Abstract
The most popular present-day public-key cryptosystems are RSA and ElGamal cryptosystems. Some practical algebraic generalization of the ElGamal cryptosystem is considered-basic modular matrix cryptosystem (BMMC) over the modular matrix ring M2(Zn). An example of computation for an artificially small number n is presented. Some possible attacks on the cryptosystem and mathematical problems, the solution of which are necessary for implementing these attacks, are studied. For a small number n, computational time for compromising some present-day public-key cryptosystems such as RSA, ElGamal, and Rabin, is compared with the corresponding time for the ВММС. Finally, some open mathematical and computational problems are formulated.
Highlights
Security of some present-day public-key cryptosystems is based on computational complexity of some numbertheoretical problems
Two of these problems are used most often: the integer factorization problem and the discrete logarithm problem. These problems ensure the security of the RSA and ElGamal cryptosystems, as well as of the corresponding digital signature schemes [1]
In [2], randomized polynomial-time algorithms for computing discrete logarithms and integer factoring were presented for the quantum computer
Summary
Security of some present-day public-key cryptosystems is based on computational complexity of some numbertheoretical problems. Two of these problems are used most often: the integer factorization problem and the discrete logarithm problem. New practical algebraic generalization of the ElGamal cryptosystem will be given, some attacks on this cryptosystem—in the Section 4, new hard computational problems—in the Section 5, comparison of the security level of classical RSA, ElGamal and Rabin cryptosystems with security level of this cryptosystem for the same small number—in the Section 7, some related open mathematical and computational problems— in the Section 8 It should be noted, that some other theoretical algebraic generalizations of the ElGamal cryptosystem are given in [13,14]
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