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Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure
Algebraic-group factorization algorithm
Algebraic-group_factorization_algorithm
Decomposition of a number into a product
called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer
Integer_factorization
Integer factorization algorithm
theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by
Williams's_p_+_1_algorithm
Probabilistic algorithm for computing discrete logarithms
for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } Using an integer factorization algorithm optimized for smooth numbers, try to factor g k mod q {\displaystyle
Index_calculus_algorithm
Accomplishments in factoring large integers
factored. In February 2020, the factorization of the 829-bit (250-digit) RSA-250 was completed. In September 2026, the factorization of the 862-bit (260-digit)
Integer_factorization_records
Algorithms for matrix decomposition
matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a
Non-negative matrix factorization
Non-negative_matrix_factorization
Field of mathematics
numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition
Numerical_linear_algebra
Algorithm for integer factorization
elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
distinct-degree factorization algorithm, Rabin's algorithm is based on the lemma stated above. Distinct-degree factorization algorithm tests every d not
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Algorithm for computing greatest common divisors
essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic
Euclidean_algorithm
Matrix decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of
QR_decomposition
Scientific area at the interface between computer science and mathematics
computer science, computer algebra, also called symbolic computation or algebraic computation, is the study and development of algorithms and software for manipulating
Computer_algebra
Mathematical software
algebraic decomposition Quantifier elimination over real numbers via cylindrical algebraic decomposition Mathematics portal List of computer algebra systems
Computer_algebra_system
the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p = p 1 p 2 2 ⋯ p k k {\displaystyle p=p_{1}p_{2}^{2}\cdots
Polynomial_root-finding
Branch of number theory
Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields
Algebraic_number_theory
Special-purpose algorithm for factoring integers
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning
Pollard's_p_−_1_algorithm
Unsolved problem in computer science
efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer
P_versus_NP_problem
Mathematical technique
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that
Matrix factorization of a polynomial
Matrix_factorization_of_a_polynomial
Branch of mathematics that studies the properties of groups
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known
Group_theory
Algebraic structure
Dedekind domains, which allows replacing unique factorization of elements with unique factorization of ideals. In particular, many Z [ ζ p ] , {\displaystyle
Principal_ideal_domain
numbers AKS primality test Pollard's p − 1 algorithm Pollard's rho algorithm Lenstra elliptic curve factorization Quadratic sieve Special number field sieve
List_of_number_theory_topics
Commutative ring with a Euclidean division
efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra. It is
Euclidean_domain
squares Dixon's algorithm Fermat's factorization method General number field sieve Lenstra elliptic curve factorization Pollard's p − 1 algorithm Pollard's
List_of_algorithms
Computer system for solving algebra problems
a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma
Magma (computer algebra system)
Magma_(computer_algebra_system)
Discrete Fourier transform algorithm
complex-number arithmetic to group theory and number theory. The best-known FFT algorithms depend upon the factorization of n, but there are FFTs with
Fast_Fourier_transform
Branch of mathematics
empirical sciences. Algebra is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure is a non-empty
Algebra
Computer hardware technology that uses quantum mechanics
cryptography systems: efficient factorization would compromise RSA, while an efficient discrete-logarithm algorithm would compromise Diffie–Hellman key
Quantum_computing
Number divisible only by 1 and itself
although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can
Prime_number
Algorithm for solving the discrete logarithm problem
Cohen, A course in computational algebraic number theory, Springer, 1996. D. Shanks, Class number, a theory of factorization and genera. In Proc. Symp. Pure
Baby-step_giant-step
Approach to public-key cryptography
in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. The use of elliptic
Elliptic-curve_cryptography
Problem of inverting exponentiation in groups
usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them proportional to
Discrete_logarithm
Algebraic structure
commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings
Polynomial_ring
Fast Fourier Transform algorithm
The prime-factor algorithm (PFA), also called the Good–Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the
Prime-factor_FFT_algorithm
Cryptanalytic attacks using a system of multivariate equations
Algebraic attack is a method of algebraic cryptanalysis by which a set of algebraic equations can be used to solve a cryptographic Boolean function that
Algebraic_attack
String that is strictly smaller in lexicographic order than all of its rotations
suffix of the given string. A factorization into a nonincreasing sequence of Lyndon words (the so-called Lyndon factorization) can be constructed in linear
Lyndon_word
efficiently than in Schoof's algorithm. In the case of an Atkin prime, we can gain some information from the factorization pattern of Φ l ( X , j ( E )
