Quadratic Residues and the Distribution of Primes

1959 ◽  
Vol 13 (68) ◽  
pp. 272 ◽  
Author(s):  
Daniel Shanks
Keyword(s):  
Distribution Of Primes ◽  
Quadratic Residues

scholarly journals Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Riemann Hypothesis ◽  
Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

Identity oriented signature scheme based on quadratic residues

2005 ◽  
Vol 168 (1) ◽  
pp. 235-242 ◽  
Author(s):  
Weidong Qiu ◽  
Kefei Chen
Keyword(s):  
Signature Scheme ◽  
Quadratic Residues

Quadratic Residues

1986 ◽  
pp. 109-156
Author(s):  
Nikolaos Kritikos
Keyword(s):  
Quadratic Residues

scholarly journals A New Symmetric Key Cryptographic Algorithm using Paley Graphs and ASCII Values

2021 ◽  
Vol 297 ◽  
pp. 01046
Author(s):  
Zhour Oumazouz ◽  
Driss Karim
Keyword(s):  
Brute Force ◽  
Strong Regularity ◽  
Quadratic Residues ◽  
Cryptographic Algorithm ◽  
Paley Graph ◽  
Encryption And Decryption ◽  
Symmetric Key ◽  
Brute Force Attack ◽  
The Moment ◽  
Paley Graphs

The main objective of the study conducted in this article is to introduce a new algorithm of encryption and decryption of a sensitive message after transforming it into a binary message. Our proposed encryption algorithm is based on the study of a particular graph constructed algebraically from the quadratic residues. We have exploited the Paley graph to introduce an abstract way of encryption of such message bit according to the other message bits by the intermidiate study of the neighborhood of a graph vertex. The strong regularity of the Paley graphs and the unknown behavior of the quadratic residues will play a very important role in the cryptanalysis part which allows to say that the brute force attack remains for the moment the only way to obtain the set of possible messages.

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