Related papers: Pythagorean Triples and Cryptographic Coding
This article presets a review of the achievements rapidly developing field of cryptography - public-key cryptography based on the lattice theory. Paper contains the necessary basic concepts and the major problems of the lattice theory, as…
In this paper, we provide properties and applications of some special integer sequences. We generalize and give some properties of Pisano period. Moreover, we provide a new application in Cryptography and applications of some quaternion…
Cryptographic systems are derived using units in group rings. Combinations of types of units in group rings give units not of any particular type. This includes cases of taking powers of units and products of such powers and adds the…
Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…
These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thi\`es, Senegal. They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting…
The goal of this chapter is to present a survey of homomorphic encryption techniques and their applications. After a detailed discussion on the introduction and motivation of the chapter, we present some basic concepts of cryptography. The…
It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that…
General cryptographic schemes are presented where keys can be one-time or ephemeral. Processes for key exchange are derived. Public key cryptographic schemes based on the new systems are easily established. Authentication and signature…
We had recently shown that every positive integer can be represented uniquely using a recurrence sequence, when certain restrictions on the digit strings are satisfied. We present the details of how such representations can be used to build…
The classification of quantum symmetric-key encryption protocol is presented. According to five elements of a quantum symmetric-key encryption protocol: plaintext, ciphertext, key, encryption algorithm and decryption algorithm, there are 32…
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such…
Nowadays, using cryptographic systems play an effective role in security and safety technologies. One of the most applied kind of cryptography is Symmetric Cryptography and its applications. New aspects of symmetric Cryptography…
Theoretical computer science has found fertile ground in many areas of mathematics. The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to…
This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers…
Quantum computers impose an immense threat to system security. As a countermeasure, new cryptographic classes have been created to prevent these attacks. Technologies such as post-quantum cryptography and quantum cryptography. Quantum…
Public-key cryptography has become a popular way to motivate the teaching of concepts in elementary number theory, abstract algebra, and introduction to proof courses, as well as in cryptography courses. Unfortunately, many experts expect…
The principal pivot transform (PPT) of a matrix A partitioned relative to an invertible leading principal submatrix is a matrix B such that A [x_1^T x_2^T]^T = [y_1^T y_2^T]^T if and only if B [y_1^T x_2^T]^T = [x_1^T y_2^T]^T, where all…
Isogenies, the mappings of elliptic curves, have become a useful tool in cryptology. These mathematical objects have been proposed for use in computing pairings, constructing hash functions and random number generators, and analyzing the…
This paper presents a recursive hiding scheme for 2 out of 3 secret sharing. In recursive hiding of secrets, the user encodes additional information about smaller secrets in the shares of a larger secret without an expansion in the size of…
In this expository article we present an overview of the current state-of-the-art in post-quantum group-based cryptography. We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic…