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In [[number theory]], a branch of [[mathematics]], the '''special number field sieve''' (SNFS) is a special-purpose [[integer factorization]] algorithm. The [[general number field sieve]] (GNFS) was derived from it.

The special number field sieve is efficient for integers of the form ''r''''e'' ± ''s'', where ''r'' and ''s'' are small (for instance [[Mersenne number]]s).

[[Heuristic]]ally, its [[Computational complexity theory|complexity]] for factoring an integer <math>n</math> is of the form:<ref>{{Citation|last=Pomerance|first=Carl|author-link=Carl Pomerance|date=December 1996|title=A Tale of Two Sieves|periodical=Notices of the AMS|volume=43|issue=12|pages=1473–1485|url=http://www.ams.org/notices/199612/pomerance.pdf}}</ref>

:<math>\exp\left(\left(1+o(1)\right)\left(\tfrac{32}{9}\log n\right)^{1/3}\left(\log\log n\right)^{2/3}\right)=L_n\left[1/3,(32/9)^{1/3}\right]</math>

in [[Big O notation|O]] and [[L-notation]]s.

The SNFS has been used extensively by NFSNet (a volunteer [[distributed computing]] effort), [http://escatter11.fullerton.edu/nfs/ NFS@Home] and others to factorise numbers of the [[Cunningham project]]; for some time the [[Integer factorization records|records for integer factorisation]] have been numbers factored by SNFS.

==Overview of method==

The SNFS is based on an idea similar to the much simpler [[rational sieve]]; in particular, readers may find it helpful to read about the [[rational sieve]] first, before tackling the SNFS.

The SNFS works as follows. Let ''n'' be the integer we want to factor. As in the [[rational sieve]], the SNFS can be broken into two steps:
*First, find a large number of multiplicative relations among a ''factor base'' of elements of [[Modular arithmetic#Ring of congruence classes|'''Z'''/''n'''''Z''']], such that the number of multiplicative relations is larger than the number of elements in the factor base.
*Second, multiply together subsets of these relations in such a way that all the exponents are even, resulting in congruences of the form ''a''2&equiv;''b''2 ([[modular arithmetic|mod]] ''n''). These in turn immediately lead to factorizations of ''n'': ''n''=[[greatest common divisor|gcd]](''a''+''b'',''n'')×gcd(''a''-''b'',''n''). If done right, it is almost certain that at least one such factorization will be nontrivial.

The second step is identical to the case of the [[rational sieve]], and is a straightforward [[linear algebra]] problem. The first step, however, is done in a different, more [[algorithmic efficiency|efficient]] way than the rational sieve, by utilizing [[algebraic number field|number fields]].

==Details of method==

Let ''n'' be the integer we want to factor. We pick an [[irreducible polynomial]] ''f'' with integer coefficients, and an integer ''m'' such that ''f''(''m'')&equiv;0 ([[modular arithmetic|mod]] ''n'') (we will explain how they are chosen in the next section). Let ''α'' be a [[Root of a function|root]] of ''f''; we can then form the [[ring (mathematics)|ring]] '''[[integer|Z]]'''[α]. There is a unique [[ring homomorphism]] φ from '''Z'''[''α''] to [[Modular arithmetic#Ring of congruence classes|'''Z'''/n'''Z''']] that maps ''α'' to ''m''. For simplicity, we'll assume that '''Z'''[''α''] is a [[unique factorization domain]]; the algorithm can be modified to work when it isn't, but then there are some additional complications.

Next, we set up two parallel ''factor bases'', one in '''Z'''[''α''] and one in '''Z'''. The one in '''Z'''[''α''] consists of all the prime ideals in '''Z'''[''α''] whose norm is bounded by a chosen value <math>N_{\max}</math>. The factor base in '''Z''', as in the rational sieve case, consists of all prime integers up to some other bound.

We then search for [[relatively prime]] pairs of integers (''a'',''b'') such that:
*''a''+''bm'' is [[smooth number|smooth]] with respect to the factor base in '''Z''' (i.e., it is a product of elements in the factor base).
*''a''+''bα'' is smooth with respect to the factor base in '''Z'''[''α'']; given how we chose the factor base, this is equivalent to the norm of ''a''+''bα'' being divisible only by primes less than <math>N_{\max}</math>.

These pairs are found through a sieving process, analogous to the [[Sieve of Eratosthenes]]; this motivates the name "Number Field Sieve".

