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- move the code from StandardFF to new files `lib/dlog.gd`, `lib/dlog.gi`, independent of the record `FFECONWAY` - call the functions `_DLog` and `_DLogShanks`, since StandardFF currently wants to bind global variables `_DLog` and `_DLogShanks` - add a few tests
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,35 @@ | ||
| ############################################################################# | ||
| ## | ||
| ## This file is part of GAP, a system for computational discrete algebra. | ||
| ## This file's authors include Frank Lübeck. | ||
| ## | ||
| ## 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 declares operations related to discrete logarithms. | ||
| ## | ||
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| ############################################################################# | ||
| ## | ||
| #F _DLog( <base>, <x>[, <m>] ) | ||
| ## | ||
| ## <ManSection> | ||
| ## <Func Name="_DLog" Arg='base, x[, m]'/> | ||
| ## | ||
| ## <Description> | ||
| ## <Index Subkey="discrete">logarithm</Index> | ||
| ## returns a discrete logarithm of <A>x</A> w.r.t. the basis <A>base</A>, | ||
| ## i. e., an integer <M>l</M> such that <M><A>base</A>^l = <A>x</A></M> | ||
| ## holds, if such a number exists. | ||
| ## Otherwise <K>fail</K> is returned. | ||
| ## <P/> | ||
| ## The order of <A>base</A> or its factorization can be given as <A>m</A>. | ||
| ## </Description> | ||
| ## </ManSection> | ||
| ## | ||
| DeclareGlobalFunction( "_DLog" ); | ||
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| DeclareGlobalFunction( "_DLogShanks" ); |
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| ############################################################################# | ||
| ## | ||
| ## This file is part of GAP, a system for computational discrete algebra. | ||
| ## This file's authors include Frank Lübeck. | ||
| ## | ||
| ## 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 methods related to discrete logarithms. | ||
| ## The code was copied from the StandardFF package. | ||
| ## | ||
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| ############################################################################# | ||
| ## | ||
| #F _DLogShanks( <base>, <x>, <r> ) | ||
| ## | ||
| ## Let <base> be a multiplicative element of order <r>. | ||
| ## Return an integer l such that <x> = <base>^l holds, | ||
| ## or 'fail' if no such l exists. | ||
| ## | ||
| InstallGlobalFunction( _DLogShanks, function(base, x, r) | ||
| local rr, baby, ord, giant, t, pos, i, j; | ||
| rr := RootInt(r, 2); | ||
| baby := [One(base)]; | ||
| if x = baby[1] then | ||
| return 0; | ||
| fi; | ||
| for i in [1..rr-1] do | ||
| baby[i+1] := baby[i]*base; | ||
| if x = baby[i+1] then | ||
| return i; | ||
| fi; | ||
| od; | ||
| giant := baby[rr]*base; | ||
| ord := [0..rr-1]; | ||
| SortParallel(baby, ord); | ||
| t := x; | ||
| for j in [1..QuoInt(r, rr)+1] do | ||
| t := t*giant; | ||
| pos := PositionSet(baby, t); | ||
| if IsInt(pos) then | ||
| return (ord[pos] - j * rr) mod r; | ||
| fi; | ||
| od; | ||
| return fail; | ||
| end ); | ||
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| ############################################################################# | ||
| ## | ||
| #F _DLog( <base>, <x>[, <m>] ) | ||
| ## | ||
| ## recursive method, <m> can be the order m of <base> or its factorization | ||
| ## Let r be the largest prime factor of m, then we use | ||
| ## <base>^e = <x> with e = a + b*r where 0 <= a < r and | ||
| ## 0 <= b < m/r, | ||
| ## and compute a with _DLogShanks and b by recursion. | ||
| ## | ||
| InstallGlobalFunction( _DLog, function(base, x, m...) | ||
| local r, mm, mp, a, b; | ||
| if Length(m) = 0 then | ||
| m := Order(base); | ||
| else | ||
| m := m[1]; | ||
| fi; | ||
| if not IsList(m) then | ||
| m := Factors(m); | ||
| fi; | ||
| if Length(m) = 1 then | ||
| return _DLogShanks(base, x, m[1]); | ||
| fi; | ||
| r := m[Length(m)]; | ||
| mm := m{[1..Length(m)-1]}; | ||
| mp := Product(mm); | ||
| a := _DLogShanks(base^mp, x^mp, r); | ||
| if a = fail then | ||
| return fail; | ||
| fi; | ||
| b := _DLog(base^r, x/(base^a), mm); | ||
| if b = fail then | ||
| return fail; | ||
| fi; | ||
| return a + b*r; | ||
| end ); | ||
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| ############################################################################# | ||
| ## | ||
| ## used as method for LogFFE | ||
| ## | ||
| BindGlobal( "DoDLog", function(x, base) | ||
| local ob, o, e; | ||
| ob := Order(base); | ||
| o := Order(x); | ||
| if ob mod o <> 0 then | ||
| return fail; | ||
| fi; | ||
| if ob <> o then | ||
| e := ob/o; | ||
| base := base^e; | ||
| else | ||
| e := 1; | ||
| fi; | ||
| return _DLog(base, x, o) * e; | ||
| end ); |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,22 @@ | ||
| #@local R, o | ||
| gap> START_TEST( "dlog.tst" ); | ||
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| # _DLog | ||
| gap> R:= Integers mod 71;; o:= One( R );; | ||
| gap> _DLog( 2*o, 4*o ); | ||
| 2 | ||
| gap> R:= Integers mod 17;; o:= One( R );; | ||
| gap> _DLog( 2*o, 3*o ); | ||
| fail | ||
| gap> R:= Integers mod 9;; o:= One( R );; | ||
| gap> _DLog( 2*o, 4*o ); | ||
| 2 | ||
| gap> _DLog( 3*o, 4*o ); | ||
| Error, <obj> is not invertible | ||
| gap> ForAll( Primes, p -> p = 2 or _DLog( Z(p^2), Z(p^2)^2 ) = 2 ); | ||
| true | ||
| gap> ForAll( Primes, p -> p = 2 or _DLog( Z(p^2)^2, Z(p^2) ) = fail ); | ||
| true | ||
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| # | ||
| gap> STOP_TEST( "dlog.tst" ); |