Double-Moduli Gaussian Encryption/Decryption with Primary Residues and Secret Controls

In this paper an encryption-decryption algorithm based on two moduli is described: one in the real field of integers and another in the field of complex integers. Also the proper selection of cryptographic system parameters is described. Several numeric illustrations explain step-by-step how to precondition a plaintext, how to select secret control parameters, how to ensure feasibility of all private keys and how to avoid ambiguity in the process of information recovery. The proposed cryptographic system is faster than most of known public key cryptosystems, since it requires a small number of multiplications and additions, and does not require exponentiations for its implementation.

Protection of Sensitive Messages Based on Quadratic Roots of Gaussians: Groups with Complex Modulus

This paper considers three algorithms for the extraction of square roots of complex integers {called Gaussians} using arithmetic based on complex modulus p + iq. These algorithms are almost twice as fast as the analogous algorithms extracting square roots of either real or complex integers in arithmetic based on modulus p, where is a real prime. A cryptographic system based on these algorithms is provided in this paper. A procedure reducing the computational complexity is described as well. Main results are explained in several numeric illustrations.

Parallel Minimax Searching Algorithm for Extremum of Unimodal Unbounded Function

In this paper we consider a parallel algorithm that detects the maximizer of unimodal function f(x) computable at every point on unbounded interval (0, ∞). The algorithm consists of two modes: scanning and detecting. Search diagrams are introduced as a way to describe parallel searching algorithms on unbounded intervals. Dynamic programming equations, combined with a series of liner programming problems, describe relations between results for every pair of successive evaluations of function f in parallel. Properties of optimal search strategies are derived from these equations. The worst-case complexity analysis shows that, if the maximizer is located on a priori unknown interval (n-1], then it can be detected after cp(n)=「2log「p/2」+1(n+1)」-1 parallel evaluations of f(x), where p is the number of processors.

Space Complexity of Algorithm for Modular Multiplicative Inverse

In certain computational systems the amount of space required to execute an algorithm is even more restrictive than the corresponding time necessary for solution of a problem. In this paper an algorithm for modular multiplicative inverse is introduced and its computational space complexity is analyzed. A tight upper bound for bit storage required for execution of the algorithm is provided. It is demonstrated that for range of numbers used in public-key encryption systems, the size of bit storage does not exceed a 2K-bit threshold in the worst-case. This feature of the Enhanced-Euclid algorithm allows designing special-purpose hardware for its implementation as a subroutine in communication-secure wireless devices.

Information Protection Based on Extraction of Square Roots of Gaussian Integers

A cryptosystem based on computation of square roots of complex integers modulo composite n is described in this paper. This paper provides an algorithm extracting a square root of Gaussian integer. Various properties of square roots and a method for finding Gaussian generators are demonstrated. The generators can be instrumental in constructing other cryptosystems. It is shown how to significantly reduce average complexity of decryption per each block of ciphertext.

Scheme for Secure Communication via Information Hiding Based on Key Exchange and Decomposition Protocols

This paper considers a decomposition framework as a mechanism for information hiding for secure communication via open network channels. Two varieties of this framework are provided: one is based on Gaussian arithmetic with complex modulus and another on an elliptic curve modular equation. The proposed algorithm is illustrated in a numerical example.

Computation of Complex Primes Using Elliptic Curves: Application for Cryptosystem Design

This paper provides several generalizations of Gauss theorem that counts points on special elliptic curves. It is demonstrated how to implement these generalizations for computation of complex primes, which are applicable in several protocols providing security in communication networks. Numerical examples illustrate the ideas discussed in this paper.

Corporate Intranet Security: Packet-Level Protocols for Preventing Leakage of Sensitive Information and Assuring Authorized Network Traffic

Securing large corporate communication networks has become an increasingly difficult task. Sensitive information routinely leaves the company network boundaries and falls into the hands of unauthorized users. New techniques are required in order to classify packets based on user identity in addition to the traditional source and destination host addresses. This paper introduces Gaussian cryptographic techniques and protocols to assist network administrators in the complex task of identifying the originators of data packets on a network and more easily policing their behavior. The paper provides numerical examples that illustrate certain basic ideas.

Integer Factorization of Semi-Primes Based on Analysis of a Sequence of Modular Elliptic Equations

In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.

Algorithms for Integer Factorization Based on Counting Solutions of Various Modular Equations

This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.

Large-Integer Multiplication Based on Homogeneous Polynomials

Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.

Astronomical Algorithms: Amended Multi-Millennia Calendar

Three new worldwide calendars are proposed and compared in this paper. None of them requires any departure from an existing tradition to divide years on lean and leap. Although all three are pretty accurate, it is demonstrated that the Julian calendar with one additional amendment is the simplest and the most suitable for implementation.

Enhanced Euclid Algorithm for Modular Multiplicative Inverse and Its Application in Cryptographic Protocols

Numerous cryptographic algorithms (ElGamal, Rabin, RSA, NTRU etc) require multiple computations of modulo multiplicative inverses. This paper describes and validates a new algorithm, called the Enhanced Euclid Algorithm, for modular multiplicative inverse (MMI). Analysis of the proposed algorithm shows that it is more efficient than the Extended Euclid algorithm (XEA). In addition, if a MMI does not exist, then it is not necessary to use the Backtracking procedure in the proposed algorithm;this case requires fewer operations on every step (divisions, multiplications, additions, assignments and push operations on stack), than the XEA. Overall, XEA uses more multiplications, additions, assignments and twice as many variables than the proposed algorithm.

Service Networks Topological Design

Topological design of service networks is studied in the paper. Quantitative model and algorithm minimizing cost of processing and delivery is described. Algorithm solving combinatorial problem of optimal design based on binary partitioning, a parametric search and dynamic programming optimization of a binary tree is described and demonstrated in numeric example.

Potential Vulnerability of Encrypted Messages: Decomposability of Discrete Logarithm Problems

This paper provides a framework that reduces the computational complexity of the discrete logarithm problem. The paper describes how to decompose the initial DLP onto several DLPs of smaller dimensions. Decomposability of the DLP is an indicator of potential vulnerability of encrypted messages transmitted via open channels of the Internet or within corporate networks. Several numerical examples illustrate the frame- work and show its computational efficiency.

Winning Strategies and Complexity of Nim-Type Computer Game on Plane

A Nim-type computer game of strategy on plane is described in this paper. It is demonstrated that winning strategies of this two-person game are determined by a system of equations with two unknown integer sequences. Properties of winning points/states are discussed and an O(loglogn) algorithm for the winning states is provided. Two varieties of the Game are also introduced and their winning strategies are analyzed.

Hybrid Authentication Cybersystem Based on Discrete Logarithm, Factorization and Array Entanglements

A hybrid cryptographic system providing digital authentication is described and analyzed in this paper. The proposed cryptosystem incorporates three features: complexity of the discrete logarithm problem, complexity of integer factorization of a product of two large primes and a combination of symmetric and asymmetric keys. In order to make the cryptosystem less vulnerable to cryptanalytic attacks a concept of digital entanglements is introduced. As a result, the proposed cryptographic system has four layers (entanglement-encryption- decryption-disentanglement). It is shown that in certain instances the proposed communication cryptocol is many times faster than the RSA cryptosystem. Examples provided in the paper illustrate details of the proposed authentication cryptocol.