For each such pair, we can apply the ring homomorphism φ to the factorization of ''a''+''bα'', and we can apply the canonical ring homomorphism from '''Z''' to '''Z'''/n'''Z''' to the factorization of ''a''+''bm''. Setting these equal gives a multiplicative relation among elements of a bigger factor base in '''Z'''/n'''Z''', and if we find enough pairs we can proceed to combine the relations and factor ''n'', as described above.

==Choice of parameters==

Not every number is an appropriate choice for the SNFS: you need to know in advance a polynomial ''f'' of appropriate degree (the optimal degree is conjectured to be <math>\left(3 \frac{\log N}{\log \log N}\right) ^{1/3}</math>, which is 4, 5, or 6 for the sizes of N currently feasible to factorise) with small coefficients, and a value ''x'' such that <math>f(x) \equiv 0 \pmod N</math> where N is the number to factorise. There is an extra condition: ''x'' must satisfy <math>ax+b \equiv 0 \pmod N</math> for a and b no bigger than <math>N^{1/d}</math>.

One set of numbers for which such polynomials exist are the <math>a^b \pm 1</math> numbers from the [[Cunningham project|Cunningham tables]]; for example, when NFSNET factored 3^479+1, they used the polynomial x^6+3 with x=3^80, since (3^80)^6+3 = 3^480+3, and <math>3^{480}+3 \equiv 0 \pmod {3^{479}+1}</math>.

Numbers defined by linear recurrences, such as the [[Fibonacci number|Fibonacci]] and [[Lucas number|Lucas]] numbers, also have SNFS polynomials, but these are a little more difficult to construct. For example, <math>F_{709}</math> has polynomial <math>n^5 + 10n^3 + 10n^2 + 10n + 3</math>, and the value of ''x'' satisfies <math>F_{142} x - F_{141} = 0</math>.<ref>{{cite web
| last = Franke
| first = Jens
| title = Installation notes for ggnfs-lasieve4
| url=http://stuff.mit.edu/afs/sipb/project/pari-gp/ggnfs/Linux/src/lasieve4/INSTALL.and.USE
| publisher =[[MIT]] Massachusetts Institute of Technology}}</ref>

If you already know some factors of a large SNFS-number, you can do the SNFS calculation modulo the remaining part; for the NFSNET example above, 3^479+1 = (4*158071*7167757*7759574882776161031) times a 197-digit composite number (the small factors were removed by [[elliptic curve method|ECM]]), and the SNFS was performed modulo the 197-digit number. The number of relations required by SNFS still depends on the size of the large number, but the individual calculations are quicker modulo the smaller number.

==Limitations of algorithm==

This algorithm, as mentioned above, is very efficient for numbers of the form ''r''''e''±''s'', for ''r'' and ''s'' relatively small. It is also efficient for any integers which can be represented as a polynomial with small coefficients. This includes integers of the more general form ''a''''r''''e''±''b''''s''''f'', and also for many integers whose binary representation has low Hamming weight. The reason for this is as follows: The Number Field Sieve performs sieving in two different fields.
The first field is usually the rationals. The second is a higher degree field. The efficiency of the algorithm strongly depends on the norms of certain elements in these fields. When an integer can be represented as a polynomial with small coefficients, the norms that arise are much smaller than those that arise when an integer is represented by a general polynomial. The reason is that a general polynomial will have much larger coefficients, and the norms will be correspondingly larger. The algorithm attempts to factor these norms over a fixed set of prime numbers. When the
norms are smaller, these numbers are more likely to factor.

== See also ==
* [[General number field sieve]]

== References ==
{{reflist}}

==Further reading==
*{{Citation |last=Byrnes |first=Steven |lastauthoramp= |date=May 18, 2005 |title=The Number Field Sieve |journal=Math 129 |volume= |issue= |pages= |url=http://modular.fas.harvard.edu/129-05/final_papers/Steve_Byrnes.pdf |issn= }}
*{{Citation |doi=10.1090/S0025-5718-1993-1182953-4 |last=Lenstra |first=A. K. |authorlink=Arjen Lenstra |last2=Lenstra |first2=H. W., Jr. |authorlink2=Hendrik Lenstra |last3=Manasse |first3=M. S. |last4=Pollard |first4=J. M. |lastauthoramp=yes |year=1993 |title=The Factorization of the Ninth Fermat Number |journal=Mathematics of Computation |volume=61 |issue=203 |pages=319–349 |url=http://www.std.org/~msm/common/f9paper.ps |issn= }}
*{{citation |editor-last=Lenstra |editor-first=A. K. |editor2-last=Lenstra |editor2-first=H. W., Jr. |title=The Development of the Number Field Sieve |series=Lecture Notes in Mathematics |volume=1554 |year=1993 |publisher=Springer-Verlag |location=New York |isbn=3-540-57013-6 |pages= |url= }}
*{{citation |last=Silverman |first=Robert D. |title=Optimal Parameterization of SNFS |journal=J. Mathematical Cryptology |volume=1 |year=2007 |publisher=de Gruyter |pages= 105–124 |doi=10.1515/JMC.2007.007}}

==External links==
* http://www.nfsnet.org/

{{number theoretic algorithms}}

[[Category:Integer factorization algorithms]]

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49.249.199.48:

{{for|a more general mathematical discussion|Conjugate variables}}
{{Thermodynamics|cTopic=[[List of thermodynamic properties|System properties]]}}

In [[thermodynamics]], the [[internal energy]] of a system is expressed in terms of pairs of '''conjugate variables''' such as temperature and entropy or pressure and volume. In fact, all [[thermodynamic potential]]s are expressed in terms of conjugate pairs.

For a mechanical system, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain [[Generalized force|generalized "forces"]] that, when unbalanced, cause certain [[Generalized coordinates|generalized "displacements"]], and the product of the two is the energy transferred as a result. These forces and their associated displacements are called conjugate variables. The thermodynamic force is always an [[intensive variable]] and the displacement is always an [[extensive variable]], yielding an extensive energy transfer. The intensive (force) variable is the derivative of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.

The [[thermodynamic square]] can be used as a tool to recall and derive some of the [[thermodynamic potential]]s based on conjugate variables.

In the above description, the product of two conjugate variables yields an energy. In other words, the conjugate pairs are conjugate with respect to energy. In general, conjugate pairs can be defined with respect to any thermodynamic state function. Conjugate pairs with respect to [[Free entropy|entropy]] are often used, in which the product of the conjugate pairs yields an entropy. Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of the [[Onsager reciprocal relations]]. The present article is concerned only with energy-conjugate variables.

== Overview ==
Just as a small increment of energy in a mechanical system is the product of a force times a small displacement, so an increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when unbalanced, cause certain generalized "displacements" to occur, with their product being the energy transferred as a result. These forces and their associated displacements are called '''conjugate variables'''.<ref name="Alberty 2001 p1353">Alberty (2001) p 1353</ref> For example, consider the ''PV'' conjugate pair. The pressure ''P'' acts as a generalized force: Pressure differences force a change in volume ''dV'', and their product is the energy lost by the system due to work. Here, pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables. In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heat transfer. The thermodynamic force is always an [[intensive variable]] and the displacement is always an [[extensive variable]], yielding an extensive energy. The intensive (force) variable is the derivative of the (extensive) internal energy with respect to the extensive (displacement) variable, with all other extensive variables held constant.

The theory of thermodynamic potentials is not complete until one considers the number of particles in a system as a variable on par with the other extensive quantities such as volume and entropy. The number of particles is, like volume and entropy, the displacement variable in a conjugate pair. The generalized force component of this pair is the [[chemical potential]]. The chemical potential may be thought of as a force which, when imbalanced, pushes an exchange of particles, either with the surroundings, or between phases inside the system. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds liquid water and water vapor, there will be a chemical potential (which is negative) for the liquid which pushes the water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate, and the chemical potential of each phase is equal, is equilibrium obtained.

The most commonly considered conjugate thermodynamic variables are (with corresponding [[SI]] units):

:Thermal parameters:
:* [[Temperature]]: ''T''  ([[Kelvin|K]])
:* [[Entropy]]: ''S''  (J K−1)

:Mechanical parameters:
:* [[Pressure]]: ''P''  ([[Pascal (unit)|Pa]]= J m−3)
:* [[volume (thermodynamics)|Volume]]: ''V''  (m3 = J Pa−1)

::or, more generally,

:* [[Cauchy stress tensor|Stress]]: <math>\sigma_{ij}\,</math> ([[Pascal (unit)|Pa]]= J m−3)
:* Volume × [[strain tensor|Strain]]: <math>V\times\varepsilon_{ij}</math> (m3 = J Pa−1)

:Material parameters:
:* [[chemical potential]]: μ (J)
:* [[particle number]]: ''N''  (particles or mole)

For a system with different types <math>i</math> of particles, a small change in the internal energy is given by:

:<math>\mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V + \sum_i \mu_i \mathrm{d}N_i\,,</math>

where ''U'' is internal energy, ''T'' is temperature, ''S'' is entropy, ''P'' is pressure, ''V'' is volume, <math>\mu_i</math> is the chemical potential of the ''i''-th particle type, and <math>N_i</math> is the number of ''i''-type particles in the system.

