Abstract: The present invention provides a method of implementing digital signatures or verification and a privacy communication using the following: (1) choose a positive integer d such that gives an imaginary quadratic field Q{(-d) 1/2 } a small class number, and choose a prime number p such that is expressed by 2 t -α (where α is a small number), such that either p+1-a or p+1+a is divisible by a prime number, and such that 4p-a 2 =d·b 2 to construct a finite field GF(p), or a definition filed of an elliptic curve, then construct an elliptic curve over the finite filed GF(p) having a root for a class polynomial H d (x)=0 modulo GF(p) as j-invariant; (2) construct elliptic curves E 1 , E 2 , . . . , E n in such a way that each will be not isomorphic but have a same number of elements to replace an elliptic curve one from the others.

Abstract: We investigate the computational power of depth two circuits consisting of MODr--gates at the bottom and a threshold gate at the top (for short, threshold--MODr circuits) and circuits with two levels of MOD gates (MODp-MODq circuits.) In particular, we will show the following results (i) For all prime numbers p and integers qr it holds that if p divides r but not q then all threshold--MODq circuits for MODr have exponentially many nodes. (ii) For all integers r all problems computable by depth two ANDORNOT --circuits of (quasi) polynomial size can be represented by threshold--MODr circuits with (quasi)poly\-no\-mially many edges. (iii) There is a problem computable by depth three ANDORNOT --circuits of linear size and constant bottom fan--in which for all r needs threshold--MODr circuits with exponentially many nodes. (iv) For pr different primes, and q2 k positive integers, where p does not divide q every MODpk-MODq circuit for MODr has exponentially many nodes...

Abstract: A new method of cryptologic attack on binary sequences is given, using their linear complexities relative to odd prime numbers. We show that, relative to a particular prime number p, the linear complexity of a binary geometric sequence is low. It is also shown that the prime p can be determined with high probability by a randomized algorithm if a number of bits much smaller than the linear complexity is known. This determination is made by exploiting the imbalance in the number of zeros and ones in the sequences in question, and uses a new statistical measure, the partial imbalance.

Abstract: Many cryptographic algorithms use number theory. They share the problem of generating large primes with a given (fixed) number n of bits. In a series of articles, Brandt, Damgard, Landrock and Pomerance address the problem of optimal use of probabilistic primality proofs for generation of cryptographic primes. Maurer proposed using the Pocklington lemma for generating provable primes. His approach loses efficiency due to involved mechanisms for generating close to uniform distribution of primes. We propose an algorithm which generates provable primes and can be shown to be the most efficient prime generation algorithm up to date. This is possible at the cost of a slight reduction of the set of primes which may be produced by the algorithm. However, the entropy of the primes produced by this algorithm is assymptotically equal to the entropy of primes with random uniform distribution. Primes are sought in arithmetic progressions and proved by recursion. Search in arithmetic progressions allows the use of Eratosthenes sieves, which leads finaly to saving 1/3 of the psuedo prime tests compared to random search.

Abstract: textIn this paper the family of elliptic curves over Q given by the equation y2 = (x + p)(x2 + p2) is studied. It is shown that for p a prime number = ±3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2, Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p = 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p = 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve arc constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication. Copyright

Abstract: We show that versions of the prime number theorem as well as equivalent statements hold in an arbitrary model ofIΔ0+exp.

Abstract: For pt. I, see ibid., vol. 41, no. 10, p. 641-55 (1994). In Part I of the research work, we introduced an extension to the well known Chinese remainder theorem for processing polynomials with coefficients defined over a finite integer ring. We term this extension as the American-Indian-Chinese extension of the Chinese remainder theorem. A systematic procedure for factorizing a monic polynomial into pairwise relatively prime monic factor polynomials over integer rings was presented. This factorization is based on the corresponding factor polynomials, monic and relatively prime, over the associated finite field containing prime number of elements. In this paper, we study the application of the theory developed in Part I to deriving computationally efficient algorithms for performing tasks having multilinear form. Especially, we focus on the cyclic and acyclic convolution as they are two of the most frequently occurring computationally intensive tasks in digital signal processing. >

Abstract: Let p be a prime number We shall show that there is a finite unramified covering C of the affine line A1, defined over the field of p elements, so that A1 is the quotient of C by the fixed-point free action of SL n (Fpk)

Abstract: In this paper, we compute the action of the mod $p$ Steenrod operations on the modular invariants of the linear groups with $p$ an odd prime number.

Abstract: In this paper we present two vectorized numerical sieve algorithms for the number theoretical functions μ(n) and τ (n). These sieve algorithms are generalizations of Eratosthenes’ sieve for finding prime numbers. We show algorithms for fast systematic computations on Mertens’ conjecture and Dirichlet’s divisor problem. We have implemented the algorithm for Mertens’ conjecture on a Cray C90 and performed a systematic computation of extremes of M(x)/ √ x up to 10. We established the bounds −0.513 < M(x)/ √ x < 0.571, valid for 200 < x ≤ 10.

Abstract: In class, students are often curious about the fact that any even integer bigger than 4 is the sum of two prime numbers. They usually cannot understand why the obvious fact cannot be proved mathematically. From the computational point of view, unless we find an even number which cannot be expressed as a sum of two prime numbers, we cannot claim the Goldbach conjecture is true. So, the most important thing is to try all the even integers to check if they conform to the Goldbach conjecture. Unfortunately, most of the modern verification methods need deep mathematical ideas which usually cannot be understood by undergraduate students. In this paper, we present a simple but elegant computer verification method for the Goldbach conjecture, and propose two implementation approaches (via C and Maple) to this method. Compared with other verification methods, ours is easy to implement and easy to understand. But of course, at present we may not expect this method, or probably any other existing methods, to be able...

