Abstract: The number field sieve is an algorithm to factor integers of the form re − s for small positive r and |s|. The algorithm depends on arithmetic in an algebraic number field. We describe the algorithm, discuss several aspects of its implementation, and present some of the factorizations obtained. A heuristic run time analysis indicates that the number field sieve is asymptotically substantially faster than any other known factoring method, for the integers that it applies to. The number field sieve can be modified to handle arbitrary integers. This variant is slower, but asymptotically it is still expected to beat all older factoring methods.

Abstract: Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is O(log2 m) . This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log2m .

Abstract: Publisher Summary This chapter discusses algorithms that solve two basic problems in computational number theory—factoring integers into prime factors and finding discrete logarithms. In the factoring problem, one is given an integer n 1 and is asked to find the decomposition of n into prime factors. It is common to split this problem into two parts. The first is called primality testing: given n , it is determined whether n is prime or composite. The second is called factorization: if n is composite, a nontrivial divisor of n is to be calculated. In the discrete logarithm problem, one is given a prime number p , and two elements h, y of the multiplicative group F* p of the field of integers modulo p. The algorithms and their analyses depend on many different parts of number theory. Number theory is considered the purest of all sciences, and within number theory the hunt for large primes and for factors of large numbers has always been remote from applications, even to other questions of a number-theoretic nature.

Abstract: CONTENTS Introduction Chapter I. The Riemann zeta-function and its connection with primes § 1. Definition of and Euler's identity § 2. Continuation of to the half-plane § 3. Continuation of to the whole plane § 4. Functional properties of and § 5. Zeros of and primes § 6. Elementary theorems on the complex zeros of § 7. Theorems of de la Vallee Poussin § 8. Elementary consequences of the Riemann hypothesis Notes Chapter II. Approximate functional equations for § 1. An approximation of an exponential sum by a sum of integrals § 2. An asymptotic calculation for a class of exponential integrals § 3. An approximation of an exponential sum by a shorter one § 4. Approximate functional equations for Notes Chapter III. Vinogradov's method in the theory of the Riemann zeta-function § 1. The mean-value of a power of the modulus of an exponential sum § 2. Simple lemmas § 3. The main recurrence inequality § 4. Vinogradov's mean-value theorem § 5. An estimate for a zeta-sum and its consequences § 6. The current boundary of the zero-free region and some consequences Notes Chapter IV. A zero-density theorem and primes in short intervals § 1. Auxiliary lemmas § 2. A zero-density theorem § 3. Primes in short intervals Notes Appendix References

Abstract: Let p be a prime number. Let k be a field of characteristic different from p and containing the p-th roots of unity. Let be a finite group. Let L/k be a finite normal extension with Galois group and let c be a 2-cocycle on with coefficients in , where acts trivially on By Emb(L/k, c) we denote the question of the existence of a finite normal extension M of k, such that M contains L, such that [M: L] = p, and such that, denoting by the Galois group of M/k, the extension is given by the class of c.

Abstract: Let l be a prime number, and let F be an algebraic closure of the prime field F l. Suppose that $$\rho :Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL(2,\mathbb{F})$$ is an irreducible (continuous) representation. We say that ρ is modular of level N, for an integer N≥ 1, if ρ arises from cusp forms of weight 2 and trivial character on Γ0(N).

Abstract: This paper describes a new method for generating primes together with a proof of their primality that is extremely efficient (for 100-digit primes the average running time is equal to the average time required for finding a “strong pseudoprime” of the same size that passes the Miller-Rabin test for only four bases), that yields primes that are nearly uniformly distributed over the set of all primes in a given interval, and that is easily modified to yield (with no additional computational effort) primes that are nearly uniformly distributed over the subset of these primes that satisfy certain security constraints for use in the RSA cryptosystem. This method is used to generate, for a given encryption exponent e, an RSA-modulus m = pq that is nearly uniformly distributed over all secure RSA-moduli in a given interval I, i.e., over the set of all integers in I that are (1) the product of exactly two primes p and q none of which is smaller than a given limit L, where (2) (p − 1, e) = (q − 1, e) = 1 and (3) p − 1 and q − 1 each contain a prime factor greater than a given limit L′, and where (4) for all but a provably (given) small fraction of plaintexts in Z*m, the minimum number of iterated encryptions with exponent e required to recover the plaintext, is provably greater than a given limit M. Our method exploits a well-known result due to Pocklington [20] that allows one to prove the primality of p when only a partial factorization of p − 1 is known. These prime factors of p − 1 are generated by recursive application of the prime generating procedure. Although the discussion is centered on the RSA system, our method can of course be used in other cryptographic systems, such as the Diffie-Hellman public key distribution system, that require large primes satisfying certain security constraints.

Abstract: Let n be an even integer not divisible by 3. Suppose that p=n 2 +n+1 is a prime and 2 n+1 ≡1(mod p). The question is asked whether this can only occur if n is a power of 2. It is noted that an affirmative answer to this question implies that a finite projective plane with a flag transitive collineation group is Desarguesian

Abstract: Given an algebraic number field with unit group and a prime number , consider the bilinear form , where is the -adic logarithm. For certain types of fields it is shown that the form is nondegenerate. We investigate the behavior of the rank of the kernel of on the family of intermediate fields in a -extension . Bibliography: 11 titles.

Abstract: The main purpose of this paper is to establish the “abstract prime number theorem” for algebraic function fields. This theorem has its original motivation in enumeration theorems on algebraic function fields and has been investigated by several authors [4,7,8,9].

