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Decomposition of a number into a product
prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using
Integer_factorization
Accomplishments in factoring large integers
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography
Integer_factorization_records
(Mathematical) decomposition into a product
For example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered
Factorization
Integers have unique prime factorizations
arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Number without repeated prime factors
square-free integers that are pairwise coprime. This is called the square-free factorization of n. To construct the square-free factorization, let n = ∏
Square-free_integer
Quantum algorithm for integer factorization
qubits. In 2012, the factorization of 15 {\displaystyle 15} was performed with solid-state qubits. Later, in 2012, the factorization of 21 {\displaystyle
Shor's_algorithm
Algorithm for public-key cryptography
factoring large integers on a classical computer has yet been found, but it has not been proven that none exists; see integer factorization for a discussion
RSA_cryptosystem
Computational method
algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product
Factorization_of_polynomials
Integer that divides another integer
Arithmetic functions Euclidean algorithm Fraction (mathematics) Integer factorization Table of divisors – A table of prime and non-prime divisors for
Divisor
Mathematical table
followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional
Table of Gaussian integer factorizations
Table_of_Gaussian_integer_factorizations
Factorization method based on the difference of two squares
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2
Fermat's_factorization_method
Number divisible only by 1 and itself
Prime factors calculator can factorize any positive integer up to 20 digits. Fast Online primality test with factorization makes use of the Elliptic Curve
Prime_number
Complex number whose real and imaginary parts are both integers
unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic
Gaussian_integer
Approach to public-key cryptography
used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. The use of elliptic
Elliptic-curve_cryptography
Complexity class used to classify decision problems
polynomial time. The decision problem version of the integer factorization problem: given integers n and k, is there a factor f with 1 < f < k and f dividing
NP_(complexity)
Problem of inverting exponentiation in groups
example). This asymmetry is analogous to the one between integer factorization and integer multiplication. Both asymmetries (and other possibly one-way
Discrete_logarithm
Complexity class
whether there is a polynomial-time algorithm for factorization, equivalently that integer factorization is in P, and hence this example is interesting as
Co-NP
IEEE standardization project for public-key cryptography
and encryption schemes using several mathematical approaches: integer factorization, discrete logarithm, and elliptic curve discrete logarithm. DL/ECKAS-DH1
IEEE_P1363
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
Unsolved problem in computer science
quasi-polynomial time. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision
P_versus_NP_problem
Integer factorization algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and
Pollard's_rho_algorithm
Inherent difficulty of computational problems
perspectives on this. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision
Computational complexity theory
Computational_complexity_theory
Integer factorization algorithm
Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success
Shanks's square forms factorization
Shanks's_square_forms_factorization
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
Cryptography
Set of large semiprimes
decimal digits (330 bits). Its factorization was announced on April 1, 1991, by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple-polynomial
RSA_numbers
Computer hardware technology that uses quantum mechanics
these results in 1994 with polynomial-time quantum algorithms for integer factorization and the discrete logarithm problem. A sufficiently large quantum
Quantum_computing
Mathematical for factoring integers
Fermat's factorization method. The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime
Euler's_factorization_method
Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued
List_of_number_theory_topics
Hypothesis in computational complexity theory
_{i}p_{i}} ). It is a major open problem to find an algorithm for integer factorization that runs in time polynomial in the size of representation ( log
Computational hardness assumption
Computational_hardness_assumption
Key agreement protocol
