SSL primality.gd
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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Jack Schmidt.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains declarations for the primality test in the integers.
##
##############################################################################
##
## Bibiliography
##
## http://www.ams.org/mathscinet-getitem?mr=572872
## http://links.jstor.org/sici?sici=0025-5718%28197504%2929%3A130%3C620%3ANPCAFO%3E2.0.CO%3B2-N
## @article{BLS1975,
## AUTHOR = {Brillhart, John and Lehmer, D. H. and Selfridge, J. L.},
## TITLE = {New primality criteria and factorizations of {$2\sp{m}\pm 1$}},
## JOURNAL = {Math. Comp.},
## FJOURNAL = {Mathematics of Computation},
## VOLUME = {29},
## YEAR = {1975},
## PAGES = {620--647},
## ISSN = {0025-5718},
## MRCLASS = {10A25},
## MRNUMBER = {MR0384673 (52 \#5546)},
## MRREVIEWER = {Jean-Marie De Koninck},
## }
##
## http://www.ams.org/mathscinet-getitem?mr=572872
## http://links.jstor.org/sici?sici=0025-5718%28198007%2935%3A151%3C1003%3ATPT%3E2.0.CO%3B2-D
## @article{PSW1980,
## AUTHOR = {Pomerance, Carl and Selfridge, J. L. and Wagstaff, Jr., Samuel
## S.},
## TITLE = {The pseudoprimes to {$25\cdot 10\sp{9}$}},
## JOURNAL = {Math. Comp.},
## FJOURNAL = {Mathematics of Computation},
## VOLUME = {35},
## YEAR = {1980},
## NUMBER = {151},
## PAGES = {1003--1026},
## ISSN = {0025-5718},
## CODEN = {MCMPAF},
## MRCLASS = {10A40 (10-04 10A25)},
## MRNUMBER = {MR572872 (82g:10030)},
## }
##
## http://www.ams.org/mathscinet-getitem?mr=583518
## http://links.jstor.org/sici?sici=0025-5718%28198010%2935%3A152%3C1391%3ALP%3E2.0.CO%3B2-N
## @article {BW1980,
## AUTHOR = {Baillie, Robert and Wagstaff, Jr., Samuel S.},
## TITLE = {Lucas pseudoprimes},
## JOURNAL = {Math. Comp.},
## FJOURNAL = {Mathematics of Computation},
## VOLUME = {35},
## YEAR = {1980},
## NUMBER = {152},
## PAGES = {1391--1417},
## ISSN = {0025-5718},
## CODEN = {MCMPAF},
## MRCLASS = {10A25},
## MRNUMBER = {MR583518 (81j:10005)},
## MRREVIEWER = {V. C. Harris},
## }
##
##############################################################################
## Section 1
#############################################################################
##
#F IsSquareInt(<n>)
##
## <#GAPDoc Label="IsSquareInt">
## <ManSection>
## <Func Name="IsSquareInt" Arg='n'/>
##
## <Description>
## <Ref Func="IsSquareInt"/> tests whether the integer <A>n</A> is the
## square of an integer or not.
## This test is much faster than the simpler <C>RootInt</C><M>(n)^2=n</M>
## because it first tests whether <A>n</A> is a square residue modulo
## some small integers.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IsSquareInt");
## Section 2
DeclareGlobalFunction("IsStrongPseudoPrimeBaseA");
DeclareGlobalFunction("IsBPSWLucasPseudoPrime");
DeclareGlobalFunction("IsLucasPseudoPrimeDP");
DeclareGlobalFunction("IsStrongLucasPseudoPrimeDP");
DeclareGlobalFunction("IsBPSWPseudoPrime");
DeclareGlobalFunction("IsBPSWPseudoPrime_VerifyCorrectness");
## Section 3
DeclareGlobalFunction("PrimalityProof_FindFermat");
DeclareGlobalFunction("PrimalityProof_FindLucas");
DeclareGlobalFunction("PrimalityProof_FindStructure");
