Abstract
Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design stage of a cryptographic system is a formidable task. Fermat primality checking is one of the simplest of all tests. Unfortunately, there are composite integers (called Carmichael numbers) that are not detectable by the Fermat test. In this paper we consider modular arithmetic based on complex integers; and provide several tests that verify the primality of real integers. Although the new tests detect most Carmichael numbers, there are a small percentage of them that escape these tests.
Highlights
Large prime numbers are at the core of every modern cryptographic protocol
These protocols rely on multidigit large primes to ensure that the cryptanalysis of an encrypted message is too complicated to break in any relevant time
The concept of prime numbers is so important that it has been generalized in different ways in various branches of mathematics
Summary
Large prime numbers are at the core of every modern cryptographic protocol. These protocols rely on multidigit large primes to ensure that the cryptanalysis of an encrypted message is too complicated to break in any relevant time. Erdös: Conjecture: If there exists an algorithm that describes an order in the sequence of primes smaller than n, it has complexity f n , where f(n) is a monotone nondecreasing function of n, [3]. There are many ways to test an integer for primality. The Sieve of Eratosthenes, able to detect all primes, has a time complexity in the order of n, [4]. The Fermat test is very simple, there exists an infinite set of composite integers, {called Carmichael numbers or CMNs, for short}, that are not detectable by the Fermat test, [5]
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