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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1277 (4 self)
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. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical
Accelerated Search for Gaussian Generator Based on Triple Prime Integers 1
"... Abstract: Problem statement: Modern cryptographic algorithms are based on complexity of two problems: Integer factorization of real integers and a Discrete Logarithm Problem (DLP). Approach: The latter problem is even more complicated in the domain of complex integers, where Public Key Cryptosystems ..."
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(generator) in a domain of complex integers modulo triple prime p was provided in this study. It showed the properties of triple primes, the frequencies of their occurrence on a specified interval and analyzed the efficiency of the proposed algorithm. Conclusion: Numerous computer experiments
EXPLICIT FACTORIZATION OF PRIME INTEGERS IN QUARTIC NUMBER FIELDS DEFINED BY X4 + aX + b
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The Divisibility of Integers and Integer Relatively Primes
, 2003
"... We introduce the following notions: 1) the least common multiple of two integers (lcm(i, j)), 2) the greatest common divisor of two integers (gcd(i, j)), 3) the relative prime integer numbers, 4) the prime numbers. A few facts concerning the above items, among them a socalled Fundamental Theorem of ..."
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Cited by 3 (0 self)
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We introduce the following notions: 1) the least common multiple of two integers (lcm(i, j)), 2) the greatest common divisor of two integers (gcd(i, j)), 3) the relative prime integer numbers, 4) the prime numbers. A few facts concerning the above items, among them a socalled Fundamental Theorem
The Divisibility of Integers and Integer Relatively Primes 1
"... Summary. We introduce the following notions: 1) the least common multiple of two integers (lcm(i, j)), 2) the greatest common divisor of two integers (gcd(i, j)), 3) the relative prime integer numbers, 4) the prime numbers. A few facts concerning the above items, among them a socalled Fundamental T ..."
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Summary. We introduce the following notions: 1) the least common multiple of two integers (lcm(i, j)), 2) the greatest common divisor of two integers (gcd(i, j)), 3) the relative prime integer numbers, 4) the prime numbers. A few facts concerning the above items, among them a socalled Fundamental
Prime factors of consecutive integers
 Math. Comp
"... A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, ..."
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Cited by 3 (1 self)
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A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem
Computationally private information retrieval with polylogarithmic communication
 Advances in Cryptology—EUROCRYPT ’99
, 1999
"... We present a singledatabase computationally private information retrieval scheme with polylogarithmic communication complexity. Our construction is based on a new, but reasonable intractability assumption, which we call the ΦHiding Assumption (ΦHA): essentially the difficulty of deciding whether a ..."
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Cited by 256 (2 self)
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a small prime> 2 divides ϕ(m), where m is a composite integer of unknown factorization. Our result also implies the existence of tworound CS proof systems under a concrete complexity assumption. Keywords: Integer factorization, Euler’s function, Φhiding assumption, private information retrieval
NTRU: A RingBased Public Key Cryptosystem
 Lecture Notes in Computer Science
, 1998
"... . We describe NTRU, a new public key cryptosystem. NTRU features reasonably short, easily created keys, high speed, and low memory requirements. NTRU encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability the ..."
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Cited by 205 (4 self)
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theory. The security of the NTRU cryptosystem comes from the interaction of the polynomial mixing system with the independence of reduction modulo two relatively prime integers p and q. Contents 0. Introduction 1. Description of the NTRU algorithm 1.1. Notation 1.2. Key Creation 1.3. Encryption 1
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