Merkle-Hellman Research Articles

Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper examines the performance of three binary metaheuristic algorithms when applied to two distinct knapsack problems (0–1 knapsack problems (KP01) and multidimensional knapsack problems (MKP)). These binary algorithms are based on the classical mantis search algorithm (MSA), the classical quadratic interpolation optimization (QIO) method, and the well-known differential evolution (DE). Because these algorithms were designed for continuous optimization problems, they could not be used directly to solve binary knapsack problems. As a result, the V-shaped and S-shaped transfer functions are used to propose binary variants of these algorithms, such as binary differential evolution (BDE), binary quadratic interpolation optimization (BQIO), and binary mantis search algorithm (BMSA). These binary variants are evaluated using various high-dimensional KP01 examples and compared to several classical metaheuristic techniques to determine their efficacy. To enhance the performance of those binary algorithms, they are combined with repair operator 2 (RO2) to offer better hybrid variants, namely HMSA, HQIO, and HDE. Those hybrid algorithms are evaluated using several medium- and large-scale KP01 and MKP instances, as well as compared to other hybrid algorithms, to demonstrate their effectiveness. This comparison is conducted using three performance metrics: average fitness value, Friedman mean rank, and computational cost. The experimental findings demonstrate that HQIO is a strong alternative for solving KP01 and MKP. In addition, the proposed algorithms are applied to the Merkle-Hellman Knapsack Cryptosystem and the resource allocation problem in adaptive multimedia systems (AMS) to illustrate their effectiveness when applied to optimize those real applications. The experimental findings illustrate that the proposed HQIO is a strong alternative for handling various knapsack-based applications.

An Efficient Binary Hybrid Equilibrium Algorithm for Binary Optimization Problems: Analysis, Validation, and Case Studies

Binary optimization problems belong to the NP-hard class because their solutions are hard to find in a known time. The traditional techniques could not be applied to tackle those problems because the computational cost required by them increases exponentially with increasing the dimensions of the optimization problems. Therefore, over the last few years, researchers have paid attention to the metaheuristic algorithms for tackling those problems in an acceptable time. But unfortunately, those algorithms still suffer from not being able to avert local minima, a lack of population diversity, and low convergence speed. As a result, this paper presents a new binary optimization technique based on integrating the equilibrium optimizer (EO) with a new local search operator, which effectively integrates the single crossover, uniform crossover, mutation operator, flipping operator, and swapping operator to improve its exploration and exploitation operators. In a more general sense, this local search operator is based on two folds: the first fold borrows the single-point crossover and uniform crossover to accelerate the convergence speed, in addition to avoiding falling into local minima using the mutation strategy; the second fold is based on applying two different mutation operators on the best-so-far solution in the hope of finding a better solution: the first operator is the flip mutation operator to flip a bit selected randomly from the given solution, and the second operator is the swap mutation operator to swap two unique positions selected randomly from the given solution. This variant is called a binary hybrid equilibrium optimizer (BHEO) and is applied to three common binary optimization problems: 0–1 knapsack, feature selection, and the Merkle–Hellman knapsack cryptosystem (MHKC) to investigate its effectiveness. The experimental findings of BHEO are compared with those of the classical algorithm and six other well-established evolutionary and swarm-based optimization algorithms. From those findings, it is concluded that BHEO is a strong alternative to tackle binary optimization problems. Quantatively, BHEO could reach an average fitness of 0.090737884 for the feature section problem and an average difference from the optimal profits for some used Knapsack problems of 2.482.

Performance Optimization and Comprehensive Analysis of Binary Nutcracker Optimization Algorithm: A Case Study of Feature Selection and Merkle–Hellman Knapsack Cryptosystem

In this paper, a binary variant of a novel nature-inspired metaheuristic algorithm called the nutcracker optimization algorithm (NOA) is presented for binary optimization problems. Because of the continuous nature of the classical NOA and the discrete nature of the binary problems, two different families of transfer functions, namely S-shaped and V-shaped, are extensively investigated for converting the classical NOA into a binary variant, namely BNOA, applicable for various binary problems. Additionally, BNOA is improved using a local search strategy based on effectively integrating some genetic operators into the BNOA’s exploitation and exploration; this additional variant is called BINOA. Both BNOA and BINOA are evaluated using three common binary optimization problems, including feature selection, 0-1 knapsack, and the Merkle–Hellman knapsack cryptosystem (MHKC), and are compared to several robust binary metaheuristic optimizers in terms of statistical information, statistical tests, and convergence speed. The experiential findings show that BINOA is better than the classical BNOA and the other rival optimizers for both the 0-1 knapsack problem and attacking MHKC and is on par with some algorithms, like the genetic algorithm for feature selection.

