Asymmetric Cryptographic Algorithms Research Articles

A knapsack-based probabilistic encryption scheme

Knapsack-based cryptosystems had been viewed as the most attractive and the most promising asymmetric cryptographic algorithms for a long time due to their NP-completeness nature and high speed in encryption/decryption. Unfortunately, most of them are broken for the low-density feature of the underlying knapsack problems. In this paper, we investigate a new easy compact knapsack problem and propose a novel knapsack-based probabilistic public-key cryptosystem in which the cipher-text is non-linear with the plaintext. For properly chosen parameters, the underlying knapsack problem enjoys a high density larger than 1.06 in the worst case. Hence, it is secure against the low-density subset-sum attacks. Our scheme can also defeat other potential attacks such as the brute force attacks and the simultaneous Diophantine approximation attack. Compared with previous knapsack-based cryptosystems, our scheme is efficient and practical.

On linguistic dynamical systems, families of graphs of large girth, and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f α : K n → K n depending on “time” α ∈ {K − 0} such that f α −1 = f −αM, the relation f α1 (x) = f α2 (x) for some x ∈ Kn implies α1 = α2, and each map f α has no invariant points. The neighborhood {f α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ a = f α1 × ⋯ × f αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.

Asymmetric Cryptographic Algorithms

The concept of asymmetric cryptographic algorithms introduced by W. Diffie and M. Hellman in 1976 is considered. The problems of factorization of large integers and finding discrete logarithms for elements of finite large-order groups are presented. It is proved that asymmetric cryptographic algorithms based on the problem of finding a discrete logarithm for points of an elliptic curve over a finite field should be used in modern information technology.

Optimized software implementations of the modular exponentiation on general purpose microprocessors

In the last few years asymmetric cryptographic algorithms have become more and more popular. They can be used for encipherment, digital signatures, authentication, key management and key distribution and for other security mechanisms. Almost all of these algorithms use the modular exponentiation of large numbers and the effective performance of the arithmetical routines is of great importance. Besides hardware solutions or solutions on special processors, such as signal processors or transputers, software implementations on general purpose microprocessors are also interesting if these are permitted or even required by the application. In this paper techniques for the software implementation of modular exponentiation are presented and analysed. It will be shown that the arithmetic routines chosen can only reach an optimal performance when the processor used and the key length are taken into consideration.