Computational number theory is a branch of number theory that focuses on the use of algorithms and computational techniques to solve problems related to integers and their properties. It encompasses a wide range of topics, including but not limited to: 1. **Primality Testing**: Developing algorithms to determine whether a given number is prime. Techniques such as the Miller-Rabin test and the AKS primality test are examples in this area.
Number theoretic algorithms are algorithms that are designed to solve problems related to number theory, which is a branch of mathematics dealing with the properties and relationships of integers. These algorithms often focus on prime numbers, divisibility, modular arithmetic, integer factorization, and related topics. They are fundamental in various fields, especially in cryptography, computer science, and computational mathematics.
ABC@Home is a program that was established by ABC Television Network to allow fans and viewers to engage with their favorite shows and provide feedback from the comfort of their homes. It typically involves activities such as viewing episodes, participating in surveys, and sometimes getting exclusive content or rewards in exchange for their feedback. Programs like this are often designed to gather audience insights, promote viewer loyalty, and enhance the overall television viewing experience.
The Algorithmic Number Theory Symposium (ANTS) is a biennial conference that focuses on the intersection of number theory and computer science, particularly the algorithmic aspects of number theory. It typically brings together researchers and practitioners who are interested in theoretical and practical problems related to algorithms in number theory, including topics like cryptography, computational arithmetic, integer factorization, and more.
A **computational hardness assumption** is a principle or conjecture in cryptography and computer science that posits certain mathematical problems are inherently difficult to solve in a reasonable amount of time, even with the best known algorithms and the most powerful computers available. These assumptions are foundational for the security of various cryptographic systems and protocols.
Evdokimov's algorithm, also known as the Evdokimov method, is primarily associated with computational mathematics and numerical analysis, particularly in the context of iterative methods for solving linear or nonlinear equations. However, there is limited widely accessible detailed documentation specifically referring to an "Evdokimov's algorithm," which may indicate it is not as well-known as other mathematical algorithms.
The Fast Library for Number Theory (FLINT) is a software library designed for efficient computation in number theory. It provides various functionalities for dealing with mathematical objects and operations related to number theory, such as integers, rational numbers, polynomials, matrices, algebraic numbers, and more. The library is optimized for performance and aims to handle large numbers and complex mathematical operations efficiently.
The Higher Residuality Problem, often referred to simply as "higher residuosity," is a concept in number theory and algebraic geometry that deals with the distribution of prime residues in modular arithmetic. Although there may not be a well-defined term widely recognized specifically as "Higher Residuosity Problem," the concept can be explored through related areas. In general, the residuosity problem examines whether certain numbers can be represented as residues modulo a prime or composite number.
The Itoh–Tsujii inversion algorithm is a mathematical method used to compute modular inverses within finite fields, particularly suitable for fields defined by irreducible polynomials over a base field. The algorithm is particularly efficient for computing inverses when dealing with fields of characteristic two, such as binary fields.
The Korkine–Zolotarev (KZ) lattice basis reduction algorithm is an important algorithm in the field of lattice theory, which is a part of number theory and combinatorial optimization. It is specifically designed to find a short basis for a lattice, which can be thought of as a discrete subgroup of Euclidean space formed by all integer linear combinations of a set of basis vectors.
The Lenstra–Lenstra–Lovász (LLL) algorithm is a polynomial-time algorithm for lattice basis reduction. It is named after its creators Arjen K. Lenstra, Hendrik W. Lenstra Jr., and László Lovász, who introduced it in 1982. The algorithm is significant in computational number theory and has applications in areas such as cryptography, coding theory, integer programming, and combinatorial optimization. ### Key Concepts 1.
The Odlyzko–Schönhage algorithm is a computational technique used for the efficient multiplication of large integers. It was developed by mathematicians Andrew Odlyzko and Arnold Schönhage in the context of number theory and computer science, particularly for applications involving large numbers, such as cryptography and scientific computing.
The Phi-hiding assumption is a concept in the field of cryptography, particularly related to public key encryption schemes and their security properties. Specifically, it pertains to the security of certain cryptographic primitives against adaptive chosen ciphertext attacks (CCA). In more detail, the Phi-hiding assumption is concerned with the difficulty of deriving information about the secret key when given a public key and a specific type of value, typically related to the encryption scheme in question.
The Quadratic Residuosity Problem (QRP) is a fundamental problem in number theory and has important implications in cryptography, particularly in the context of certain cryptographic protocols and security mechanisms. ### Definition The Quadratic Residuosity Problem can be defined as follows: Let \( p \) be a prime number, and let \( a \) be an integer such that \( 1 \leq a < p \).
The "Table of costs of operations in elliptic curves" typically refers to a comparative analysis of the computational costs associated with various operations when working with elliptic curves in cryptographic contexts. These costs can vary based on number representation (e.g., binary or affine coordinates), the underlying field (prime or binary fields), and the specific algorithms used.
