials are in a monoid. Recently, Bokut and al. gave a new extension of Grobner–Shirshov bases over a field by choosing the monomials in a semi-ring rather in a monoid. In this paper, we study Grobner–Shirshov bases where the monomials are in a semi-ring and the coefficients are in a noetherian valuation ring and we establish the relation between weak and strong Grobner bases.

Feedback Shift Register (FSR) is generally the basic element of pseudo random generators used to generate cryptographic channel or set of sequences for encryption keys. This type of generator is widely used in stream cipher and communication systems such as C.D.M.A (Code Division Multiple Access), mobile communication systems, ranging and navigating systems, spread spectrum communication systems. The objective of the present paper is to propose a method for determining linear recurring sequences generating linear feedback shift register (LFSR) from primitive polynomials (and vice-versa). The linear recurring sequences facilitate the construction of maximum length LFSR. It also insists, in the last part, on the cryptographic security of LFSR and indicates some open problems in the area of nonlinear feedback shift registers (NLFSR) based pseudo random generators.

In this work, we propose a method for computing noncommutative Grobner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Grobner bases is generalized to Buchberger's algorithm over R = 𝒱 ⟨ x 1,…, x m ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ. The proposed process generalizes previous known techniques for the computation of commutative Grobner bases over a naetherian valuation ring and/or noncommutative Grobner bases over a field.

This paper introduces a new public key cryptosystem based on two hard problems : the cube root extraction modulo a composite moduli (which is equivalent to the factorisation of the moduli) and the discrete logarithm problem. These two hard problems are combined during the key generation, encryption and decryption phases. By combining the IFP and the DLP we introduce a secure and efficient public key cryptosystem. To break the scheme, an adversary may solve the IFP and the DLP separately which is computationally infeasible. The key generation is a simple operation based on the discrete logarithm modulo a composite moduli. The encryption phase is based both on the cube root computation and the DLP. These operations are computationally efficient., Something was not correct in the paper

Copyright c � 2013 Abdoul Aziz Ciss and Djiby Sow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linear recurrences sequences are widely studied over fields rings and modules. In this paper, we introduce the notion of linear recurrence sequence over semirings. After investigating some basics properties, we give a characterization of linear recurrence sequence over zero-sum semirings.

Let R be a ring with identity. A right R-module M has the com- plete max-property if the maximal submodules of M are completely coindepen- dent (i.e., every maximal submodule of M does not contain the intersection of the other maximal submodules of M). A right R-module is said to be a good module provided every proper submodule of M containing Rad(M) is an intersection of maximal submodules of M. We obtain a new characterization of good modules. Also, we study good modules which have the complete max- property. The second part of this paper is devoted to investigate supplements in a coatomic module which has the complete max-property.

In this paper, we do a cryptanalyse of the so called ``Strong Diffie-Hellman-DSA Key Exchange (briefly: SDH-DSA-KE)" and after we propose ``Strong Diffie-Hellman-Exponential-Schnnor Key Exchange (briefly: SDH-XS-KE)" which is an improvement for efficiency and security. SDH-XS-KE protocol is secure against Session State Reveal (SSR) attacks, Key independency attacks, Unknown-key share (UKS) attacks and Key-Compromise Impersonation (KCI) attacks. Furthermore, SDH-XS-KE has Perfect Forward Secrecy (PFS) property and a key confirmation step. The new proposition is not vulnerable to Disclosure to ephemeral or long-term Diffie-Hellman exponents. We design our protocol in finite groups therefore this protocol can be implemented in elliptic curves.

In our paper paper we propose a new binary elliptic curve of the form $a[x^2+y^2+xy+1]+(a+b)[x^2y+y^2x]=0$. If $m\geq 5$ we prove that each ordinary elliptic curve $y^{2}+xy=x^{3}+\alpha x^2+\beta, \beta\neq 0$ over $\mathbb{F}_{2^m}$, is birationally equivalent over $\mathbb{F}_{2^m}$ to our curve. This paper also presents the formulas for the group law.

A deterministic extractor for an elliptic curve, that converts a uniformly random point on the curve to a random k-bit-string with a distribution close to uniform, is an important tool in cryptography. Such extractors can be used for example in key derivation functions, in key exchange protocols and to design cryptographically secure pseudorandom number generator. In this paper, we present a simple and efficient deterministic extractor for an elliptic curve E defined over Fq,n, where q is prime and n is a positive integer. Our extractor, denoted by Dk, for a given random point P on E, outputs the k-first Fq-coordinates of the abscissa of the point P. This extractor confirms the two conjectures stated by R. R. Farashahi and R. Pellikaan in and by R. R. Farashahi, A. Sidorenko and R. Pellikaan in, related to the extraction of bits from coordinates of a point of an elliptic curve.

A deterministic extractor for an elliptic curve, that converts a uniformly random point on the curve to a random bit-string with a uniform distribution, is an important tool in cryptography. Such extractors can be used for example in key derivation functions, in key exchange protocols and to design cryptographically secure pseudorandom number generator. In this paper, we present a simple and efficient deterministic extractor for an elliptic curve E defined over a non prime finite field. Our extractor, for a given random point P on the curve, outputs the k-first coefficients of the abscissa of the point P. This extractor confirms the two conjectures stated by Farashahi and Pellikaan (2007) and Farashahi et al. (2008), related to the extraction of bits from coordinates of a point of an elliptic curve. As applications of our extractor, we show under the decisional Diffie-Hellman problem on an elliptic curve defined over a finite field of characteristic two, that the k-first or the k-last bits of the abscissa of a random point on the curve are indistinguishable from a random bit-string of the same length.