Related papers: Average Case Error Estimates of the Strong Lucas T…
The strong Lucas test is a widely used probabilistic primality test in cryptographic libraries. When combined with the Miller-Rabin primality test, it forms the Baillie-PSW primality test, known for its absence of false positives,…
The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the…
It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we…
We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior…
The strong probable primality test is an important practical tool for discovering prime numbers. Its effectiveness derives from the following fact: for any odd composite number $n$, if a base $a$ is chosen at random, the algorithm is…
Prime numbers play a very vital role in modern cryptography and especially the difficulties involved in factoring numbers composed of product of two large prime numbers have been put to use in many modern cryptographic designs. Thus, the…
Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy. Developing new algorithms or improving upon existing ones could…
Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…
We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$,…
In this paper we present and expand upon procedures for obtaining large d digit prime number to an arbitrary probability. We use a layered approach. The first step is to limit the pool of random number to exclude numbers that are obviously…
In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…
This paper studies grading algorithms for randomized exams. In a randomized exam, each student is asked a small number of random questions from a large question bank. The predominant grading rule is simple averaging, i.e., calculating…
Primality generation is the cornerstone of several essential cryptographic systems. The problem has been a subject of deep investigations, but there is still a substantial room for improvements. Typically, the algorithms used have two parts…
This work develops central limit theorems for cross-validation and consistent estimators of its asymptotic variance under weak stability conditions on the learning algorithm. Together, these results provide practical, asymptotically-exact…
This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear…
We revisit the fundamental question of simple-versus-simple hypothesis testing with an eye towards computational complexity, as the statistically optimal likelihood ratio test is often computationally intractable in high-dimensional…
In this paper, a random primality proving algorithm is proposed, which generates prime certificate of length O(log n). The certificate can be verified in deterministic time O(log^4 n). The algorithm runs in heuristical time tilde{O}(log^4…
Primality testing is an especially useful topic for public-key cryptography. In this paper, a novel primality test algorithm based on the Pell's cubic will be introduced, and its necessary primality conditions will be proved using three…
Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most $\frac 14$ of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius…
The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…