Abstract: A public-key data encryption system employing RSA public-key data encryption including a message encrypter capable of encrypting messages using a non-secret encryption key, a transmitter-receiver coupled to the message encrypter which transmits or receives an encrypted message to or from a remote location, the transmitter-receiver also being coupled to a decrypter capable of decrypting a received encrypted message using a decryption key which is a secret input to the decrypter, and an encryption-decryption key generator, including a microprocessor or other large-scale integrated circuit or circuits formed to generate a sequence of prime numbers beginning with a selected known prime number having a length relatively short with respect to the desired length of the last in the sequence of prime numbers, and which is constructed to form the sequence of prime numbers in the form hP+1 where P is the preceding prime number in the sequence, and to test hP+1 for primality by first determining if hP+1 has a GCD of 1 with x, wherein x is a composite number consisting of the product of all known prime numbers less than or equal to a pre-selected known prime number and if the GCD is not equal to 1, incrementing h to form a new hP+1 to be tested for a GCD equal to 1, and when a GCD is found to be 1, performing the primality tests to determine whether 2 hP ≡1 [mod (hP+1)] and 2 h ≢1 [mod (hP+1)], and if either 2 hP ≢1 [mod (hP+1)] or 2 h ≡1 [mod (hP+1)] further incrementing h and so on until a prime is found in this manner and then determining if the length of the prime number is of or greater than the desired length. If the hP+1 which has been determined to be prime is not of the desired length, hP+1 is placed in the sequence of prime numbers and a new h selected to be used to find the next prime number in the sequence in accordance with the above described procedure by forming a new hP+1 in which P is the previously determined prime number in the sequence of prime numbers. When a prime number in the sequence of prime numbers is found which is of the desired length it is input into the encryption-decryption key generator for generating the RSA public-key encryption and decryption keys.

Abstract: This paper discusses a technique for multiplying numbers, modulo a prime number, using look-up tables stored in read-only memories. The application is in the computation of number theoretic transforms implemented in a ring which is isomorphic to a direct sum of several Galois fields, parallel computations being performed in each field.

Abstract: A new algorithm for testing primality is presented. The algorithm is distinguishable from the lovely algorithms of Solvay and Strassen [36], Miller [27] and Rabin [32] in that its assertions of primality are certain (i.e., provable from Peano's axioms) rather than dependent on unproven hypothesis (Miller) or probability (Solovay-Strassen, Rabin). An argument is presented which suggests that the algorithm runs within time c1ln(n)c2ln(ln(ln(n))) where n is the input, and C1, c2 constants independent of n. Unfortunately no rigorous proof of this running time is yet available.

Abstract: An asynchronous data-processing system for multiplying two binary numbers, by a use of read-only memories storing tables of data for transforming the numbers into exponents of a prime number. The exponents are added and then transformed back into the terms of the original numbers system. The transforms produce the product of the two numbers merely by addressing read-only memories and, therefore, accomplish the multiplication at a very high speed. Sophistications of the system compensate for variations in the bit patterns of input signals, inabilities to handle some numbers in the numbering systems, and the like. A use of Fermat prime numbers eliminates rounding errors which occur in systems using logarithmic transforms.

Abstract: For calculating a product of a first and a second integer, each given by even digits, a multiplier comprises a first unit for calculating a first residue congruent to the product modulo a prime number by the use of one-to-one correspondence of each integer to an exponent of a primitive root of the prime number, a second unit for calculating a second residue congruent to the product modulo an even number equal to the prime number less one, and a third unit for processing the first and the second residues to a processed result. A higher and a lower half of digits of the product are given by the processed result and the second residue. Each integer may be given on the basis of a predetermined radix, such as 10 or 2, by h digits with the prime number given by an h-th power of the radix plus one. Preferably, the second residue is calculated by multiplying a higher and a lower half of the digits of the first integer by a higher and a lower half of the second integer and by combining the product in a predetermined manner.

Abstract: In this paper a class of abelian groups (SKT-modules), which includes the torsion totally projective groups, S-groups, and balanced projectives is shown to be a subclass of a projective class of groups with respect to a naturally defined class of short exact sequences called the ch-projective modules and ch-pure sequences, respectively. Every ZP-module has a ch-pure projective resolution and every reduced ch-projective module is a summand of a SKT-module. It is finally shown that a ZP-module M is ch-projective if and only if, for every ordinal a, the two ZP-modules paM and M/p 'M are both ch-projective. 1. Definitions. If p is a prime number then the ring of all rational numbers a/b with b relatively prime top will be denoted by Z. The category of Z4-modules are those abelian groups with the property that multiplication by a prime other than the prime p is an automorphism of the group. If M is a Zp-module then (i) the torsion submodule of M is the maximal torsion subgroup of M; (ii) M is torsion-free if and only if the torsion submodule of M is (0); (iii) M is reduced if and only if M is a reduced group; and (iv) M is divisible if and only if M is a divisible group. Z and Q will denote the groups of integers and rational numbers respectively. The group Ext(Q/Zp, *) will be denoted by c(*). The limit ordinal X is a limit ordinal cofinal with w if there is a sequence of smaller ordinals fAi such that X = sup A.f Otherwise X is said to be not cofinal with

Abstract: For n E Z with \ \\ > l we denote the number of distinct prime divisors of n by ω (n) and the gfeatest prime divisor of n by P (n). Ramachandra, Shorey, Tijdeman, Langevin and the author have given lower estimates for the greatest prime divisor and the number of prime divisors of the product of k consecutive integers. In this paper we derive similar bounds for the case of the product of the values of a polynomial at k consecutive integers. We shall prove the following theorems.