Knapsack Cipher-based metaheuristic optimization algorithms for cryptanalysis in blockchain-enabled internet of things systems

The integration of the Internet of Things (IoT) and blockchain demand the use of public-key cryptography systems to secure network communications. In this study, one of those public-key algorithms, known as Merkle–Hellman Knapsack Cryptosystem (MHKC), is employed to cryptographically analyze its utilization for blockchain technology using a metaheuristic algorithm. To do so, eight well-known metaheuristic algorithms are employed to determine the trustworthiness of MHKC against cryptoanalysis attacks using various knapsack lengths, ranging from 8 to 32 bits. The experimental findings showed that pathfinder algorithm (PFA) and slime mold optimizer (SMA) could exploit MHKC under 8-bit ASCII code, and their performance gradually deteriorates with higher bit representations, while the performance of manta ray foraging optimization (MRFO) could be superior for the knapsack lengths higher than 8-bit. Additionally, MRFO would not attack MHKC under 32-bit; thus, some genetic operators have been integrated to manipulate the binary solutions obtained by this algorithm to promote its exploration capability in a variant, namely HMRFO. The experimental findings revealed that HMRFO is a better alternative to the existing ones for attacking the MHKC with knapsack lengths higher than 8-bit to appear their fragility points, while both SMA and PFA are competitive for 8-bit ASCII code and superior to the other algorithms.

Attacking the Knapsack Public-key Cryptosystem

Merkle–Hellman knapsack cryptosystem is a public-key cryptosystem, which means, two keys are used, a public key for the encryption and a private key for the decryption. This cryptosystem is not secure against the cryptosystem attacks, such as LLL Algorithm. In this paper, we introduce a new study to attack the knapsack problem. This study is based on discussing certain facts regarding the public key to extract the multiplier. Also, based on some facts we can find the modulus that is used in the cryptosystem. Therefore, the experimental outcome shows that this method is easier and faster to attack the knapsack cryptosystem.

Cryptanalysis of Merkle-Hellman cipher using ant colony optimization

<p class="Abstract"><span id="docs-internal-guid-d3fe8e21-7fff-17fc-df0e-00893428243c"><span>The Merkle-Hellman (MH) cryptosystem is one of the earliest public key cryptosystems, which is introduced by Ralph Merkle and Martin Hellman in 1978 based on an NP-hard problem, known as the subset-sum problem. Furthermore, ant colony optimization (ACO) is one of the most nature-inspired meta-heuristic optimization, which simulates the social behaviour of ant colonies. ACO has demonstrated excellent performance in solving a wide variety of complex problems. In this paper, we present a novel ant colony optimization (ACO) based attack for cryptanalysis of MH cipher algorithm, where two different search techniques are used. Moreover, experimental study is included, showing the effectiveness of the proposed attacking scheme. The results show that ACO based attack is more suitable than many other algorithms like genetic algorithm (GA) and particle swarm optimization (PSO).</span></span></p>

A Recursive and Parallelized Dynamic Programming Implementation of Hard Merkle-Hellman Knapsack System for Public Key Cryptography

Abstract Merkle-Hellman public key cryptosystem is a long-age old algorithm used in cryptography. Despite being computationally fast, for very large input sizes it may operate slower due to thread creation overhead or reaching a deadlock situation. In this paper, we discuss the working principles of the Traditional Merkle-Hellman knapsack cryptosystem, which is an Easy knapsack. The challenges of Hard Knapsack and how it overcomes the shortcomings of the Traditional Easy Knapsack, are also discussed. The Hard knapsack variant of Merkle-Hellman is solved first using plain recursion and then improvised using a dynamic programming approach to the problem. Parallelism and Concurrency has been achieved on the dynamic programming implementation using OpenMP API which further has enhanced the performance time. A comparative study of both variants of Hard Knapsack for messages of different lengths has shown that the latter is faster.

