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We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Z p [X]/(C r (X))) * and state the modified conjecture that the set {X-1, X+2} generate big enough subgroup of this group.

For given N=pq with p and q different odd primes and natural m Li introduced the public key cryptosystem. In the case m=1 the system is just the famous RSA system. We answer the Li's question about correctness of the system.

We propose to verify the AKS algorithm identities not for sequential integers, but for integers which are sequentially squared. In that case a number of elements, for which the identities are valid, doubles. 1. Introduction Prime numbers are of fundamental importance in mathematics in general, and cryptography theory in particular. Efficient primality tests… (More)

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