consisting of a private key d {\displaystyle d} (a randomly selected integer in the interval [ 1 , n − 1 ] {\displaystyle [1,n-1]} ) and a public key
Elliptic-curve_Diffie–Hellman
Challenge for factoring large semiprimes
The factoring challenge was intended to track the cutting edge in integer factorization. A primary application is for choosing the key length of the RSA
RSA_Factoring_Challenge
Integer factorization algorithm
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field
Quadratic_sieve
Cryptography secured against quantum computers
rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm problem, or the elliptic-curve discrete
Post-quantum_cryptography
Cryptographic algorithm for digital signatures
Bézout's identity). Alice creates a key pair, consisting of a private key integer d A {\displaystyle d_{A}} , randomly selected in the interval [ 1 , n −
Elliptic Curve Digital Signature Algorithm
Elliptic_Curve_Digital_Signature_Algorithm
Study of analyzing information systems in order to discover their hidden aspects
constructed problems in pure mathematics, the best-known being integer factorization. In encryption, confidential information (called the "plaintext")
Cryptanalysis
Concept in number theory
theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials
Aurifeuillean_factorization
Integer that is a perfect square modulo some integer
composite moduli whose prime factorization is known. In the case of a composite modulus with unknown prime factorization, the problem of identifying quadratic
Quadratic_residue
Type of integral domain
unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a unique factorization domain
Unique_factorization_domain
Congruence used in integer factorization algorithms
is a congruence commonly used in integer factorization algorithms. Given a positive integer n, Fermat's factorization method relies on finding numbers
Congruence_of_squares
Method of exchanging cryptographic keys
base g = 5 (which is a primitive root modulo 23). Alice chooses a secret integer a = 4, then sends Bob A = ga mod p A = 54 mod 23 = 4 (in this example both
Diffie–Hellman_key_exchange
Algorithm in number theory
theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it
Dixon's_factorization_method
sections 5.3 and 5.4. Lenstra elliptic-curve factorization Galbraith, Steven (2012). "Primality Testing and Integer Factorisation using Algebraic Groups". Mathematics
Algebraic-group factorisation algorithm
Algebraic-group_factorisation_algorithm
Complexity class
PIGEON. There exist polynomial-time randomized reductions from the integer factorization problem to WEAK-PIGEON. Additionally, under the generalized Riemann
PPP_(complexity)
Algebraic curve in mathematics
also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective
Elliptic_curve
a hypothetical integer factorization device described in 1999 by Adi Shamir and purported to be capable of factoring 512-bit integers. It is also a pun
TWINKLE
Quantum-safe key encapsulation mechanism
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
ML-KEM
Prime number of the form 2^n – 1
– Factorization of Mersenne numbers Mn (n up to 1280) Factorization of completely factored Mersenne numbers The Cunningham project, factorization of
Mersenne_prime
Natural number
highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base
2
American mathematician (1947–2020)
use of Montgomery curves in applications of elliptic curves to integer factorization and other problems, and the Montgomery ladder, which is used to
Peter Montgomery (mathematician)
Peter_Montgomery_(mathematician)
Algorithm for determining whether a number is prime
Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is
Primality_test
Group of units of the ring of integers modulo n
, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Integer having a non-trivial divisor
Mathematics portal Canonical representation of a positive integer Integer factorization Sieve of Eratosthenes Table of prime factors Divisor function
Composite_number
Number theory library written in C
various ring arithmetics as well as derived functionality such as integer factorization using a quadratic sieve. The library is designed to be compiled
Fast Library for Number Theory
Fast_Library_for_Number_Theory
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
Euclidean_algorithm
Digital verification standard
1 {\displaystyle p-1} is a multiple of q {\displaystyle q} . Choose an integer h {\displaystyle h} randomly from { 2 … p − 2 } {\displaystyle \{2\ldots
Digital_Signature_Algorithm
In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning
Continued fraction factorization
Continued_fraction_factorization
Number-theoretical function
function that gives the number of representations for a given positive integer n {\displaystyle n} as the sum of k {\displaystyle k} squares, where representations
