Merkle-Hellman Research Articles

A Comparison between Memetic algorithm and Genetic algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm

Genetic algorithms are a population-based Meta heuristics. They have been successfully applied to many optimization problems. However, premature convergence is an inherent characteristic of such classical genetic algorithms that makes them incapable of searching numerous solutions of the problem domain. A memetic algorithm is an extension of the traditional genetic algorithm. It uses a local search technique to reduce the likelihood of the premature convergence. The cryptanalysis of simplified data encryption standard can be formulated as NP-Hard combinatorial problem. In this paper, a comparison between memetic algorithm and genetic algorithm were made in order to investigate the performance for the cryptanalysis on simplified data encryption standard problems(SDES). The methods were tested and various experimental results show that memetic algorithm performs better than the genetic algorithms for such type of NP-Hard combinatorial problem. This paper represents our first effort toward efficient memetic algorithm for the cryptanalysis of SDES. This paper proposes the cryptanalysis of simplified encryption standard algorithm using memetic and genetic algorithm. The cryptanalysis of simplified data encryption standard can be formulated as NP-Hard combinatorial problem. Solving such problems requires effort (e.g., time and/or memory requirement) which increases with the size of the problem. Techniques for solving combinatorial problems fall into two broad groups - traditional optimization techniques (exact algorithms) and non traditional optimization techniques (approximate algorithms). A traditional optimization technique guarantees that the optimal solution to the problem will be found. The traditional optimization techniques like branch and bound, simplex method, brute force search algorithm etc methodology is very inefficient for solving combinatorial problem because of their prohibitive complexity (time and memory requirement). Non traditional optimization techniques are employed in an attempt to find an adequate solution to the problem. A non traditional optimization technique - memetic algorithm, genetic algorithm, simulated annealing and tabu search were developed to provide a robust and efficient methodology for cryptanalysis. The aim of these techniques to find sufficient solution efficiently with the characteristics of the problem, instead of the global optimum solution, and thus it also provides attractive alternative for the large scale applications. These nontraditional optimization techniques demonstrate good potential when applied in the field of cryptanalysis and few relevant studies have been recently reported. In 1993 Spillman (16) for the first time presented a genetic algorithm approach for the cryptanalysis of substitution cipher using genetic algorithm. He has explored the possibility of random type search to discover the key (or key space) for a simple substitution cipher. In the same year Mathew (12) used an order based genetic algorithm for cryptanalysis of a transposition cipher. In 1993, Spillman (17), also successfully applied a genetic algorithm approach for the cryptanalysts of a knapsack cipher. It is based on the application of a directed random search algorithm called a genetic algorithm. It is shown that such a algorithm could be used to easily compromise even high density knapsack ciphers. In 1997 Kolodziejczyk (11) presented the application of genetic algorithm in cryptanalysis of knapsack cipher .In 1999 Yaseen (18) presented a genetic algorithm for the cryptanalysis of Chor-Rivest knapsack public key cryptosystem.

COMPACT GENETIC ALGORITHM FOR CRYPTANALYSIS TRAPDOOR 0-1 KNAPSACK CIPHER

Security is a broad topic and covers a multitude of sins. In its simplest form, it is concerned with people trying to access remote services that they are not authorized to use. Cryptology is the science and studies of systems for secrete communication. It consists of two complementary fields of study: Cryptography and Cryptanalysis. The application of genetic algorithm in cryptanalysis of knapsack cipher is suggested by Spillman. This paper considers a new approach to cryptanalysis knapsack cipher based on the representation of the population as a probability distribution over the set of solutions; this is called compact Genetic Algorithm (cGA). Tests have been presented to clarify the results obtained. The results show that cGA achieve the breaking of the ciphertext. Moreover, the comparison among Spillman results, simple GA (sGA) and our results are also provided. The results show that cGA is worth to be considered for the attack of trapdoor 0-1 knapsack cipher.