ВДОСКОНАЛЕНА ПОЛІНОМІАЛЬНО-СКЛАДНА АТАКА ВІДНОВЛЕННЯ ВІДКРИТОГО ТЕКСТУ НА РАНЦЕВИЙ ШИФР НА ОСНОВІ МАТРИЦЬ

Asymmetric ciphers are widely used to ensure the confidentiality of data transmission via insecure channels. These cryptosystems allow the interacting parties to create a shared secret key for a symmetric cipher in such a way that an eavesdropper gets no information useful for cryptanalysis. Network security protocols that use asymmetric ciphers include TLS, S/MIME, OpenPGP, Tor, and many others. Some of the asymmetric encryption schemes are homomorphic, that is, that they allow calculations on encrypted data to be performed without preliminary decryption. The aforesaid property makes possible using these cryptosystems not only for symmetric key establishment but also in several areas of application, in particular in secret voting protocols and cloud computing. The matrix-based knapsack cipher is a new additively homomorphic asymmetric encryption scheme, which is based on the properties of isomorphic transformations of the inner direct product of diagonal subgroups of a general linear group over a Galois field. Unlike classic knapsack encryption schemes, the cryptographic strength of this cipher depends on the computational complexity of the multidimensional discrete logarithm problem. Despite some useful properties, further research into the cryptographic strength of the matrix-based knapsack cipher has found serious drawbacks inherent in this cryptographic scheme. In the given paper an improved polynomial-time plaintext-recovery attack on the matrix-based knapsack cipher is proposed. Applying this cryptanalytic method requires only public information and has time complexity O(t1.34), where t denotes the decryption time of the attacked cryptosystem. The aforementioned attack is more productive and easier to implement in software in comparison with the original one. The advantages of the proposed method are due to using in its algorithm the simple and relatively fast matrix trace operation instead of more complex and slower transformations.

POLYNOMIAL-TIME PLAINTEXT-RECOVERY ATTACK ON THE MATRIX-BASED KNAPSACK CIPHER

The aim of the present paper is to propose a polynomial-time plaintext-recovery attack on the matrix-based knapsack cipher. The aforesaid algorithm uses only public information and has time complexity O(t1.34), where t is the decryption time of the attacked cryptosystem. The matrix-based knapsack cipher is a novel additively homomorphic asymmetric encryption scheme, which is a representative of group-based knapsack ciphers. This cryptosystem is based on the isomorphic transformation’s properties of the inner direct product of diagonal subgroups of a general linear group over a Galois field. Unlike the classical knapsack cryptoschemes, the cryptographic strength of the aforesaid cipher depends on the computational complexity of the multidimensional discrete logarithm problem. Due to the attack proposed in the given paper, the matrix-based knapsack cipher can be considered broken and should not be used as a privacy tool. However, this cryptosystem is still suitable for educational purposes as an example of the application of linear and abstract algebras in asymmetric cryptography.

Soft public key cipher

Encryption in general is the process of keeping up the secrecy of data (both fixed and versatile) utilizing programs that can Convert and make an interpretation of that data into codes so that whenever got to by unapproved people doesn't They can comprehend anything since what appears to them is a blend of images, numbers and unlimited characters, the file is encrypted And decryption by password, which must be known to both parties (sender and receiver) This is called With symmetric encryption, Decryption means decryption. The quality and viability of encryption rely upon two key factors: the algorithm, and the key length evaluated by Bit, the higher the bit, the greater the security and difficulty of decrypting. The concept of soft set was studied and applied on the public key cryptography like simple Knapsack and Trapdoor Knapsack ciphers. Significant results were presented that was the measure of the evaluate encryption using the SNR and PSNR.

Deterministic lattice reduction on knapsacks with collision‐free properties

The knapsack problem is an important problem in computer science and had been used to design public key cryptosystems. Low-density subset sum algorithms are powerful tools to reduce the security of trapdoor knapsacks to the shortest vector problem (SVP) over lattices. Several knapsack ciphers Chor–Rivest, Okamoto–Tanaka–Uchiyama, and Kate–Goldberg were proposed to defend low-density attacks by utilising low-weight knapsack problems. Some evidence was also found on the vulnerabilities of the above three knapsack ciphers to lattice attacks. However, previous lattice-based cryptanalytic results have been established via a probabilistic approach. The authors investigate some collision-free properties and derive from the properties a deterministic reduction from the knapsack problems in the Chor–Rivest, Okamoto–Tanaka–Uchiyama, and Kate–Goldberg knapsack ciphers to SVP without imposing any restriction and assumption. To the best of the authors' knowledge, the proposed reduction is the first deterministic reduction from public key cryptographic knapsacks to SVP.

Cryptanalysis of Knapsack Cipher Using Ant Colony Optimization

Ant Colony Optimization is a search metaheuristic inspired by the behavior of real ant colonies and shown their effectiveness, robustness to solve a wide variety of complex problems. In this paper, we present a novel Ant Colony Optimization (ACO) based attack for cryptanalysis of knapsack cipher algorithm. A Cipher-text only attack is used to discover the plaintext from the cipher-text. Moreover, our approach allows us to break knapsack cryptosystem in a minimum search space when compared with other techniques. Experimental results prove that ACO can be used as an effective tool to attack knapsack cipher.