Sum_of_squares_function
Computation modulo a fixed integer
used to limit the size of integer coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known
Modular_arithmetic
Mathematical scheme for verifying the authenticity of digital documents
Digitalized Signatures and Public Key Functions as Intractable as Factorization (PDF) (Technical report). Cambridge, MA, United States: MIT Laboratory
Digital_signature
Integer factorization algorithm
understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n, the integer to be factored, can
Trial_division
Special-purpose integer factorization algorithm
integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of
Special_number_field_sieve
Mechanism for authenticating cryptographic keys
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Web_of_trust
Unsolved problem in cryptography
sufficiently large (see integer factorization). The RSA key setup routine already turns the public exponent e, with this prime factorization, into the private
RSA_problem
Decomposition of an integer as a sum of positive integers
related to Integer partitions. Rank of a partition, a different notion of rank Crank of a partition Dominance order Factorization Integer factorization Partition
Integer_partition
On finding a repeating loop in a sequence
are possible. The classic example is Pollard's rho algorithm for integer factorization, which searches for a factor p of a given number n by looking for
Cycle_detection
Cross-platform reverse-Polish calculator program
3, 2019. "Advanced Bash-Scripting Guide, Chapter 16, Example 16-52 (Factorization)". Retrieved 2020-09-20. Adam Back. "Diffie–Hellman in 2 lines of Perl"
Dc_(computer_program)
Generalization of the Legendre symbol in number theory
primality testing and integer factorization; these in turn are important in cryptography. For any integer a and any positive odd integer n, the Jacobi symbol
Jacobi_symbol
Special-purpose algorithm for factoring integers
integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with
Pollard's_p_−_1_algorithm
Numbers k where x - phi(x) = k has many solutions
many highly cototient numbers. Computations become harder, since integer factorization becomes harder as the numbers get larger. The cototient of x {\displaystyle
Highly_cototient_number
Algebraic structure
different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is
Polynomial_ring
Field of knowledge
mathematics traces its roots back to Ancient Greece. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical
Mathematics
System that can issue, distribute and verify digital certificates
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Public_key_infrastructure
Natural number
This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization. Among
288_(number)
Algorithm for generating numbers coprime with first few primes
thus be used for an improvement of the trial division method for integer factorization, as none of the generated numbers need be tested in trial divisions
Wheel_factorization
Integer factorization algorithm
In computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms
Williams's_p_+_1_algorithm
Digital signature scheme
keys are elements of G 2 {\displaystyle G_{2}} , and the secret key is an integer in [ 0 , q − 1 ] {\displaystyle [0,q-1]} . Working in an elliptic curve
BLS_digital_signature
Project by NIST to standardize post-quantum cryptography
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
NIST Post-Quantum Cryptography Standardization
NIST_Post-Quantum_Cryptography_Standardization
Smallest positive number divisible by two integers
algorithm for integer factorization. The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the
Least_common_multiple
Factorization algorithm
classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits)
General_number_field_sieve
Mathematical polynomial factorization
irreducible polynomial, so this factorization of infinitely many of its values cannot be extended to a factorization of Φ 4 {\displaystyle \Phi _{4}}
Sophie_Germain's_identity
Algebraic construction
ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example
Ring_of_integers
Number in {..., –2, –1, 0, 1, 2, ...}
Canonical factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued
Integer
Natural number
{\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]} whose ring of integers has a unique factorization, or class number of 1. 9 is the largest single-digit number
9
Computational problem possibly useful for post-quantum cryptography
fundamental base for public-key cryptography in the future just as the integer factorization and discrete logarithm problem have served as the base for public
Ring_learning_with_errors
Public-key cryptosystem