COMBINATORIAL OPTIMIZATION AND THE KNAPSACK CIPHER

Recent suggestions in [8] that optimization techniques such as the genetic algorithm can be used to successfully solve knapsack ciphers are somewhat optimistic. The inability to assign an appropriate fitness to an arbitrary solution of the knapsack cipher is the downfall with this method. In this paper a detailed analysis of the proposed fitness function is undertaken and numerical results are presented displaying the futility of using this fitness function in a genetic algorithm for solving knapsack ciphers of any reasonable size.

CRYPTANALYSIS OF KNAPSACK CIPHERS USING GENETIC ALGORITHMS

This paper is the second in a series of papers which examine a new method of cryptanalysis. This work focuses on the cryptanalysis of a knapsack cipher. It is based on the application of a directed random search algorithm called a genetic algorithm. It is shown that such a algorithm could be used to easily compromise even high density knapsack ciphers.

Elementary Number Theory and Its Applications.

P. What is Number Theory? 1. The Integers. Numbers and Sequences. Sums and Products. Mathematical Induction. The Fibonacci Numbers. 2. Integer Representations and Operations. Representations of Integers. Computer Operations with Integers. Complexity of Integer Operations. 3. Primes and Greatest Common Divisors. Prime Numbers. The Distribution of Primes. Greatest Common Divisors. The Euclidean Algorithm. The Fundemental Theorem of Arithmetic. Factorization Methods and Fermat Numbers. Linear Diophantine Equations. 4. Congruences. Introduction to Congruences. Linear Congrences. The Chinese Remainder Theorem. Solving Polynomial Congruences. Systems of Linear Congruences. Factoring Using the Pollard Rho Method. 5. Applications of Congruences. Divisibility Tests. The perpetual Calendar. Round Robin Tournaments. Hashing Functions. Check Digits. 6. Some Special Congruences. Wilson's Theorem and Fermat's Little Theorem. Pseudoprimes. Euler's Theorem. 7. Multiplicative Functions. The Euler Phi-Function. The Sum and Number of Divisors. Perfect Numbers and Mersenne Primes. Mobius Inversion. 8. Cryptology. Character Ciphers. Block and Stream Ciphers. Exponentiation Ciphers. Knapsack Ciphers. Cryptographic Protocols and Applications. 9. Primitive Roots. The Order of an Integer and Primitive Roots. Primitive Roots for Primes. The Existence of Primitive Roots. Index Arithmetic. Primality Tests Using Orders of Integers and Primitive Roots. Universal Exponents. 10. Applications of Primitive Roots and the Order of an Integer. Pseudorandom Numbers. The EIGamal Cryptosystem. An Application to the Splicing of Telephone Cables. 11. Quadratic Residues. Quadratic Residues and nonresidues. The Law of Quadratic Reciprocity. The Jacobi Symbol. Euler Pseudoprimes. Zero-Knowledge Proofs. 12. Decimal Fractions and Continued. Decimal Fractions. Finite Continued Fractions. Infinite Continued Fractions. Periodic Continued Fractions. Factoring Using Continued Fractions. 13. Some Nonlinear Diophantine Equations. Pythagorean Triples. Fermat's Last Theorem. Sums of Squares. Pell's Equation. 14. The Gaussian Integers. Gaussian Primes. Unique Factorization of Gaussian Integers. Gaussian Integers and Sums of Squares.

CIPHERTEXT ONLY ATTACK ON THE MERKLE-HELLMAN PUBLIC-KEY SYSTEM UNDER BROADCAST SITUATIONS

A method is presented by which a ciphertext only attack can be mounted against the Merkle-Hellman (MH) system based on trapdoor knapsacks if a number of copies of the same message are broadcast in different keys.

Fast decryption algorithm for the knapsack cipher

A fast decryption algorithm is described which permits use of the knapsack cipher (a public-key cryptosystem) at data rates in the neighborhood of 10 Mbit/sec. This high-speed capability can be used to incorporate the security and flexibility of public-key cryptosystems into a wide variety of real-time communications applications. Implementation of the algorithm using Very Large Scale Integration appears attractive: The circuit functions required are approximately 56 kilobits of memory and a small amount of arithmetic logic.