FastFlow tutorial

FastFlow is a structured parallel programming framework targeting shared memory multicores. Its layered design and the optimized implementation of the communication mechanisms used to implement the FastFlow streaming networks provided to the application programmer as algorithmic skeletons support the development of efficient fine grain parallel applications. FastFlow is available (open source) at SourceForge (http://sourceforge.net/projects/mc-fastflow/). This work introduces FastFlow programming techniques and points out the different ways used to parallelize existing C/C++ code using FastFlow as a software accelerator. In short: this is a kind of tutorial on FastFlow.Comment: 49 pages + cove

A representation of the natural numbers by means of cycle-numbers, with consequences in number theory

In this paper we give rules for creating a number triangle T in a manner analogous to that for producing Pascal's arithmetic triangle; but all of its elements belong to {0, 1}, and cycling of its rows is involved in the creation. The method of construction of any one row of T from its preceding rows will be defined, and that, together with starting and boundary conditions, will suffice to define the whole triangle, by sequential continuation. We shall use this triangle in order to define the so-called cycle-numbers, which can be mapped to the natural numbers. T will be called the 'cyclenumber triangle'. First we shall give some theorems about relationships between the cyclenumbers and the natural numbers, and discuss the cycling of patterns within the triangle's rows and diagonals. We then begin a study of figures (i.e. (0,1)- patterns, found on lines, triangles and squares, etc.) within T. In particular, we shall seek relationships which tell us something about the prime numbers. For our later studies, we turn the triangle onto its side and work with a doubly-infinite matrix C. We shall find that a great deal of cycling of figures occurs within T and C, and we exploit this fact whenever we can. The phenomenon of cycling patterns leads us to muse upon a 'music of the integers', indeed a 'symphony of the integers', being played out on the cycle-number triangle or on C. Like Pythagoras and his 'music of the spheres', we may well be the only persons capable of hearing it!

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

On Generating Prime Numbers Efficiently

The prime numbers can be considered as the building blocks of natural numbers, having innumerable applications in number theory and cryptography. There exist multiple different sieving algorithms for the generation of prime numbers. In this thesis, an elementary modular result is utilized to construct an analytically useful generator function and its inverse function. The functions are used to generate a (log)log-linear time complexity prime sieving algorithm which is further optimized to be of linear time complexity. The constructed algorithms and their operation are studied and the linear implementations in JS, Python and C++ are compared to other prime sieves.Alkulukuja voidaan pitää luonnollisten lukujen rakennuspalikoina joilla on lukemattomia sovelluksia lukuteoriassa ja kryptografiassa. Alkulukujen luomiseen on olemassa useita erilaisia seulonta-algoritmeja. Tässä opinnäytetyössä käytetään modulaarista perustulosta analyyttisesti hyödyllisten kehitysfunktion ja sen käänteisfunktion luomiseen. Funktioiden avulla luodaan aikakompleksisuudeltaan (log)log-lineaarinen alkulukuseula, joka optimoidaan lineaariseksi. Rakennettuja algoritmeja ja niiden toimintaa tarkastellaan ja lineaarista implementaatiota JS, Python ja C++ ohjelmointikielillä verrataan toisiin alkulukuseuloihin

Apollonian circle packings: Dynamics and Number theory

We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the fall of 2013.Comment: To appear in Japanese Journal of Mat

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