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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra intfact.spad}
\author{Michael Monagan}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package PRIMES IntegerPrimesPackage}
<<package PRIMES IntegerPrimesPackage>>=
)abbrev package PRIMES IntegerPrimesPackage
++ Author: Michael Monagan
++ Date Created: August 1987
++ Date Last Updated: 31 May 1993
++ Updated by: James Davenport
++ Updated Because: of problems with strong pseudo-primes
++ and for some efficiency reasons.
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: integer, prime
++ Examples:
++ References: Davenport's paper in ISSAC 1992
++ AXIOM Technical Report ATR/6
++ Description:
++ The \spadtype{IntegerPrimesPackage} implements a modification of
++ Rabin's probabilistic
++ primality test and the utility functions \spadfun{nextPrime},
++ \spadfun{prevPrime} and \spadfun{primes}.
IntegerPrimesPackage(I:IntegerNumberSystem): with
prime?: I -> Boolean
++ \spad{prime?(n)} returns true if n is prime and false if not.
++ The algorithm used is Rabin's probabilistic primality test
++ (reference: Knuth Volume 2 Semi Numerical Algorithms).
++ If \spad{prime? n} returns false, n is proven composite.
++ If \spad{prime? n} returns true, prime? may be in error
++ however, the probability of error is very low.
++ and is zero below 25*10**9 (due to a result of Pomerance et al),
++ below 10**12 and 10**13 due to results of Pinch,
++ and below 341550071728321 due to a result of Jaeschke.
++ Specifically, this implementation does at least 10 pseudo prime
++ tests and so the probability of error is \spad{< 4**(-10)}.
++ The running time of this method is cubic in the length
++ of the input n, that is \spad{O( (log n)**3 )}, for n<10**20.
++ beyond that, the algorithm is quartic, \spad{O( (log n)**4 )}.
++ Two improvements due to Davenport have been incorporated
++ which catches some trivial strong pseudo-primes, such as
++ [Jaeschke, 1991] 1377161253229053 * 413148375987157, which
++ the original algorithm regards as prime
nextPrime: I -> I
++ \spad{nextPrime(n)} returns the smallest prime strictly larger than n
prevPrime: I -> I
++ \spad{prevPrime(n)} returns the largest prime strictly smaller than n
primes: (I,I) -> List I
++ \spad{primes(a,b)} returns a list of all primes p with
++ \spad{a <= p <= b}
== add
smallPrimes: List I := [2::I,3::I,5::I,7::I,11::I,13::I,17::I,19::I,_
23::I,29::I,31::I,37::I,41::I,43::I,47::I,_
53::I,59::I,61::I,67::I,71::I,73::I,79::I,_
83::I,89::I,97::I,101::I,103::I,107::I,109::I,_
113::I,127::I,131::I,137::I,139::I,149::I,151::I,_
157::I,163::I,167::I,173::I,179::I,181::I,191::I,_
193::I,197::I,199::I,211::I,223::I,227::I,229::I,_
233::I,239::I,241::I,251::I,257::I,263::I,269::I,_
271::I,277::I,281::I,283::I,293::I,307::I,311::I,_
313::I]
productSmallPrimes := */smallPrimes
nextSmallPrime := 317::I
nextSmallPrimeSquared := nextSmallPrime**2
two := 2::I
tenPowerTwenty:=(10::I)**20
PomeranceList:= [25326001::I, 161304001::I, 960946321::I, 1157839381::I,
-- 3215031751::I, -- has a factor of 151
3697278427::I, 5764643587::I, 6770862367::I,
14386156093::I, 15579919981::I, 18459366157::I,
19887974881::I, 21276028621::I ]::(List I)
PomeranceLimit:=27716349961::I -- replaces (25*10**9) due to Pinch
PinchList:= [3215031751::I, 118670087467::I, 128282461501::I, 354864744877::I,
546348519181::I, 602248359169::I, 669094855201::I ]
PinchLimit:= (10**12)::I
PinchList2:= [2152302898747::I, 3474749660383::I]
PinchLimit2:= (10**13)::I
JaeschkeLimit:=341550071728321::I
rootsMinus1:Set I := empty()
-- used to check whether we detect too many roots of -1
count2Order:Vector NonNegativeInteger := new(1,0)
-- used to check whether we observe an element of maximal two-order
primes(m, n) ==
-- computes primes from m to n inclusive using prime?