over any cyclic group G {\displaystyle G} , like multiplicative group of integers modulo n if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime
ElGamal_encryption
Natural number
totients. It follows 4,294,967,294 and precedes 4,294,967,296. It has a factorization of 3 ⋅ 5 ⋅ 17 ⋅ 257 ⋅ 65537 {\displaystyle 3\cdot 5\cdot 17\cdot 257\cdot
4,294,967,295
Non-federated cryptographic protocol
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Signal_Protocol
Statement in complex analysis
= g {\displaystyle \rho =g} is an integer. The proof below follows Conway's treatment of Hadamard's factorization theorem. Let f {\displaystyle f} be
Hadamard factorization theorem
Hadamard_factorization_theorem
About products of primitive polynomials
polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to
Gauss's_lemma_(polynomials)
Estimate of time taken for running an algorithm
sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the general number field sieve, which runs in time about 2 O (
Time_complexity
Branch of pure mathematics
composite numbers. Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition
Number_theory
Theorem in complex analysis
and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly
Weierstrass factorization theorem
Weierstrass_factorization_theorem
Subfield of computer science and mathematics
theoretic computations. The best known problem in the field is integer factorization. Cryptography is the practice and study of techniques for secure
Theoretical_computer_science
Integer factorization algorithm
b2 (mod n), which can be turned into a factorization of n = gcd(a + b, n) × gcd(a − b, n). This factorization might turn out to be trivial (i.e. n = n
Rational_sieve
Quantum algorithm for solving systems of linear equations
The Harrow–Hassidim–Lloyd (HHL) algorithm is a quantum algorithm for obtaining certain limited information about the solution to a system of linear equations
HHL_algorithm
List of unsolved computational problems
possible? Log-rank conjecture Hartmanis–Stearns conjecture Can integer factorization be done in polynomial time on a classical (non-quantum) computer
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Asymmetric key encryption algorithm
solved given the factorization of N, while new quadratic residues may be generated by any party, even without knowledge of this factorization. The GM cryptosystem
Goldwasser–Micali cryptosystem
Goldwasser–Micali_cryptosystem
Set of cryptographic algorithms by the NSA
Mode of operation Padding v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali
Commercial National Security Algorithm Suite
Commercial_National_Security_Algorithm_Suite
Irreducible polynomial whose roots are nth roots of unity
p-adic integers, since Hensel's lemma allows lifting a factorization over the field with p elements to a factorization over the p-adic integers. If x takes
Cyclotomic_polynomial
Integer having only small prime factors
factorization of integers. 2-smooth numbers are simply the powers of 2, while 5-smooth numbers are also known as regular numbers. A positive integer is
Smooth_number
INTEGER FACTORIZATION
INTEGER FACTORIZATION
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Boy/Male
Muslim
To wait
Boy/Male
Norse
Son's army.
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Boy/Male
Arabic, Muslim
To Wait
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
INTEGER FACTORIZATION
INTEGER FACTORIZATION
Surname or Lastname
English
English : variant spelling of Dart.
Male
Russian
Variant spelling of Russian Sergei, possibly SERGEY means "sergeant."
Female
English
Variant spelling of English Cherise, CHERICE means "cherry."
Boy/Male
Hindu, Indian
God
Girl/Female
Tamil
Vaidurya | வைதà¯à®°à¯à®¯à®¾
A gem stone
Boy/Male
Bengali, Hindu, Indian
The Enlightened
Girl/Female
Hindu
Exciting
Female
Irish
Pet form of Irish CaitrÃona, CAITIE means "pure."
Girl/Female
African, Arabic, Australian, Indian, Malayalam, Muslim, Sindhi, Swahili
Star; Success; Safety
Boy/Male
Australian, Greek, Latin
Steady; Steadfast; Constant; Diminutive of Constantine
INTEGER FACTORIZATION
INTEGER FACTORIZATION
INTEGER FACTORIZATION
INTEGER FACTORIZATION
INTEGER FACTORIZATION
n.
One who gathers the vintage.
n.
One who intends.
p. pr. & vb. n.
of Inter
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
v. t.
To inhume; to bury; to inter.
v. t.
To inter.
v. t.
To deposit or inter in a chapel; to enshrine.
v. t.
To inter with funeral rites; to bury.
n.
One who makes an index.
imp. & p. p.
of Inter
n.
One who makes an entrance or beginning.
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
v. t.
To place in a tomb; to bury; to inter; to entomb.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
v. t.
To inter again.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
n.
One who inters.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.