Computationally efficient search for large primes

To satisfy the speed of communication and to meet the demand for the continuously larger prime numbers, the primality testing and prime numbers generating algorithms require continuous advancement. To find the most efficient algorithm, a need for a survey of methods arises. Concurrently, an urge for the analysis of algorithms\u27 performances emanates. The critical criteria in the analysis of the prime numbers generation are the number of probes, number of generated primes, and an average time required in producing one prime. Hence, the purpose of this thesis is to indicate the best performing algorithm. The survey the methods, establishment of the comparison criteria, and comparison of approaches are the required steps to find the best performing algorithm. In the first step of this research paper the methods were surveyed and classified using the approach described in Menezes [66]. Wifle chapter 2 sorted, described, compared, and summarized primality testing methods, chapter 3 sorted, described, compared, and summarized prime numbers generating methods. In the next step applying a uniform technique, the computer programs were written to the selected algorithms. The programs were installed on the Unix operating system, running on the Sun 5.8 server to perform the computer experiments. The computer experiments\u27 results pertaining to the selected algorithms, provided required parameters to compare the algorithms\u27 performances. The results from the computer experiments were tabulated to compare the parameters and to indicate the best performing algorithm. Survey of methods indicated that the deterministic and randomized are the main approaches in prime numbers generation. Random number generation found application in the cryptographic keys generation. Contemporaneously, a need for deterministically generated provable primes emerged in the code encryption, decryption, and in the other cryptographic areas. The analysis of algorithms\u27 performances indicated that the prime nurnbers generated through the randomized techniques required smaller number of probes. This is due to the method that eliminates the non-primes in the initial step, that pre-tests randomly generated primes for possible divisibility factors. Analysis indicated that the smaller number of probes increases algorithm\u27s efficiency. Further analysis indicated that a ratio of randomly generated primes to the expected number of primes, generated in the specific interval is smaller than the deterministically generated primes. In this comparison the Miller-Rabin\u27s and the Gordon\u27s algorithms that randomly generate primes were compared versus the SFA and the Sequences Containing Primes. The name Sequences Containing Primes algorithm is abbreviated in this thesis as 6kseq. In the interval [99000,1000001 the Miller Rabin method generated 57 out of 87 expected primes, the SFA algorithm generated 83 out of 87 approximated primes. The expected number of primes was computed using the approximation n/ln(n) presented by Menezes [66]. The average consumed time of originating one prime in the [99000, 100000] interval recorded 0.056 [s] for Miller-Rabin test, 0.0001 [s] for SFA, and 0.0003 [s] for 6kseq. The Gordon\u27s algorithm in the interval [1,100000] required 100578 probes and generated 32 out of 8686 expected number of primes. Algorithm Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers invented by Doctor Verkhovsky [1081 verifies and generates primes and quasi primes using special mathematical constructs. This algorithm indicated best performance in the interval [1,1000] generating and verifying 3585 variances of provable primes or quasi primes. The Parametric Representation of Composite Twins algorithm consumed an average time per prime, or quasi prime of 0.0022315 [s]. The Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers algorithm implements very unique method of testing both primes and quasi-primes. Because of the uniqueness of the method that verifies both primes and quasi-primes, this algorithm cannot be compared with the other primality testing or prime numbers generating algorithms. The ((a!)^2)*((-1^b) Function In Generating Primes algorithm [105] developed by Doctor Verkhovsky was compared versus extended Fermat algorithm. In the range of [1,10001 the [105] algorithm exhausted an average 0.00001 [s] per prime, originated 167 primes, while the extended Fermat algorithm also produced 167 primes, but consumed an average 0.00599 [s] per prime. Thus, the computer experiments and comparison of methods proved that the SFA algorithm is deterministic, that originates provable primes. The survey of methods and analysis of selected approaches indicated that the SFA sieve algorithm that sequentially generates primes is computationally efficient, indicated better performance considering the computational speed, the simplicity of method, and the number of generated primes in the specified intervals

Survey of RSA Vulnerabilities

Rivest et al. patented (US) RSA. RSA forms the basis of most public encryption systems. It describes a public key encryption algorithm and certification process, which protects user data over networks. The patent expired in September 2000 and now is available for general use. According to Marketsandmarkets.com, the global network encryption market size is expected to grow from USD 2.9 billion in 2018 to USD 4.6 billion by 2023, at a compound annual growth rate (CAGR) of 9.8%. Major growth drivers for the market include increasing adoption of optical transmission, an increasing demand to meet various regulatory compliances and a growing focus on shielding organizations from network security breaches. In short, RSA forms the basis of almost all public encryption systems. This, however, is not without risk. This chapter explores some of these vulnerabilities in a mathematical context and provides the reader with an appreciation of the strength of RSA

Proximity Gaps for Reed-Solomon Codes

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are $\delta$-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are $\delta$-close to the property. In particular, no set in the collection has roughly half of its members $\delta$-close to the property and the others $\delta$-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed--Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any $\delta$ smaller than the Johnson/Guruswami-Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for $\delta$ smaller than the unique decoding radius. Concretely, if $\delta$ is smaller than half the minimal distance of an RS code $V\subset {\mathbb F}_q^n$, every affine space is either entirely $\delta$-close to the code, or alternatively at most an $(n/q)$-fraction of it is $\delta$-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed--Solomon codes (due to Berlekamp-Welch and Guruswami-Sudan) on a formal element of an affine space. This involves working with Reed-Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests

On the computation of discrete logarithms in finite prime fields

In this thesis we write about practical experience when solving congruences of the form a^x = b mod p, a,b,p,x Element Z, p prime. This is referred to as the discrete logarithm problem in (Z/pZ)*. Many cryptographic protocols such as signature schemes, message encryption, key exchange and identification depend on the difficulty of this problem. We are concerned with the practicability of different index calculus variants, which are the asymtotically fastest known algorithms at present to solve this problem. We present computations for p having up to 85 decimal digits. We include a partial solution to McCurley\u27s challenge with a 129-digit p, which has a special form.In dieser Arbeit berichten wir über praktische Erfahrungen mit der Lösung von Kongruenzen der Form a^x = b mod p, a,b,p,x Element Z, p Primzahl. Dies ist das Problem der Diskreten Logarithmen in (Z/pZ)*. Zahlreiche kryptographische Protokolle wie digitale Unterschriften, Verschlüsselung von Nachrichten, Schlüsselaustausch und Identifikation basieren auf der Schwierigkeit dieses Problems. In dieser Arbeit befassen wir uns mit der Praktikabilität verschiedener Index-Calculus Verfahren, die zur Zeit die asymptotisch schnellsten Algorithmen liefern, um dieses Problem zu lösen. Wir präsentieren Berechnungen mit bis zu 85-stelligem p und legen eine partielle Lösung zu McCurley\u27s Challenge vor, die ein 129-stelliges p von spezieller Form benutzt