Abstract
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.
Highlights
An analogue of the Solovay–Strassen primality test in general quadratic Euclidean domains is obtained
Из предложения 2 вытекает, что a 2 можно найти с помощью бинарного алгоритма возведения в степень за O(log3|N|m) двоичных операций
Shortest division chains in unique factorization domains / M
Summary
Для любого N ∈ OK с нечетной нормой и a ∈ OK, взаимно простого с N, определяем символ. Якоби a N pk – простые числа кольца OK взаимно простых нечетного b (к∈паок лпаригаоцеимезлв оедNгaое нaи=е∈1,сие смлвчиоелрNоезв–Л оbaебж раабнтуиддмреаым∏йоnkбэ=ло1ез мнpаaеkчна тт, ькгдоселиьNмцва=оO∏лKЯnk).=к1Доpблkия, в кольце.
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