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Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described. The Navier-Stokes equations (NSEs) u t + (u · ∇) u − u + ∇p = 0, div u = 0 (1) which describe the motion of an incompressible viscous fluid are the basic equations of… (More)

- Roman Popovych
- Finite Fields and Their Applications
- 2012

- Roman Popovych
- Finite Fields and Their Applications
- 2013

- R. Popovych
- Proceedings of the International Conference…
- 2004

Possible approaches to cryptoanalysis RSA system of enciphering with public key are considered, especially with using of Euler function value. RSA algorithm is used in computer payment system of national bank of Ukraine national payment system, which provides implementation of accounts amongst bank institutions, boards of state treasury on the territory of… (More)

- Roman Dunets, Roman Popovych, Bogdan Popovych
- 2015 Xth International Scientific and Technical…
- 2015

Using of extended finite fields for cryptographic information protection is considered. In particular, explicit construction in finite fields elements of high multiplicative order is described.

- Roman Popovych
- IACR Cryptology ePrint Archive
- 2009

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.

- Roman Popovych
- IACR Cryptology ePrint Archive
- 2006

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)

- Roman Popovych
- IACR Cryptology ePrint Archive
- 2009

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.

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