Abstract: The correlation described over twenty years ago by Matula between the prime factorization of integers and the class of alkanes is re-examined with a view to explaining the probable reason why there have, to date, been no major extensions of this idea. By considering the class of alkanes as a one-dimensional one-parameter system, a new perspective on the method is gained that is amenable to extension, but in a different direction than originally anticipated. With this new perspective, the classes of polybenzenes and polymantanes are seen to be the representatives of two- and three-dimensional one-parameter systems, respectively. A “nomenclature”, comparable to one that Matula used for alkanes, is created that gives a unique canonical name to all possible combinations of either polybenzene or polymantane modules. Such a “nomenclature” contains a “built-in” means of positioning the molecule in the field of interest in accordance with arbitrary pre-selected criteria, such as Patterson's rules, and also coding that indicates symmetries inherent in the structure of this “molecule”.

Abstract: The topological centralizers of Toeplitz flows satisfying a condition (Sh) and their Z2-extensions are described. Such Toeplitz flows are topologically coalescent. If {q0, q1, ...} is a set of all except at least one prime numbers and I0, I1, ... are positive integers then the direct sum Ai=08 Zqi|i A Z can be the topological centralizer of a Toeplitz flow.

Abstract: After summarizing some results on pseudoprimes, a characterization is presented of Fibonacci pseudoprimes of the bth kind for all integers b. Subsequently, some generalizations of the concept of strong pseudoprimes are established.

Abstract: A great deal of work has been carried out in recent years into the construction of computationally efficient small discrete Fourier transform (DFT) algorithms. Most small-DFT algorithms exploit the equivalence of prime number DFT computation with that of circular convolution, as well as Winograd's complexity theory results relating to the optimal computation of small circular convolutions, to achieve reduced-complexity solutions. The paper extends these results to the case of medium/large prime number DFT computation by means of the Agarwal—Cooley technique, whereby a multidimensional index mapping, combined with Winograd's results, converts the associated one-dimensional circular convolution into a multidimensional nested circular convolution. The resulting computational structure is then expressed in the form of an input addition phase, an output addition phase and, in between, a number of independent circular convolutions, which in hardware can be implemented in parallel, via both word-level and bit-level arithmetic techniques, to provide high-throughput solutions to the original prime number DFT computation.

Abstract: Let q be a power of a prime number, Fq the finite field with q elements, n an integer dividing q−1, n≥2, and χ a character of order n of the multiplicative group F*q. If X is an algebraic curve defined over Fq and if G is a divisor on X, we define a non linear code Γ(q, X, G, n, χ) on an alphabet with n+1 letters. We compute the parameters of this code, through the consideration of some character sums.

Abstract: The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind ζ-functions of all fields\(\mathbb{Q}\left( {\left( {\frac{{1 + \sqrt 5 }}{2}} \right)^{1/k} , 1^{1/k} } \right)\) Then there exists a constant C>0 such that on the interval p≤x one can find at least Cx log−1 x prime numbers p for which h(5p2)=2. Here h(d) is the number of proper equivalence classes of primitive binary quadratic forms of discriminant d. In addition, it is proved that\(\sum\limits_{p \leqslant x} {h (5p^2 )} log p = O (x^{3/2} )\). For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes.

Abstract: We make an attempt to compare the speed of some primality testing algorithms for certifying 100-digit prime numbers.

Abstract: An algorithm for computing the two-dimensional discrete Hartley transform (2D-DHT) of size pn×pn, where p is an odd prime number and n a positive integer, is proposed The algorithm is based on linear congruences and a local ring structure The 2D-DHT is calculated by the sum of the cores of the discrete Hartley transform, CHTs

Abstract: Conventional addressing schemes for prime factor mapping (PFM) often involve two equations, one for data loading and one for data retrieval. In the letter we show that only one equation is enough in the realisation of PFM for the discrete Fourier transform with lengths equal to products of prime squares. Hence the realisation is truly in-place, in-order.

Abstract: Let p be an odd prime number and n a natural number. Let K be a (2,...,2)-extension of the pn−th cyclotomic number field obtained by adjoining $$\sqrt {m_1 } $$ ,..., $$\sqrt {m_t } $$ , where m1,...,mt are rational integers. We get a system of generators of the minus part of the Stickelberger ideal of K, and calculate its index. This index is described as a product of some determinants of rational integer components. From this result, it is shown that the relative class number of K is expressed as this index.

Abstract: We give an intuitive account of the concepts in the title, by considering the following simple number-theoretic example. Imagine two distant players who communicate by exchanging binary messages (bits). One player is given a prime number x, and the second a composite number y, where x,y < 2". The players' task is to find a prime number p, with p < 2n, such that x ^ y (mod p). The existence of such a small prime p is guaranteed by the prime number theorem and the Chinese remainder theorem. The players agree beforehand on a "protocol" for exchanging messages. The protocol dictates to each player what message to send at each point, based on his input and the messages he received so far. It also dictates when to stop, and how do determine the answer from the information received. There is no limit on the computational complexity of these decisions, which are free of charge. The cost of the protocol is the number of bits they have to exchange on the worst case choice of inputs. We shall be interested in the cost of the best protocol under this measure, which we denote by t(n). There is a trivial protocol in which one player sends his input to the second (n bits), who computes the answer and sends it (log n bits) back to the first. This shows that t(n) <,n + log n. How small can t(n) be? Is it possible that t(n) = 0(log n), which is (essentially) the trivial lower bound? At present, these trivial upper and lower bounds are the best known ! Why should anyone take the time to think about this problem, besides its innocently simple statement and the challenge of the exponential gap in our knowledge? The reason is that this information theoretic problem encodes the computational difficulty of primality testing! Answering it is extremely important for computational number theory and theoretical computer science as follows: