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Algorithm to solve the discrete logarithm problem
mathematics, the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic
Function_field_sieve
Topics referred to by the same term
Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function
Function_field
Best results achieved to date
variant of the medium-sized base field function field sieve, for binary fields, to compute a discrete logarithm in a field of 21971 elements. In order to
Discrete_logarithm_records
Ways to estimate the size of sifted sets of integers
sophisticated sieves also do not work directly with sets per se, but instead count them according to carefully chosen weight functions on these sets (options
Sieve_theory
theorem Brun sieve Function field sieve General number field sieve Large sieve Larger sieve Quadratic sieve Selberg sieve Sieve of Atkin Sieve of Eratosthenes
List_of_number_theory_topics
Decomposition of a number into a product
completed with a highly optimized implementation of the general number field sieve run on hundreds of machines. No algorithm has been published that can
Integer_factorization
Problem of inverting exponentiation in groups
the size of the group). Baby-step giant-step Function field sieve Index calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm
Discrete_logarithm
Integer factorization algorithm
quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve).
Quadratic_sieve
Probabilistic algorithm for computing discrete logarithms
{\displaystyle p} is large compared to q {\displaystyle q} , the function field sieve, L q [ 1 / 3 , 32 / 9 3 ] {\textstyle L_{q}\left[1/3,{\sqrt[{3}]{32/9}}\
Index_calculus_algorithm
Pre-generalisation of the fundamental lemma of sieve theory
In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers
Brun_sieve
giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve Greatest common divisor Binary Euclidean Extended Euclidean Lehmer's
Quadratic_Frobenius_test
Natural number
ISBN 0198503415. Gaitsgory, Dennis; Lurie, Jacob (2019). Weil's Conjecture for Function Fields (Volume I). Annals of Mathematics Studies. Vol. 199. Princeton: Princeton
1
Branch of pure mathematics
important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular
Number_theory
Function representing relative sizes of particles in a system
normally only collect very large particles, those that can be separated using sieve trays. Centrifugal collectors will normally collect particles down to about
Particle-size_distribution
Group of similar cells performing a specific function
cells that are nestled between sieve-tube members that function in some manner bringing about the conduction of food. Sieve-tube members that are alive contain
Tissue_(biology)
Number divisible only by 1 and itself
depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization
Prime_number
Norwegian mathematician (1917–2007)
turned to sieve theory, a previously neglected topic which Selberg's work brought into prominence. In a 1947 paper he introduced the Selberg sieve, a method
Atle_Selberg
Integer having only small prime factors
factorization algorithms, for example: the general number field sieve), the VSH hash function is another example of a constructive use of smoothness to
Smooth_number
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Mathematics award
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical
Fields_Medal
Polish-American mathematician (born 1947)
He has made deep contributions to the field of analytic number theory, mainly in modular forms on GL(2) and sieve methods." He became a fellow of the American
Henryk_Iwaniec
American mathematician
correlation conjecture on the zeros of the Riemann zeta function, is known for his development of large sieve methods, and is the author of multiple books on
Hugh_Lowell_Montgomery
Notation describing limiting behavior in computational number theory
c=(64/9)^{1/3}\approx 1.923} . The best such algorithm prior to the number field sieve was the quadratic sieve which has running time L n [ 1 / 2 , 1 ] = e ( 1 + o ( 1
L-notation
Topics referred to by the same term
theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern integer factorization algorithm Quadratic convergence, in which
Quadratic
2 ≡ q ( mod n ) . {\displaystyle x^{2}\equiv q{\pmod {n}}.} sieve of Eratosthenes Sieve of Eratosthenes square-free integer A square-free integer is
Glossary_of_number_theory
Diameter of a sphere of the same volume as an irregularly-shaped subject
the equivalent sieve diameter, or the diameter of a sphere that just passes through a defined sieve pore. Of note, the equivalent sieve diameter can be
Equivalent_spherical_diameter
Provides an asymptotic formula for counting the number of prime ideals of a number field
Alina Carmen Cojocaru; M. Ram Murty (8 December 2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts
Landau_prime_ideal_theorem
wound signaling also function in signaling other defense responses. Cross-talk events regulate the activation of different roles. Sieve elements are very
Wound_response_in_plants
number theory and was "known for his work in sieve methods". He is the namesake of the Buchstab function, which he wrote about in 1937. Buchstab was born
Alexander_Buchstab
Formal power series
function Generating function transformation Stanley's reciprocity theorem Integer partition Combinatorial principles Cyclic sieving Z-transform Umbral
Generating_function
Problem easily dividable into parallel tasks
particle physics. The marching squares algorithm. Sieving step of the quadratic sieve and the number field sieve. Tree growth step of the random forest machine
Embarrassingly_parallel
British mathematician
(1974). Sieve Methods. London: Academic Press. ISBN 0-12-318250-6. MR 0424730. Zbl 0298.10026.Halberstam, Heini; Richert, Hans-Egon (2011). Sieve Methods
Heini_Halberstam
Technique in cryptography
curve from 676 bits to 923 bits. In 2016, the Extended Tower Number Field Sieve algorithm allowed to reduce the complexity of finding discrete logarithm
Pairing-based_cryptography
Potential counterexample to the generalized Riemann hypothesis
generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields. Roughly speaking, these are possible zeros very near
Siegel_zero
dry sieving A method of sifting artefacts from excavated sediments by shaking it through sieves or meshes of varying sizes. As opposed to wet sieving, which
Glossary_of_archaeology
Agricultural machine
A stone picker (or rock picker) is an implement to sieve through the top layer of soil to separate and collect rocks and soil debris from good topsoil
Stone_picker
Number
zero function (or zero map) on a domain D. This is the constant function with 0 as its only possible output value, that is, it is the function f defined
0
Natural number
number which is generated by a certain "sieve". 127 is the 8th term in the prime number variation of Flavius's sieve. 127 is a Fortunate number which is linked
127_(number)
result of S. D. Cohen, based on the large sieve method, extends this result, counting points by height function and showing, in a strong sense, that a thin
Thin_set_(Serre)
Algorithm for generating numbers coprime with first few primes
the halfway point. Sieve of Sundaram Sieve of Atkin Sieve of Pritchard Sieve theory Pritchard, Paul, "Linear prime-number sieves: a family tree," Sci
Wheel_factorization
are proteins occurring in the sieve tubes of Fabaceae. They are synthesised in companion cells and later moved to sieve elements where they are surrounded
Forisome
American mathematician, cryptologist and computer scientist (born 1971)
fast small prime sieve with low memory footprint based on the sieve of Atkin (rather than the more usual sieve of Eratosthenes). Sieve of Atkin was co-authored
Daniel_J._Bernstein
Function whose domain is the positive integers
prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value
Arithmetic_function
Prime number with a certain relationship to an elliptic curve
0}(x)\ll {\frac {x}{\left(\log x\right)^{2-\varepsilon }}}} by incorporating sieve-theoretic techniques. Despite this rarity, Noam Elkies proved in 1987 that
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Arithmetic operation
groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition
Exponentiation
Bone of the facial skeleton
ethmoid bone (/ˈɛθmɔɪd/; from Ancient Greek: ἡθμός, romanized: hēthmós, lit. 'sieve') is an unpaired bone in the skull that separates the nasal cavity from
Ethmoid_bone
British-American mathematician
Bernstein, he developed the sieve of Atkin. Atkin is also known for his work on properties of the integer partition function and the monster module. He
A._O._L._Atkin
German forestry biologist (1805–1880)
the first to discover and name the sieve tube element cells (as Siebfasern - sieve fibres and Siebröhren - sieve tubes) in 1837. His zoologist author
Theodor_Hartig
Method of exchanging cryptographic keys
exponent. An attacker can exploit both vulnerabilities together. The number field sieve algorithm, which is generally the most effective in solving the discrete
Diffie–Hellman_key_exchange
Chinese economist
constraints on. They exam the unknown habit function and compare the internal habit and external habit formation by Sieve Minimum Distance (SMD) procedure. By
Xiaohong_Chen
Agricultural machine
or the walkers, meets a set of sieves mounted on an assembly called a shoe, which is shaken mechanically. The top sieve has larger openings and serves
Threshing_machine
Numbers obtained by adding the two previous ones
using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number. More generally, F k n + c = ∑ i = 0
Fibonacci_sequence
Number theory library written in C
as derived functionality such as integer factorization using a quadratic sieve. The library is designed to be compiled with the GNU Multi-Precision Library
Fast Library for Number Theory
Fast_Library_for_Number_Theory
Algorithm for public-key cryptography
5 gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process. Rivest, Shamir, and Adleman noted that Miller has shown that –
RSA_cryptosystem
Agricultural machinery manufacturer
pre-sieve and top sieve so that each element can operate as its optimum efficiency. The cascade distance between the grain pan and the pre-sieve is increased
New_Holland_Agriculture
Programming language by David Turner
are filtered out of the list of candidates. > primes = sieve [2..] > sieve (p:x) = p : sieve [n | n <- x; n mod p ~= 0] Here, we have some more examples
Miranda (programming language)
Miranda_(programming_language)
Finite sum formed using the exponential function
of stationary phase, and the later Vinogradov method (c.1930). The large sieve method (c.1960), the work of many researchers, is a relatively transparent
Exponential_sum
number field sieve Lenstra elliptic curve factorization Pollard's p − 1 algorithm Pollard's rho algorithm prime factorization algorithm Quadratic sieve Shor's
List_of_algorithms
Integer factorization algorithm
such cases other methods are used such as the quadratic sieve and the general number field sieve (GNFS). Because these methods also have superpolynomial
Trial_division
Region around an astronomical object
scientists were able to study the variations in Earth's magnetic field as functions of both time and latitude and longitude. Beginning in the late 1940s
Magnetosphere
Describes approximate behavior of a function
notation is a mathematical notation that describes the approximate size of a function on a domain. Big O is a member of a family of notations invented by the
Big_O_notation
Theory in number theory
related geometric object, can help to recover X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi
Anabelian_geometry
Estimate of time taken for running an algorithm
best-known classical algorithm for integer factorization, the general number field sieve, which runs in time about 2 O ( n 1 / 3 ( log n ) 2 / 3 ) {\displaystyle
Time_complexity
Exploring properties of the integers with complex analysis
(1986), The Theory of the Riemann Zeta Function (2nd ed.), Oxford University Press H. Halberstam and H. E. Richert, Sieve Methods R. C. Vaughan, The Hardy–Littlewood
Analytic_number_theory
Quantum algorithm for integer factorization
the most scalable classical factoring algorithm, the general number field sieve, which works in sub-exponential time: O ( e 1.9 ( log N ) 1 / 3 ( log
Shor's_algorithm
Mathematician
the theory of the Riemann zeta function, random multiplicative functions, S-unit equations, smooth numbers, the large sieve, and the recent highly innovative
Adam_Harper
Machine that harvests grain crops
grains from the stalks in a more efficient manner. They then go into the sieve and are filtered by weight and size, and cleaned grains go into an auger
Combine_harvester
Counting technique in combinatorics
be viewed as an example of the sieve method extensively used in number theory and is sometimes referred to as the sieve formula. As finite probabilities
Inclusion–exclusion_principle
Number of subsets of a given size
space over a finite field and counting the number of subsets of {1, 2, ..., n} with certain symmetries (an instance of the cyclic sieving phenomenon). The
Binomial_coefficient
Prime number of the form 2^n – 1
primes. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm
Mersenne_prime
Process in quantum mechanics
definition of pointer states has been introduced. This is the "predictability sieve" criterion, based on an intuitive idea: Pointer states can be defined as
Einselection
American mathematician (1935–2019)
2019) was an American mathematician who pioneered large sieve theory and invented the larger sieve. Patrick Ximenes Gallagher was born on January 2, 1935
Patrick_X._Gallagher
Cluster of genes
original variant. Turner calls this a "sieve" explanation, and the Turner explanation might be called the "Turner sieve" hypothesis. Maynard Smith agreed with
Supergene
American mathematician (1936–2020)
functions with analogous properties to Riemann's function, as well as more recent work on the large sieve and density estimates. Advanced Calculus: A Differential
Harold Edwards (mathematician)
Harold_Edwards_(mathematician)
Overview of and topical guide to combinatorics
Reduction to linear algebra Sparsity Weight function Minimax algorithm Alpha–beta pruning Probabilistic method Sieve methods Analytic combinatorics Symbolic
Outline_of_combinatorics
Signal-processing operation
"removing the foot") is the modification of the shape of a mathematical function. The function may represent an electrical signal, an optical transmission, or
Apodization
Number raised to the third power
n × n × n. The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as (−n)3 = −(n3). The
Cube_(algebra)
Programming paradigm
consists of filtering, with the only action being "capture". Less extremely, sieve has filters and actions, but in the base standard has no variables or loops
Data-driven_programming
Ability of a plant to retain information
travel over a longer distance, sieve tubes are used since they have pores and a continuous plasma membrane which makes sieve tubes low resistance. With a
Plant_memory
connect local and global properties of geometric objects. Sheaf cohomology Sieve theory Single operator theory deals with the properties and classifications
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Manner in which fluids behave when flowing through a porous medium
of mass in order to express the capillary force or fluid velocity as a function of various other parameters including the effective pore radius, liquid
Fluid flow through porous media
Fluid_flow_through_porous_media
Used to count, measure, and label
elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they
Number
Inherent difficulty of computational problems
best known algorithm for integer factorization is the general number field sieve, which takes time O ( e ( 64 9 3 ) ( log n ) 3 ( log log n ) 2
Computational complexity theory
Computational_complexity_theory
Polynomial with reversed root positions
are conjugate-reciprocal for n > 1. This is used in the special number field sieve to allow numbers of the form x11 ± 1, x13 ± 1, x15 ± 1 and x21 ± 1 to
Reciprocal_polynomial
Notion for comparing dimensions of particles in different states of matter
based on light, other on ultrasound, or electric field, or gravity, or centrifugation. The use of sieves is a common measurement technique, however this
Particle_size
Priestesses of the Roman goddess Vesta
England was portrayed holding a sieve to evoke Tuccia, the Vestal who proved her virtue by carrying water in a sieve. Tuccia herself had been a subject
Vestal_Virgin
Country in South Asia
Barbara N. (1999). "Women in South Asia". In Barbara N. Ramusack; Sharon L. Sievers (eds.). Women in Asia: Restoring Women to History. Indiana University Press
India
and finding square roots c. 300 BC – Euclid's algorithm c. 200 BC – the Sieve of Eratosthenes 263 AD – Gaussian elimination described by Liu Hui 628 –
Timeline_of_algorithms
Unsolved problem in computer science
efficient known algorithm for integer factorization is the general number field sieve, which takes expected time O ( exp ( ( 64 n 9 log ( 2 ) ) 1 3 ( log
P_versus_NP_problem
Even integers as sums of two primes
Goldbach conjecture is much less than X1⁄2 + c for small c. In 1948, using sieve theory methods, Alfréd Rényi showed that every sufficiently large even number
Goldbach's_conjecture
Emergency medical process
triage tools to be used in major incidents, replacing the NASMeD Triage Sieve. These new tools resulted from a multi-stakeholder review led by the NHS
Triage
Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step;
1000_(number)
Set of polynomials where any two are orthogonal to each other
Lévy processes. Sieved orthogonal polynomials, such as the sieved ultraspherical polynomials, sieved Jacobi polynomials, and sieved Pollaczek polynomials
Orthogonal_polynomials
Analysis of soil
engineering soil tests. Water content Specific gravity Grain size analysis (sieve analysis or hydrometer method) Atterberg limits Free swell index Swelling
Soil_test
Swedish American mathematician (1923–2016)
(particularly the order-constrained estimation of cumulative distribution functions using his sieve estimator). In recent decades, Grenander contributed to computational
Ulf_Grenander
Microporous, aluminosilicate mineral group
24 December 2025. The classic reference for the field has been Breck's book Zeolite Molecular Sieves: Structure, Chemistry, And Use. Sheppard RA (1973)
Zeolite
About simultaneous modular congruences
a field K . {\displaystyle K.} For getting the theorem for a general Euclidean domain, it suffices to replace the degree by the Euclidean function of
Chinese_remainder_theorem
More than 20 years later, Heath-Brown closed on the problem, giving a new sieve method, and conjectured that it could be improved to obtain the predicted
Kummer_sum
is represented by the pore size distribution. It is represented by the function f(r), whose value is proportional to the total volume of all pores whose
Pore_structure
Species of marine plant
chiama così perché ha una forma arrotondata). "Seagrass 'Neptune balls' sieve millions of plastic particles from water, study finds". The Guardian. Agence
Posidonia_oceanica
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
Boy/Male
English
Gathering field; meeting field.
Boy/Male
English
Pasture; field.
Boy/Male
English
Pasture; field.
Girl/Female
Tamil
Hay field
Boy/Male
Anglo, British, English
Field with Ferns; Fern Field
Boy/Male
English
Fern field.
Surname or Lastname
English
English : topographic name from Middle English feldes, plural or possessive of feld ‘open country’. This name is also found as a translation of equivalent names in other languages, in particular French Deschamps, Duchamp.
Boy/Male
English
In the field.
Surname or Lastname
English
English : topographic name for someone who lived on land which had been cleared of forest, but not brought into cultivation, from Old English feld ‘pasture’, ‘open country’, as opposed on the one hand to æcer ‘cultivated soil’, ‘enclosed land’ (see Acker) and on the other to weald ‘wooded land’, ‘forest’ (see Wald).Possibly also Scottish or Irish : reduced form of McField (see McPhail).Jewish (American) : Americanized and shortened form of any of the many Jewish surnames containing Feld.
Boy/Male
Australian, British, English
A Field
Girl/Female
Indian
Hay field
Girl/Female
Japanese American
Valley field.
Boy/Male
Indian
Friction
Girl/Female
Hindu, Indian
Fraction of the Cosmos
Boy/Male
Anglo, British, English
Field with Ferns; Fern Field
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Boy/Male
African, American, Anglo, Australian, British, Christian, English, Jamaican
Battlefield; Spear Field; Triangular Field
Girl/Female
Afghan, Arabic, Australian, Indian, Muslim
Fiction; Romance; Story
Girl/Female
Bengali, Indian
Fraction of Time
Surname or Lastname
English
English : variant of Field.
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
Girl/Female
Tamil
Desire
Surname or Lastname
English
English : occupation name for a net-maker, from Old French retier.German : from a Germanic personal name composed with rÄd, rÄt ‘counsel’ + hari, heri ‘army’.
Boy/Male
Assamese, Hindu, Indian, Sanskrit
Early Winter; Name of a Season
Boy/Male
Danish Norse Swedish Latin
Great.
Girl/Female
Tamil
Chethanya | சேதாநà¯à®¯
Girl/Female
Latin
Beautiful Christian.
Girl/Female
Greek
Untamed.
Boy/Male
Hindi Indian
Soldier. Also, in Hindu mythology, the monkey king who can weaken enemies with a wish.
Boy/Male
Greek Hungarian
Farmer.
Boy/Male
Hindu, Indian, Sanskrit
To Heat Up
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
FUNCTION FIELD-SIEVE
v. t.
To sell by auction.
v. t.
To permit; to grant; as, to yield passage.
n.
The things sold by auction or put up to auction.
a.
Pertaining to, or connected with, a function or duty; official.
v. t.
The act of uniting, or the state of being united; junction.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
v. t.
To use with full command or power, as a thing not too heavy for the holder; to manage; to handle; hence, to use or employ; as, to wield a sword; to wield the scepter.
n.
The act of anointing, smearing, or rubbing with an unguent, oil, or ointment, especially for medical purposes, or as a symbol of consecration; as, mercurial unction.
n.
The whole surface of an escutcheon; also, so much of it is shown unconcealed by the different bearings upon it. See Illust. of Fess, where the field is represented as gules (red), while the fess is argent (silver).
a.
Open, like a field.
a.
Pertaining to the function of an organ or part, or to the functions in general.
v. t.
To give sanction to; to ratify; to confirm; to approve.
a.
Relating to an open fields; drowing in a field; growing in a field, or open ground.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
v. t.
To supply with an organ or organs having a special function or functions.
adv.
To, in, or on the field.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
v. i.
To take the field.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
v. i.
To stand out in the field, ready to catch, stop, or throw the ball.