A heuristic asymptotic formula concerning the distribution of prime numbers
Paul T. Bateman,Roger A. Horn +1 more
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In this article, the authors consider polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **, hk respectively, and assume that each polynomial is irreducible over the field of rational numbers and no two of them differ by a constant factor.Abstract:
Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these polynomials is irreducible over the field of rational numbers and no two of them differ by a constant factor. Let Q(fi , f2, ... , fk ; N) denote the number of positive integers n between 1 and IV inclusive such that fi(n) , f2(n), , fk(n) are all primes. (We ignore the finitely many values of n for which some fi(n) is negative.) Then heuristically we would expect to have for N largeread more
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Design and Analysis of Practical Public-Key Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack.
Ronald Cramer,Victor Shoup +1 more
TL;DR: A new public-key encryption scheme, along with several variants, is proposed and analyzed that appear to be the first public- key encryption schemes in the literature that are simultaneously practical and provably secure.
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A Computational Introduction to Number Theory and Algebra
TL;DR: The introductory book as discussed by the authors emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience, and it alternates between theory and applications in order to motivate and illustrate the mathematics.
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A Taxonomy of Pairing-Friendly Elliptic Curves
TL;DR: This paper gives a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature and provides recommendations as to which pairing- friendly curves to choose to best satisfy a variety of performance and security requirements.
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Signature schemes based on the strong RSA assumption
Ronald Cramer,Victor Shoup +1 more
TL;DR: In this article, a new digital signature scheme is proposed, which is quite efficient, does not require the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the so-called strong RSA assumption.
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Primes in tuples I
TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
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Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
G. H. Hardy,J. E. Littlewood +1 more
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Sur certaines hypothèses concernant les nombres premiers
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Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers"
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On the conjecture of Hardy & Littlewood concerning the number of primes of the form ²+
TL;DR: The conjecture that there are infinitely many odd primes has been shown to be true only for the trivial case of a = -k2 as discussed by the authors, where the Legendre symbol does not divide a. But for no a has this been proven.