Тестирование чисел на простоту: теория и практика

Наводиться класифікація та огляд основних алгоритмів тестування чисел на простоту, а також їх порівняльний аналіз та рекомендації з побудови практичних засобів.Classification, review of main primality test algorithms, comparative analysis and recommendation of mean building are given in the article

Miller's primality test

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Primality Testing

This tutorial describes the Miller-Rabin method for testing the primality of large integers. The method is illustrated by a Pascal algorithm. The performance of the algorithm was measured on a Computing Surface

Parallel Algorithms for Depth-First Search

In this paper we examine parallel algorithms for performing a depth-first search (DFS) of a directed or undirected graph in sub-linear time. this subject is interesting in part because DFS seemed at first to be an inherently sequential process, and for a long time many researchers believed that no such algorithms existed. We survey three seminal papers on the subject. The first one proves that a special case of DFS is (in all likelihood) inherently sequential; the second shows that DFS for planar undirected graphs is in NC; and the third shows that DFS for general undirected graphs is in RNC. We also discuss randomnized algorithms, P-completeness and matching, three topics that are essential for understanding and appreciating the results in these papers

Assessment of VLSI resources requirement for a sliced trusted platform module

Recent increases in cybercrime suggest questions such as: How can one trust a secure system? How can one protect private information from being stolen and maintain security? Trust in any system requires a foundation or root of trust. A root of trust is necessary to establish confidence that a machine is clean and that a software execution environment is secure. A root of trust can be implemented using the Trusted Platform Module (TPM), which is promising for enhancing security of general-purpose computing systems. In cloud computing, one of the proposed approaches is to use homomorphic encryption to create k program slices to be executed on k different cloud nodes. The TPM at the cloud node can then also be distributed or sliced along the lines presented in this thesis. In this work, we propose to increase TPM efficiency by distributing the TPM into multiple shares using Residue Number Systems (RNS). We then perform an evaluation of the silicon area, and execution time required for a sliced-TPM implementation and compares it to a single TPM. We characterize the execution time required by each TPM command using measurements obtained on ModelSim simulator. Finally, we show that the proposed scheme improves TPM efficiency and that execution time of TPM commands was noticeably improved. In the case of 4 shares the required execution time of the TPM commands that involving RSA operation in each slice was decreased by 93%, and the area of each slice was decreased by 2.93% while the total area was increased by 74%. In the case of 10 shares the required execution time of the TPM commands that involving RSA operations in each slice was decreased by 99%, and the area of each slice was decreased by 3.3% while the total area was increased by 85%

Certifying Giant Nonprimes

GIMPS and PrimeGrid are large-scale distributed projects dedicated to searching giant prime numbers, usually of special forms like Mersenne and Proth. The numbers in the current search-space are millions of digits large and the participating volunteers need to run resource-consuming primality tests. Once a candidate prime $N$ has been found, the only way for another party to independently verify the primality of $N$ used to be by repeating the expensive primality test. To avoid the need for second recomputation of each primality test, these projects have recently adopted certifying mechanisms that enable efficient verification of performed tests. However, the mechanisms presently in place only detect benign errors and there is no guarantee against adversarial behavior: a malicious volunteer can mislead the project to reject a giant prime as being non-prime. In this paper, we propose a practical, cryptographically-sound mechanism for certifying the non-primality of Proth numbers. That is, a volunteer can -- parallel to running the primality test for $N$ -- generate an efficiently verifiable proof at a little extra cost certifying that $N$ is not prime. The interactive protocol has statistical soundness and can be made non-interactive using the Fiat-Shamir heuristic. Our approach is based on a cryptographic primitive called Proof of Exponentiation (PoE) which, for a group $\mathbb{G}$, certifies that a tuple $(x,y,T)\in\mathbb{G}^2\times\mathbb{N}$ satisfies $x^{2^T}=y$ (Pietrzak, ITCS 2019 and Wesolowski, J. Cryptol. 2020). In particular, we show how to adapt Pietrzak\u27s PoE at a moderate additional cost to make it a cryptographically-sound certificate of non-primality

Fast multiplication of multiple-precision integers

Multiple-precision multiplication algorithms are of fundamental interest for both theoretical and practical reasons. The conventional method requires 0(n2) bit operations whereas the fastest known multiplication algorithm is of order 0(n log n log log n). The price that has to be paid for the increase in speed is a much more sophisticated theory and programming code. This work presents an extensive study of the best known multiple-precision multiplication algorithms. Different algorithms are implemented in C, their performance is analyzed in detail and compared to each other. The break even points, which are essential for the selection of the fastest algorithm for a particular task, are determined for a given hardware software combination

Applications of additive combinatorics methods to some multiplicative problems

Wydział Matematyki i InformatykiGłównym celem pracy jest badanie różnych sposobów, w jakie kombinatoryka addytywna może być wykorzystana do radzenia sobie z pewnymi zagadnieniami pojawiającymi się w multiplikatywnej teorii liczb. Najważniejsza część pracy dotyczy następującego problemu: dla pewnej liczby naturalnej n i pewnej liczby pierwszej p jest nam dany zbiór reszt modulo p wszystkich dzielników liczby n i chcielibyśmy stwierdzić, które z nich odpowiadają jej czynnikom pierwszym. Przedstawiony jest algorytm rozwiązujący ten problem dla p i n spełniających pewne naturalne warunki i zostaje pokazane, że jest wiele takich liczb. Interesującą cechą przedstawionego dowodu jest to, że wymaga on użycia kombinatoryki addytywnej. W kolejnej części pracy rozważana jest suma wyrażeń exp(a2r/q ) dla wszystkich r należących do podgrupy multiplikatywnej reszt modulo q generowanej przez element 2. Podajemy górne oszacowanie wartości bezwzględnej z lepszą stałą niż dotychczas znana. W ostatniej części pracy rozważane są oszacowania na wielkość zbioru wszystkich sum postaci c1a1+c2a2+…+ckak, gdzie ci są ustalonymi współczynnikami, zaś ai są elementami zbioru A. Seria oszacowań górnych wielkości tego zbioru jest udowodniona dla A spełniającego |A+A| < K |A|. Najlepsze oszacowania dostajemy w przypadkach, gdy K jest znacznie mniejsze niż h oraz gdy zbiór współczynników ci ma pewną strukturę addytywną.The main aim of this dissertation is the study of different ways in which additive combinatorics may be used to tackle some problems arising in multiplicative number theory. The main part of the thesis deals with the following problem: Suppose that for some natural number n and some prime number p we are given the set of residues mod p of all its divisors and we would like to know which of those residues correspond to prime factors of n. An algorithm which approximately solves this problem for p and n satisfying some natural conditions is presented and it is proved that there are plenty of such numbers. One interesting feature of the proof is that it relies on additive combinatorics. In the next part of the thesis the sum of expressions exp(a2r/q ) over r belonging to multiplicative subgroup of residues modulo q generated by element 2. absolute value of this sum is estimated. The result we obtained in this line of research is the following. We give an upper-bound of absolute value of this sum with a better constant than previously known. In the last part of the thesis bounds for the size of sets of all the sums of the form c1a1+c2a2+…+ckak, where ci are coefficients and ai are elements of the set A. Series of results giving upper-bounds on the size of this set is proved for A satisfying |A+A|<K|A|. The best bounds are obtained in cases when K is much smaller than h and when the set of ci coefficients has some additive structure