Abstract
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.
Highlights
Asymmetric encryption schemes are widely used to ensure the confidentiality of communication via insecure channels
The matrix-based knapsack cipher is a novel additively homomorphic asymmetric encryption scheme, which is a representative of group-based knapsack ciphers [10]
This cryptosystem is based on the isomorphic transformation properties of the inner direct product of diagonal subgroups of a general linear group over a Galois field [11]
Summary
Asymmetric encryption schemes are widely used to ensure the confidentiality of communication 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 [1, 2]. Some of asymmetric ciphers are homomorphic meaning that they allow calculations on encrypted data to be performed without preliminary decryption This property makes it possible to use the given cryptosystems in several areas of applications besides symmetric key establishment. The approach to building this cryptosystem over a Galois field with a multiplicative group of a large smooth order was proposed in [12] Another approach, in which the aforesaid cipher is built over a small Galois field, was used in [10], where the property of additive homomorphism was proven for this cryptoscheme. This algorithm uses only public information and has computational complexity polynomial in the time required for decryption by the attacked cryptoscheme
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