#############################################################################
##
#F IsPrimeInt( <n> ) . . . . . . . . . . . . . . . . . . . test for a prime
#F IsProbablyPrimeInt( <n> ) . . . . . . . . . . . . . . . test for a prime
##
## <#GAPDoc Label="IsPrimeInt">
## <ManSection>
## <Func Name="IsPrimeInt" Arg='n'/>
## <Func Name="IsProbablyPrimeInt" Arg='n'/>
##
## <Description>
## <Ref Func="IsPrimeInt"/> returns <K>false</K> if it can prove that
## the integer <A>n</A> is composite and <K>true</K> otherwise.
## By convention <C>IsPrimeInt(0) = IsPrimeInt(1) = false</C>
## and we define
## <C>IsPrimeInt(-</C><A>n</A><C>) = IsPrimeInt(</C><A>n</A><C>)</C>.
## <P/>
## <Ref Func="IsPrimeInt"/> will return <K>true</K> for every prime <A>n</A>.
## <Ref Func="IsPrimeInt"/> will return <K>false</K> for all composite
## <A>n</A> <M>< 10^{18}</M> and for all composite <A>n</A> that have
## a factor <M>p < 1000</M>. So for integers <A>n</A> <M>< 10^{18}</M>,
## <Ref Func="IsPrimeInt"/> is a proper primality test. It is conceivable that
## <Ref Func="IsPrimeInt"/> may return <K>true</K> for some composite
## <A>n</A> <M>> 10^{18}</M>, but no such <A>n</A> is currently known.
## So for integers <A>n</A> <M>> 10^{18}</M>, <Ref Func="IsPrimeInt"/>
## is a probable-primality test. <Ref Func="IsPrimeInt"/> will issue a
## warning when its argument is probably prime but not a proven prime.
## (The function <Ref Func="IsProbablyPrimeInt"/> will do a similar
## calculation but not issue a warning.) The warning can be switched off by
## <C>SetInfoLevel( InfoPrimeInt, 0 );</C>, the default level is <M>1</M>
## (also see <Ref Oper="SetInfoLevel"/> ).
## <P/>
## If composites that fool <Ref Func="IsPrimeInt"/> do exist, they would be extremely
## rare, and finding one by pure chance might be less likely than finding a
## bug in &GAP;. We would appreciate being informed about any example of a
## composite number <A>n</A> for which <Ref Func="IsPrimeInt"/> returns <K>true</K>.
## <P/>
## <Ref Func="IsPrimeInt"/> is a deterministic algorithm, i.e., the computations involve
## no random numbers, and repeated calls will always return the same result.
## <Ref Func="IsPrimeInt"/> first does trial divisions by the primes less than 1000.
## Then it tests that <A>n</A> is a strong pseudoprime w.r.t. the base 2.
## Finally it tests whether <A>n</A> is a Lucas pseudoprime w.r.t. the smallest
## quadratic nonresidue of <A>n</A>. A better description can be found in the
## comment in the library file <File>primality.gi</File>.
## <P/>
## The time taken by <Ref Func="IsPrimeInt"/> is approximately proportional to the third
## power of the number of digits of <A>n</A>. Testing numbers with several
## hundreds digits is quite feasible.
## <P/>
## <Ref Func="IsPrimeInt"/> is a method for the general operation <Ref Oper="IsPrime"/>.
## <P/>
## Remark: In future versions of &GAP; we hope to change the definition of
## <Ref Func="IsPrimeInt"/> to return <K>true</K> only for proven primes (currently, we lack
## a sufficiently good primality proving function). In applications, use
## explicitly <Ref Func="IsPrimeInt"/> or <Ref Func="IsProbablyPrimeInt"/>
## with this change in mind.
## <Example><![CDATA[
## gap> IsPrimeInt( 2^31 - 1 );
## true
## gap> IsPrimeInt( 10^42 + 1 );
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
UnbindGlobal( "IsPrimeInt" );
DeclareGlobalFunction( "IsPrimeInt" );
DeclareGlobalFunction( "IsProbablyPrimeInt" );
#############################################################################
##
#F PrimalityProof(<n>)
##
## <#GAPDoc Label="PrimalityProof">
## <ManSection>
## <Func Name="PrimalityProof" Arg='n'/>
##
## <Description>
## Construct a machine verifiable proof of the primality of (the probable
## prime) <A>n</A>, following the ideas of <Cite Key="BLS1975"/>.
##
## The proof consists of various Fermat and Lucas pseudoprimality tests,
## which taken as a whole prove the primality. The proof is represented
## as a list of witnesses of two kinds. The first kind, <C>[ "F", divisor,
## base ]</C>, indicates a successful Fermat pseudoprimality test, where
## <A>n</A> is a strong pseudoprime at <K>base</K> with order not divisible by
## <M>(<A>n</A>-1)/divisor</M>. The second kind, <C>[ "L", divisor,
## discriminant, P ]</C> indicates a successful Lucas pseudoprimality test,
## for a quadratic form of given <K>discriminant</K> and middle term <K>P</K>
## with an extra check at <M>(<A>n</A>+1)/divisor</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("PrimalityProof");
## Section 4
DeclareGlobalFunction("PrimalityProof_VerifyWitness");
DeclareGlobalFunction("PrimalityProof_VerifyStructure");
DeclareGlobalFunction("PrimalityProof_Verify");
## Section 5
DeclareGlobalVariable("PrimesProofs");
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2026-03-28
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