Abstract: A new algorithm for testing primality is presented. The algorithm is distinguishable from the lovely algorithms of Solvay and Strassen [36], Miller [271 and Rabin [32] in that its assertions of primality are certain (i.e., provable from Peano·s axioms) rather than dependent on unproven hypothesis (Miller) or probability (Solovay-Strassen, Rabin). An argument is presented which suggests that the algorithm runs with~n time

Abstract: We now apply the last result of the preceding section to obtain approximations to $$ \psi (x;q,a) = \sum\limits_{{}_{n \equiv a(\bmod q)}n \le x} {\Lambda (n)} $$ (1) .

Abstract: In this paper a class of Abelian groups which includes the torsion totally projective groups, S-groups, and balanced projectives is studied. It is shown that this class of groups has a complete set of invariants. If p is a prime number then the ring of all rational numbers a/b with b relatively prime top will be denoted by Z . The category of ZP-modules are those Abelian groups with the property that multiplication by a prime other than the prime p is an automorphism of the group. If M is a Zp-module then (i) the torsion submodule of M is the maximal torsion subgroup of M; (ii) M is torsion-free if and only if the torsion submodule of M is (0); (iii) M is reduced if and only if M is a reduced group; and (iv) M is divisible if and only if M is a divisible group. Z and Q will denote the groups of integers and rational numbers respectively. The group Ext(Q/ZP, *) will be denoted by c(*). The limit ordinal X is a limited ordinal cofinal with w if there is a sequence of smaller ordinals /A, such that X = sup A3i Otherwise X is said to be not cofinal with

Abstract: A new algorithm for testing primality is presented. The algorithm is distinguishable from the lovely algorithms of Solvay and Strassen [36], Miller [271 and Rabin [32] in that its assertions of primality are certain (i.e., provable from Peano·s axioms) rather than dependent on unproven hypothesis (Miller) or probability (Solovay-Strassen, Rabin). An argument is presented which suggests that the algorithm runs with~n time

Abstract: In “New Directions in Cryptography”, Diffie and Hellman propose a public key distribution (PKD) system based on exponentiation in a discrete arithmetic system. The security of this technique is crucially dependent on the difficulty of computing discrete logarithms (the inverse of the discrete exponential function). Until recently, the best known method for computing discrete logs required running time which grew exponentially in the word size. However, Adleman has recently observed that certain fast algorithms for factoring integers are also applicable to computing discrete logs over GF(q), the Galois field with q elements (q denotes a prime number). He also noted that the running time for the modified algorithm should be of the same form as for factoring, namely

Abstract: Renyi used the large sieve to show that prime numbers are well distributed in arithmetic progressions (mod q) for most q; his rather complicated result allowed him to show that every large even number is representable in the form $$p + {p_1}{p_2} \ldots {p_r},$$ where r is bounded by some absolute constant. The subsequent refinements of Bombieri1 and A. I. Vinogradov2 enable one to take r = 3, and recently Chen3 has added an ingenious new idea to obtain r = 2.

Abstract: PURPOSE:To report the number of times of access to each addresses by multiplying a prime number, set specifying an assigned address, on every passing of each assigned address. CONSTITUTION:Addresses (a), (b), (c) and (d) in memory 10 are assigned to compare address register 12 and the initial value of register 19 is set to ''1''. As for a program flow, access to address (a) is attained three times and that to (b)-(d) is attained similarly. When access to address (a) is attained, a coincidence output is applied from comparator circuit 13a, and consequently multiplexer 14 selects prime signal generating circuit 15a to send a prime signal of ''2'' to multiplying register 16, thereby storing a value of one by two in register 19. Jobs as to (b)-(c) are performed similarly. The final contents of register 19 are easily resolved into the product of respective prime numbers according to the integral prime factor theorem.

Abstract: The survey covers works on the additive number theory during the period 1954–1977. Results pertaining to the classical problems of Goldbach, Hardy-Littlewood, and analogous problems are considered.

Abstract: Definition: (a)G is called hypercyclic «iff each epimorphic imageH≠1 ofG possesses a cyclic normal subgroupA≠1». (b)G is called hypercentral «iff each epimorphic imageH≠1 ofG hasZ(H)≠1». (c) the set of prime numbers which divide the orders of the torsion elements (≠1) ofG is called «the characteristic ofG». Baer has shown that each hypercyclic groupG is a subdirect product of hypercyclic groups of finite characteristic. In this note we will characterize hypercentral groups by abelian torsion groups of finite exponent.

Abstract: Using a generalised duality transformation and symmetry considerations, the authors obtain the phase diagram for Z(N) spin models. Using known properties of the Villain model, they conclude that for N>or=4 there are at least three phases, one of them being soft. For N a prime number, N>3, there are only three types of phases, two being characterised by symmetry arguments, whereas the third one is soft and has all powers of the order and disorder parameters vanishing. For N not a prime number, N>4, there are in addition to this soft phase, phases characterised by non-vanishing powers of the order or disorder parameter, with Z(N') symmetries being broken, where N' is a divisor of N.