Here, the temperature, pressure, and chemical potential are the generalized forces, which drive the generalized changes in entropy, volume, and particle number respectively. These parameters all affect the [[internal energy]] of a thermodynamic system. A small change <math>\mathrm{d}U</math> in the internal energy of the system is given by the sum of the flow of energy across the boundaries of the system due to the corresponding conjugate pair. These concepts will be expanded upon in the following sections.

While dealing with processes in which systems exchange matter or energy, classical thermodynamics is not concerned with the [[derivative|rate]] at which such processes take place, termed [[Kinetics (physics)|kinetics]]. For this reason, the term ''thermodynamics'' is usually used synonymously with ''equilibrium thermodynamics''. A central notion for this connection is that of [[quasistatic process]]es, namely idealized, "infinitely slow" processes. Time-dependent thermodynamic processes far away from equilibrium are studied by [[non-equilibrium thermodynamics]]. This can be done through linear or non-linear analysis of [[irreversible process]]es, allowing systems near and far away from equilibrium to be studied, respectively.

== Pressure/volume and stress/strain pairs ==

As an example, consider the PV conjugate pair. The [[pressure]] acts as a generalized force – pressure differences force a change in [[volume (thermodynamics)|volume]], and their product is the energy lost by the system due to [[mechanical work]]. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.

The above holds true only for non-viscous fluids. In the case of [[viscosity|viscous fluids]], [[Plasticity (physics)|plastic]] and [[Elasticity (physics)|elastic]] solids, the pressure force is generalized to the [[stress tensor]], and changes in volume are generalized to the volume multiplied by the [[strain tensor]] {{harv|Landau|Lifshitz|1986}}. These then form a conjugate pair. If <math>\sigma_{ij}</math> is the ''ij'' component of the stress tensor, and <math>\varepsilon_{ij}</math> is the ''ij'' component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain <math>d\varepsilon_{ij}</math> is:

:<math>\delta w = -V\sum_{ij}\sigma_{ij}d\varepsilon_{ij}</math>

or, using [[Einstein notation]] for the tensors, in which repeated indices are assumed to be summed:

:<math>\delta w = -V\sigma_{ij}d\varepsilon_{ij}</math>

In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times the [[Kronecker delta|unit tensor]] so that

:<math>\delta w = -V\,(-P\delta_{ij})\,d\varepsilon_{ij}=PVd\varepsilon_{kk}</math>

The [[Trace (linear algebra)|trace]] of the strain tensor (<math>\varepsilon_{kk}</math>) is just the fractional change in volume so that the above reduces to <math>\delta w = PdV</math> as it should.

== Temperature/entropy pair ==

In a similar way, [[temperature]] differences drive changes in [[entropy]], and their product is the energy transferred by [[heat]]ing. Temperature is the driving force, entropy is the associated displacement, and the two form a pair of conjugate variables. The temperature/entropy pair of conjugate variables is the only '''heat''' term; the other terms are essentially all various forms of '''work'''.

== Chemical potential/particle number pair ==

The [[chemical potential]] is like a force which pushes an increase in [[particle number]]. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds water and water vapor, there will be a chemical potential (which is negative) for the liquid, pushing water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate is equilibrium obtained.

== See also ==

* [[Generalized coordinate]] and [[generalized force]]: analogous conjugate variable pairs found in classical mechanics.

== References ==
{{reflist}}
* {{cite journal
| author=Alberty, R. A.
| url = http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf
| title = Use of Legendre transforms in chemical thermodynamics
| journal=Pure Appl. Chem.
| year=2001 | volume= 73 | issue=8 | pages=1349–1380
| doi = 10.1351/pac200173081349
}}
* {{cite book
| author=Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo
| title=Thermodynamics
| publisher= McGraw-Hill Book Co.
| location = New York, NY USA
| edition=2nd
| year=1961
| isbn =0-07-113809-9
}}
* {{cite book
| last1 = Landau
| first1 = L. D.
| authorlink1 = Lev Landau
| last2 = Lifshitz
| first2 = E. M.
| authorlink2 = Evgeny Lifshitz
| others = (Translated from Russian by J.B. Sykes and W.H. Reid)
| year = 1986
| title = Theory of Elasticity (Course of Theoretical Physics Volume 7)
| edition = Third
| publisher = Butterworth Heinemann
| location = Boston, MA
| isbn = 0-7506-2633-X
| ref = harv
}}

{{DEFAULTSORT:Conjugate Variables (Thermodynamics)}}
[[Category:Thermodynamics]]