Abstract

The AKS algorithm is an important breakthrough in showing that primality testing of an integer can be done in polynomial time. In this paper, we study the optimization of its runtime. Namely, given a finite cardinality set of alphabets of a deterministic polynomial runtime Turing machine and the number of strings of an arbitrary input integer whose primality is to be tested as the system parameters, we consider the randomized AKS primality testing function as the objective function. Under randomization of the system parameters, we have shown that there are definite signatures of the local and global instabilities in the AKS algorithm. We observe that instabilities occur at the extreme limits of the parameters. It is worth mentioning that Fermat’s little theorem and Chinese remaindering help with the determination of the underlying stability domains. On the other hand, in the realm of the randomization theory, our study offers fluctuation theory structures of the AKS primality testing of an integer through its maximum number of irreducible factors. Finally, our optimization theory analysis anticipates a class of real-world applications for future research and developments, including optimal online security, system optimization and its performance improvements, (de)randomization techniques, and beyond, e.g., polynomial time primality testing, identity testing, machine learning, scientific computing, coding theory, and other stimulating optimization problems in a random environment.

Highlights

  • The Agrawal-Kayal-Saxena (AKS) algorithm plays an important role in determining the primality of an integer [1]

  • Considering the fact that it is among the best primality testing methods of an arbitrary integer, we examine how the AKS algorithm behaves under variations of the input integer and the machine parameters that are used in its primality testing

  • It is noted that we find the signature of possible instabilities in execution of the AKS primality testing algorithm of stability rises in the limit of increasing values of

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Summary

Introduction

The Agrawal-Kayal-Saxena (AKS) algorithm plays an important role in determining the primality of an integer [1]. In the realm of modern computer science, it finds significance in cryptography, viz. From the perspective of the formal theory of computation, an apt design of protocols is achieved via a suitable algorithm to efficiently perform a given computational task [4,5]. The AKS primality testing finds further importance, as well. See [6] concerning its elementary description, correctness, and asymptotic analysis towards the primality testing of an integer. From the inception of the AKS algorithm, the prime factoring problem, presumed to be an NP-type problem, has become a P-type problem in the realm of the randomization theory. In the light of Cryptography 2019, 3, 12; doi:10.3390/cryptography3020012 www.mdpi.com/journal/cryptography

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