Knapsack Public Key Cryptosystem Research Articles

Analysis of two knapsack public key cryptosystems

Two knapsack-based public key cryptosystems were proposed recently, in which the entries of the secret knapsack sequences are composed of products of random integers and the knapsack problems have nonbinary solutions. These features make the cryptosystems to be secure against low density attacks. In this study, the authors present lattice-based complete private key recovery attacks on these two schemes. The authors attacks firstly find short vectors related to the public key and with some orthogonality to the secret knapsack sequences and then recover some components of the private key by computing common factors of the entries of the short vectors. Especially, the authors attack can both completely recover the unique key for these two schemes. The attacks are of practical complexities and verified by experiments.

Cryptanalysis of a quadratic compact knapsack public-key cryptosystem

Recently, Wang and Hu have proposed a high-density quadratic compact knapsack public-key cryptosystem using the Chinese remainder theorem to disguise two secret cargo vectors. The system is claimed to be secure against certain known attacks; however, it has not been demonstrated to fulfill any provable security goals. In this work, we show that this system is not secure. Exploiting the special structure of system parameters, we first show that a candidate list for the secret modulus can be obtained by solving linear equations with small solutions. Next, we show that with this candidate list, all other secrets can be recovered in succession with lattice-based methods by solving certain modular linear equations with small solutions. As a result, recovering a private key can be done in about 11 h for the proposed system parameter n = 100 . We also discuss a method to thwart the proposed attack.

Cryptanalysis of a quadratic knapsack cryptosystem

Wang and Hu [B. Wang and Y. Hu, Quadratic compact knapsack public-key cryptosystem, Comput. Math. Appl. 59 (1) (2010) 194–206] proposed a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem and then using the Chinese remainder theorem to disguise the easy knapsack instant. In this paper, we present a heuristic stereotyped message attack that allows the cryptanalyst to recover the plaintext message when partial information about the original message is known. In particular, as shown by our experiments, for the proposed system parameter n = 100 which corresponds to a block length of 400 bits, exposing 60% of the plaintext allows the cryptanalyst to recover the remaining 160 bits of the message with a success probability of about 90% in about 2 hours.

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.

A New Knapsack Public-Key Cryptosystem Based on Permutation Combination Algorithm

A new secure knapsack cryptosystem based on the Merkle-Hellman public key cryptosystem will be proposed in this paper. Although it is common sense that when the density is low, the knapsack cryptosystem turns vulnerable to the low-density attack. The density d of a secure knapsack cryptosystem must be larger than 0.9408 to avoid low-density attack. In this paper, we investigate a new Permutation Combination Algorithm. By exploiting this algorithm, we shall propose a novel knapsack public-key cryptosystem. Our proposed scheme can enjoy a high density to avoid the low-density attack. The density d can also exceed 0.9408 to avoid the low-density attack.

Security of a Class of Knapsack Public-Key Cryptosystems against Low-Density Attack

In 2003, Kobayashi et al. proposed a new class of knapsack public-key cryptosystems over Gaussian integer ring. This scheme using two-sequences as the public key. In 2005, Sakamoto and Hayashi proposed an improved version of Kobayashi's scheme. In this paper, we propose the knapsack PKC using l-sequences as the public key and present the low-density attack on it. We have described Schemes R and G for l = 2, in which the public keys are constructed over rational integer ring and over Gaussian integer ring, respectively. We discusses on the difference of the security against the low-density attack. We show that the security levels of Schemes R and G differ only slightly.

A knapsack public-key cryptosystem based on elliptic curve discrete logarithm

There is no question that knapsack systems still warrant continuous researches, as a result of the NP-completeness nature, the faster speed, and a desire to have a wide variety of available cryptosystems. This paper presents a new knapsack-type public key cryptosystem based on a novel application of the elliptic curve logarithm problem. The difficulty can be made sufficient to frustrate attacks against the survey major knapsack public-key cryptosystems developed in the past more than 20 years, and hope-fully shed some light on the future of the knapsack cryptosystems. The proposed new cryptosystem can be found some theoretic support on confidentiality.

Linearly shift knapsack public-key cryptosystem

Two algorithms are proposed to improve the Merkle-Hellman knapsack public-key cryptosystem. an approach to transform a superincreasing sequence to a high-density knapsack sequence is proposed. The algorithm is easy to implement and eliminates the redundancy of many knapsack cryptosystems. A linear shift method is used to improve the security of the knapsack public-key cryptosystem. It is shown that several knapsacks (e.g., the so-called useless knapsack), which cannot be generated by using the Merkle-Hellman scheme, can be generated by the linear shift method. Thus A. Shamir's (1982, 1984) attack to the original knapsack, as well as the low-density attack to the iterated knapsack, cannot be applied to this system successfully. It is interesting to note that the concept of the requirement of being one-to-one in practical enciphering keys is not necessary for this system. >

The first ten years of public-key cryptography

The development of public-key cryptography is described, and its principles are elucidated. The discussion covers exponential key exchange, the trap-door knapsack public-key cryptosystem, the Rivest-Shamir-Adleman (RSA) system, and the breaking of the knapsack cryptosystem. Early responses to public-key systems and the problem of key management are examined. Applications and implementations are described. Significant development in multiplying, factoring, and finding prime numbers which have resulted from public-key research are sketched. Directions in public-key research are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Solving low-density subset sum problems

The subset sum problem is to decide whether or not the 0-l integer programming problem Σ n i=l a i x i = M , ∀I , x I = 0 or 1, has a solution, where the a i and M are given positive integers. This problem is NP-complete, and the difficulty of solving it is the basis of public-key cryptosystems of knapsack type. An algorithm is proposed that searches for a solution when given an instance of the subset sum problem. This algorithm always halts in polynomial time but does not always find a solution when one exists. It converts the problem to one of finding a particular short vector v in a lattice, and then uses a lattice basis reduction algorithm due to A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz to attempt to find v. The performance of the proposed algorithm is analyzed. Let the density d of a subset sum problem be defined by d = n /log 2 (max i a i ). Then for “almost all” problems of density d &lt; 0.645, the vector v we searched for is the shortest nonzero vector in the lattice. For “almost all” problems of density d &lt; 1/ n , it is proved that the lattice basis reduction algorithm locates v. Extensive computational tests of the algorithm suggest that it works for densities d &lt; d c ( n ), where d c ( n ) is a cutoff value that is substantially larger than 1/ n . This method gives a polynomial time attack on knapsack public-key cryptosystems that can be expected to break them if they transmit information at rates below d c ( n ), as n → ∞.

Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme

The basic Merkle-Hellman additive trapdoor knapsack public-key cryptosystem was recently shown to be insecure, and attacks have also been developed on stronger variants of it, such as the Graham-Shamir system and the iterated knapsack cryptosystem. It is shown that some simple variants of another Merkle-Hellman system, the multiplicative knapsack cryptosystem, are insecure. It is also shown that the Shamir fast signature scheme can be broken quickly. Similar attacks can also be used to break the Schöbi-Massey authentication scheme. These attacks have not been rigorously proved to succeed, but heuristic arguments and empirical evidence indicate that they work on systems of practical size.