Schoof–Elkies–Atkin_algorithm
Branch of mathematical statistics
Algebraic statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric, and combinatorial methods in statistics. While
Algebraic_statistics
Algorithm to multiply matrices
"Communication-optimal parallel 2.5D matrix multiplication and LU factorization algorithms" (PDF). Proceedings of the 17th International Conference on Parallel
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Algorithm to be run on quantum computers
theory. Quantum algorithms may also be grouped by the type of problem solved; see, e.g., the survey on quantum algorithms for algebraic problems. The quantum
Quantum_algorithm
Algebraic study of differential equations
Horowitz-Ostrogradsky algorithm, squarefree factorization and splitting factorization to special and normal polynomials. Differential algebra can determine if
Differential_algebra
Algebra of formal sums
combination is its exponent in the factorization of a {\displaystyle a} , or the negation of its exponent in the factorization of b {\displaystyle b} . The
Free_abelian_group
Computer algebra system
Algebraic Computation (SIGSAM '89). ACM. pp. 207–211. Claire Dicrescenzo; Dominique Duval (1989). P. Gianni (ed.). Algebraic extensions and algebraic
Axiom (computer algebra system)
Axiom_(computer_algebra_system)
Algorithm in the theory of elliptic curves
Tate's algorithm takes as input an integral model of an elliptic curve E over Q {\displaystyle \mathbb {Q} } , or more generally an algebraic number field
Tate's_algorithm
Commutative algebra studies commutative rings, their ideals, and modules over such rings
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings
List of commutative algebra topics
List_of_commutative_algebra_topics
Algorithm used in modular arithmetic
computational problem equivalent to integer factorization. An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli
Tonelli–Shanks_algorithm
Branch of mathematics that studies algebraic structures
Multiplicative group Primitive element (field theory) Field extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure
List of abstract algebra topics
List_of_abstract_algebra_topics
Digital verification standard
The Digital Signature Algorithm (DSA) is a public-key cryptosystem and Federal Information Processing Standard for digital signatures, based on the mathematical
Digital_Signature_Algorithm
Group of symmetries of an n-dimensional hypercube
element. An element c of a reflection group W is said to be a quasi-Coxeter element if there is a factorization c = t 1 ⋅ t 2 ⋯ t k {\displaystyle c=t_{1}\cdot
Hyperoctahedral_group
Area of mathematics
computational algebraic topology Cryptography and computer security, which involve, in particular, research on primality testing, factorization, elliptic
Computational_mathematics
American mathematician (1947–2020)
elliptic curve method of factorization, which include a method for speeding up the second stage of algebraic-group factorization algorithms using FFT techniques
Peter Montgomery (mathematician)
Peter_Montgomery_(mathematician)
Orthonormalization of a set of vectors
mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or
Gram–Schmidt_process
Curve defined as zeros of polynomials
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in
Algebraic_curve
Study of the properties of codes and their fitness
computationally secure; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these solutions to be
Coding_theory
Mathematical function, inverse of an exponential function
relation aids in analyzing the performance of algorithms such as quicksort. Real numbers that are not algebraic are called transcendental; for example, π
Logarithm
Matrix decomposition in mathematics
general linear group is replaced by some other reductive algebraic group, due to Alexander Grothendieck (1957). Birkhoff factorization follows from the
Birkhoff_factorization
Generalization of Gaussian integers to quaternions
quaternion can be factored as a product of irreducible quaternions. This factorization is not in general unique, even up to units and order, because a positive
Hurwitz_quaternion
Theorem classifying finite simple groups
finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since
Classification of finite simple groups
Classification_of_finite_simple_groups
Set with associative invertible operation
general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups. The
Group_(mathematics)
Algebra based on a vector space with a quadratic form
the spin group is not simply connected. In this case the algebraic group Spinp,q is simply connected as an algebraic group, even though its group of real
Clifford_algebra
Theorems that help decompose a finite group based on prime factors of its order
recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system. Frattini's argument
Sylow_theorems
Topics referred to by the same term
a matrix group with integer entries Congruence of squares, in number theory, a congruence commonly used in integer factorization algorithms Matrix congruence
Congruence
Practice and study of secure communication techniques
"computationally secure". Theoretical advances (e.g., improvements in integer factorization algorithms) and faster computing technology require these designs to be continually
Cryptography
Complex number whose real and imaginary parts are both integers
and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers
Gaussian_integer
Infinite dimensional Lie group
Birkhoff factorization of loops in the character group of the associated Hopf algebra. The models considered by Kreimer (1999) had Hopf algebra H and character
Butcher_group
Branch of algebra
commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are
Ring_theory
Number without repeated prime factors
square-free factorization is not only easier to compute than the complete factorization, but it is the first step of all standard factorization algorithms. The