A Novel Whale Optimization Algorithm for Cryptanalysis in Merkle-Hellman Cryptosystem

With the advance of the communication technology and the massive flow of information across the internet, it is becoming urgent to keep the confidentiality of the transmitted information. Using the internet has been extended to several fields such as e-mail, e-commerce, e-learning, health and medicine, shopping, and so on. Cryptography is the study of different techniques for securing the communication between the sender and the receiver. One of the most known cryptosystems is Merkle–Hellman Knapsack Cryptosystem (MHKC). It is one of the earliest Public Key Cryptosystem (PKC) that is used to secure the messages between the sender and the receiver. Developing a powerful cryptosystem comes after studying the fragility points of the current cryptosystems. The Whale Optimization Algorithm (WOA) is one of the most recent nature-inspired meta-heuristic optimization algorithms, which simulates the social behavior of humpback whales. WOA has validated excellent performance in solving the continuous problems and the engineering optimization problems. This paper introduces a novel Modified version of WOA (MWOA) for cryptanalysis of MHKC. The sigmoid function is used to map the continuous values into discrete one. A penalty function is added to the evaluation function to deal with the infeasible solutions. The mutation operation is employed for improving the solutions. The results show that MWOA is more effective and robust than other algorithms in the literature.

Contravening Esotery: Cryptanalysis of Knapsack Cipher using Genetic Algorithms

Cryptanalysis of knapsack cipher is a fascinating problem which has eluded the computing fraternity for decades. However, in most of the cases either the time complexity of the proposed algorithm is colossal or an insufficient number of samples have been taken for verification. The present work proposes a Genetic Algorithm based technique for cryptanalysis of knapsack cipher. The experiments conducted prove the validity of the technique. The results prove that the technique is better than the existing techniques. An extensive review has been carried out in order to find the gaps in the existing techniques. The work paves the way of the application of computational intelligence techniques to the discipline of cryptanalysis.

Using Genetic Algorithm to Break Knapsack Cipher with Sequence Size 16

With the growth of networked system and applications such as eCommerce, the demand for effective internetsecurity is increasing. Cryptology is the science and study of systems for secret communication. It consists of twocomplementary fields of study: cryptography and cryptanalysis.The genetic algorithm is one of the search methods, whichfinds the optimal solution. It is one of the methods, which is used to decrypt cipher.This work focuses on using GeneticAlgorithms to cryptanalyse knapsack cipher. The knapsack cipher is with a knapsack sequence of size 16 to encrypt twocharacters together. Different values of parameters have been used: Population size, mutation rate, number of generation.

Four Way Encryption Method for Secure Message Coding by use of CBC, Merkle-Hellman, Randomizers and Discrete Logarithmics

The Cipher Block Chaining (CBC) and the Merkle-Hellman algorithm are two different cryptosystems for encrypting messages. But using these cryptosystems separately has many drawbacks. However there are several ways of getting rid of these drawbacks. This paper demonstrates how to use the combination of both the cryptographic algorithms along with randomized vectors, keys and super-increasing sequence to encrypt messages. The encrypted message is stored in secret buffers and these positions are encrypted by use of discrete logarithmics and sent. This not only strengthens the encrypted message but also makes the transmission more secure, so that only the intended recipient of the message is able to decipher the message. This paper focuses on enhancing the confidentiality of the message transfer. General Terms Cryptography; Discrete Logarithmics; Merkle-Hellman Knapsack Cryptosystem; RSA algorithm

Cryptanalysis of the Knapsack Cipher 0/255

The knapsack cipher 0/255 is cryptanalyzed. It is shown that given the public key of the knapsack cipher 0/255, one can recover the corresponding modular multiplier by using the continued fraction algorithm. Hence, the secret key is reconstructed from the recovered multiplier and the public key. So the knapsack cipher 0/255 is totally breakable.

Breaking Knapsack Cipher Using Population Based Incremental Learning

The demand for effective internet security is increasing exponentially day by day. Businesses have an obligation to protect sensitive data from loss or theft. Such sensitive data can be potentially damaging if it is altered, destroyed or if it falls into the wrong hands. So they need to develop a scheme that guarantees to protect the information from attacker. Cryptology which is the science and study of systems for secret communication is at the heart of providing such guarantee. Breaking knapsack cipher using Population Based Incremental Learning (PBIL) is sug gested in this paper. Results of this implementation are compared with results of Poonam Grag, Aditya Shastri, and D.C. Agarwal (2006) which had made an enhancement on the efficiency of genetic algorithm attack on knapsack cipher suggested by Spillman in 1 993. The result of comparison shows that Population Based Incremental Learning has less complexity and enjoys high speed (i.e. correct results will be generated with less number of generation compared to that generated using genetic algorithm) also the overhead for genetic algorithm operations is significantly higher than for Population Based Incremental Learning.

Integrating Merkle-Hellman knapsack cryptosystem into undergraduate computer science curriculum

In classical symmetric or private-key cryptosystems the encryption and decryption keys are either the same or can be easily found from each other. A new type of cryptosystem, call a public-key cryp...

The improvement of the knapsack cipher

In this paper, I propose an improvement of knapsack cipher. This improvement makes the cipher more high security and efficient. With the enhancement, the new knapsack cipher can be applied widely in the systems which need high security.