Abstract: Here {cn}n=1 and {rn}n=0 are given sequences of complex numbers on which we shall impose some mild growth conditions. The equation (1.1) expresses each term of the sequence {an} as an average of the previous terms, weighted by the coefficients ck, plus a remainder term rn. One might expect that this repeated averaging process induces some degree of regularity on the behavior of an. We shall show that this is indeed the case, even if we impose no regularity conditions on the behavior of cn. Our motivation for this work came from two directions. The first is the theory of multiplicative arithmetic functions, where Wirsing [Wi], Halasz [Ha1, Ha2] and others have developed deep and powerful techniques to study the asymptotic behavior of the averages m(x) := e−x ∑ n ex f(n) of a multiplicative function f . As was observed by Wirsing, these averages satisfy integral equations of the type

Abstract: We investigate the computational power of depth-2 circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold-MOD r circuits) and circuits with two levels of MOD gates ( MOD p -MOD q circuits). In particular, we will show the following results: 1. (i) For all prime numbers p and integers q , r , it holds that if p divides r but not q then all threshold-MOD q circuits for MOD r have exponentially many nodes. 2. (ii) For all integers r , all problems computable by depth-2 AND,OR,NOT circuits of polynomial size have threshold-MOD r circuits with polynomially many edges. 3. (iii) There is a problem computable by depth 3 AND,OR,NOT circuits of linear size and constant bottom fan-in which for all r needs threshold-MOD r circuits with exponentially many nodes. 4. (iv) For p , r different primes, and q ⩾ 2, k positive integers, where r does not divide q , every MOD p k -MOD q circuit for MOD r has exponentially many nodes. Results (i) and (iii) imply the first known exponential lower bounds on the number of nodes of threshold-MOD r circuits, r ≠ 2. They are based on a new method for estimating the minimum length of threshold realizations over predefined function bases, which, in contrast to previous related techniques (Goldmann et al., 1992; Bruck and Smolensky, 1990; Kailath et al., 1991; Goldmann, 1993; Grolmusz, 1993) works even if the weight of the realization is allowed to be unbounded, and if the bases are allowed to be nonorthogonal. The special importance of result (iii) consists of the fact that the known spectral-theoretically based lower bound methods for threshold-XOR circuits (Bruck and Smolensky, 1990; Kailath et al., 1991) can provably not be applied to AC 0 functions. Thus, by (ii), result (iii) is sharp. It gives a partial negative answer to the open question whether there exist simulations of AC 0 -circuits by small depth threshold circuits which are more efficient than that given by Yao's important result that ACC functions have depth-3 threshold circuits of quasipolynomial weight (Yao, 1990). Finally we observe that our method works also for MOD p -MOD q circuits, if p is a power of a prime ((iv) above); see (Barrington et al., 1990; Krause and Waack, 1991; Yan and Parberry, 1994) for related results. A preliminary version of this paper appeared in (Krause and Pudlak, 1993).

Abstract: There are three famous unsolved mathematical problems in number theory, namely the theory of partitions, Fermat's 'Last Theorem', and the prime number theorem A geometrical method and an analytical method of predicting the distribution of primes and non-primes based on visual information obtainable from a matrix map of divisibles is described Past investigations tend to concentrate on properties of the prime numbers The author feels that much information could be gathered by studying the distribution of both prime and non-prime numbers using a matrix map Both methods give accurate, deterministic mathematical models of the distribution of primes and non-primes globally The only problem is that there is no end to the prime number series and thus the prime number theorem remains unsolved

Abstract: We prove in this paper a structure theorem for prime rings whose symmetric ring of quotients has nonzero socle. Then this result is applied to prime rings satisfying a generalized identity, and to prime rings having an alternate involution.

Abstract: A new large amicable pair (a, b) = (59554936495441481044788091271148664944796300859243635311219048448, 59554936495441891385123332422108719776971992921810832072976105472) = (2 47 · 9288811670405087 · 145135534866431 · 313887523966328699903, 2 47 · 9288811670405087 · 45556233678753109045286896851222527) was found on 11 August 1993 at the University of York Computing Centre in England by using an analogue of Thabit's well-known rule. Both a and b are 65-digit integers.

Abstract: (1994). Prime Number Records. The College Mathematics Journal: Vol. 25, No. 4, pp. 280-290.

Abstract: Several results of number theory can be expressed in probabilistic terms and, for others, the simplest proof is by probabilistic methods. Simply take the uniform distribution on the consecutive integers 1, 2,…, N. Then arithmetic functions, when restricted to the integers 1 through N, become random variables and arithmetic means are expectations. The power of probabilistic methods lies in the fact that divisibility by distinct primes are almost independent events. On the other hand, most problems remain challenging since the errors generated by the not exact independence can be dominating in a problem when one faces an increasing number of primes. The best example is the study of large prime divisors where the results do not resemble those which one would get for independent random variables.

Abstract: A “theoretical” distribution of prime number gaps is proposed and compared with the actual distribution. Some probabilistic discussions are given.

Abstract: Completely entangled quantum states are shown to factorize into tensor products of entangled states whose dimensions are powers of prime numbers. The entangled states of each prime-power dimension transform among themselves under a finite Heisenberg group. We are thus led to examine processes in which factors are exchanged between entangled states and so consider canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension.

Abstract: PURPOSE:To obtain a signature/certification/secret communication system using an elliptical curve capable of reducing data amount and computing amount compared with a conventional system. CONSTITUTION:A positive integer (d) is taken so as to decrease the class number of an imaginary quadratic field Q((-d) ). A prime number (p) is taken to be representable by p=db +db+(d+1)/4. An elliptical curve E on a finite field GF (p) and a base are taken. A cryptographic system based on a discrete logarithmic problem can be comprised by using such elliptical curve and base point.