Square-free_integer
Algebraic structure with addition and multiplication
influenced by problems and ideas of algebraic number theory and algebraic geometry. In turn, commutative algebra is a fundamental tool in these branches
Ring_(mathematics)
Operations: TF – triangular factorizations (LU, Cholesky) OF – orthogonal factorizations (QR, QL, generalized factorizations) EVP – eigenvalue problems
Comparison of linear algebra libraries
Comparison_of_linear_algebra_libraries
Best results achieved to date
computation was performed simultaneously with the factorization of RSA-240, using the Number Field Sieve algorithm and the open-source CADO-NFS software. The
Discrete_logarithm_records
Branch of pure mathematics
in the product. The unique factorization theorem is the fundamental theorem of arithmetic that relates to prime factorization. The theorem states that every
Number_theory
Algorithmic runtime requirements for common math procedures
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Computation modulo a fixed integer
calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic. It is used by the
Modular_arithmetic
Representation of a matrix as a product
the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices
Matrix_decomposition
Type of mathematical expression
algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are
Polynomial
Cryptography secured against quantum computers
Most widely used public-key algorithms rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm
Post-quantum_cryptography
Cholesky factorization — sparse approximation to the Cholesky factorization Incomplete LU factorization — sparse approximation to the LU factorization Uzawa
List of numerical analysis topics
List_of_numerical_analysis_topics
Clustering methods
Sibylle; Morik, Katharina (2018). The Relationship of DBSCAN to Matrix Factorization and Spectral Clustering (PDF). LWDA. pp. 330–334. Kannan, Ravi; Vempala
Spectral_clustering
Dutch mathematician (born 1949)
ISBN 0-262-22038-5. Algorithms in Algebraic Number Theory. Bulletin of the AMS, vol. 26, 1992, pp. 211–244. Primality testing algorithms. Séminaire Bourbaki
Hendrik_Lenstra
Arithmetic operation, inverse of nth power
414213562… All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers. A square root of a number x is a number
Nth_root
computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Factorization algorithm
the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity
General_number_field_sieve
Arithmetical operation
The numbers to be multiplied are generally called the factors (as in factorization). The number to be multiplied is the multiplicand, and the number by
Multiplication
Operation in graph theory
corresponding graphs cannot be decomposed. In this sense, the failure of unique factorization on (possibly disconnected) graphs is akin to the statement that polynomials
Cartesian_product_of_graphs
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
develop earliest known algorithms for multiplying two numbers c. 1600 BC – Babylonians develop earliest known algorithms for factorization and finding square
Timeline_of_algorithms
History of a branch of mathematics
various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Joseph Louis Lagrange
History_of_group_theory
Complex number whose mapping on a coordinate plane produces a triangular lattice
Z[ω] of algebraic integers in the algebraic number field Q(ω) – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers
Eisenstein_integer
In mathematics, invariant of square matrices
theorem in several variables. In algebraic geometry, the Hessian determinant allows finding inflexion points of an algebraic plane curve. Square matrices
Determinant
Class of mathematical software
sparse tensor factorization. SPLATT ships a stand-alone executable, C/C++ library, and Octave/MATLAB API. Cadabra is a computer algebra system (CAS) designed
Tensor_software
Array of numbers
ISBN 9783540307174 Maxwell, E. A. (1969), Algebraic Structure and Matrices, Being Part II of Advanced Algebra, Cambridge University Press McHugh, Andrew
Matrix_(mathematics)
Field of knowledge
(not only algebraic ones). At its origin, it was introduced, together with homological algebra, to allow the algebraic study of non-algebraic objects such
Mathematics
theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List
List_of_theorems
Concept in modular arithmetic
known and the most efficient known computation requires m's factorization. Factorization is widely believed to be a computationally hard problem. However
Modular multiplicative inverse
Modular_multiplicative_inverse
Type of group in abstract algebra
abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation
Symmetric_group
Ring ideal generated by a single element of the ring
every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem
Principal_ideal
Product of numbers from 1 to n
multiplication algorithm, and a third comes from the divide and conquer. Even better efficiency is obtained by computing n! from its prime factorization, based
Factorial
Area of discrete mathematics
that the automorphism group of each of them is isomorphic to a finite group. Algebraic graph theory also studies the algebraic invariants, chromatic polynomial
Graph_theory
Topics referred to by the same term
Domain of holomorphy of a function Domain of an algebraic structure, the set on which the algebraic structure is defined Domain of discourse, the set
Domain
Subfield of computer science and mathematics
Computer algebra, also called symbolic computation or algebraic computation is a scientific area that refers to the study and development of algorithms and
Theoretical_computer_science
Multiplication algorithm
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen
Schönhage–Strassen_algorithm
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