Primes in quadratic unique factorization domains

The present paper is devoted to extension of a number of well-known results on natural primes for prime elements in quadratic UFD. We obtain analogues of Miller's, Euler's, Lucas' and Pocklington's criterions of primality in quadratic UFD. There is proved that an analogue of the Miller–Rabin test can be realized in quadratic UFD and extended the Rabin result on probability of successful work of the Miller–Rabin test. We construct RSA-cryptosystem in quadratic domains and prove that there hold similar properties to RSA-cryptosystem on integers.