l:List(I) :=
m <= two => [two]
empty()
n < two or n < m => empty()
if even? m then m := m + 1
ll:List(I) := [k::I for k in
convert(m)@Integer..convert(n)@Integer by 2 | prime?(k::I)]
reverse! concat!(ll, l)
rabinProvesComposite : (I,I,I,I,NonNegativeInteger) -> Boolean
rabinProvesCompositeSmall : (I,I,I,I,NonNegativeInteger) -> Boolean
rabinProvesCompositeSmall(p,n,nm1,q,k) ==
-- probability n prime is > 3/4 for each iteration
-- for most n this probability is much greater than 3/4
t := powmod(p, q, n)
-- neither of these cases tells us anything
if not (one? t or t = nm1) then
for j in 1..k-1 repeat
oldt := t
t := mulmod(t, t, n)
one? t => return true
-- we have squared someting not -1 and got 1
t = nm1 =>
leave
not (t = nm1) => return true
false
rabinProvesComposite(p,n,nm1,q,k) ==
-- probability n prime is > 3/4 for each iteration
-- for most n this probability is much greater than 3/4
t := powmod(p, q, n)
-- neither of these cases tells us anything
if t=nm1 then count2Order(1):=count2Order(1)+1
if not (one? t or t = nm1) then
for j in 1..k-1 repeat
oldt := t
t := mulmod(t, t, n)
one? t => return true
-- we have squared someting not -1 and got 1
t = nm1 =>
rootsMinus1:=union(rootsMinus1,oldt)
count2Order(j+1):=count2Order(j+1)+1
leave
not (t = nm1) => return true
# rootsMinus1 > 2 => true -- Z/nZ can't be a field
false
prime? n ==
n < two => false
n < nextSmallPrime => member?(n, smallPrimes)
not one? gcd(n, productSmallPrimes) => false
n < nextSmallPrimeSquared => true
nm1 := n-1
q := (nm1) quo two
k : NonNegativeInteger
for k in 1.. while not odd? q repeat q := q quo two
-- q = (n-1) quo 2**k for largest possible k
n < JaeschkeLimit =>
rabinProvesCompositeSmall(2::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(3::I,n,nm1,q,k) => return false
n < PomeranceLimit =>
rabinProvesCompositeSmall(5::I,n,nm1,q,k) => return false
member?(n,PomeranceList) => return false
true
rabinProvesCompositeSmall(7::I,n,nm1,q,k) => return false
n < PinchLimit =>
rabinProvesCompositeSmall(10::I,n,nm1,q,k) => return false
member?(n,PinchList) => return false
true
rabinProvesCompositeSmall(5::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(11::I,n,nm1,q,k) => return false
n < PinchLimit2 =>
member?(n,PinchList2) => return false
true
rabinProvesCompositeSmall(13::I,n,nm1,q,k) => return false
rabinProvesCompositeSmall(17::I,n,nm1,q,k) => return false
true
rootsMinus1:= empty()
count2Order := new(k,0) -- vector of k zeroes
mn := minIndex smallPrimes
for i in mn+1..mn+10 repeat
rabinProvesComposite(smallPrimes i,n,nm1,q,k) => return false
import IntegerRoots(I)
q > 1 and perfectSquare?(3*n+1) => false
((n9:=n rem (9::I))=1 or n9 = -1) and perfectSquare?(8*n+1) => false
-- Both previous tests from Damgard & Landrock
currPrime:=smallPrimes(mn+10)
probablySafe:=tenPowerTwenty
while count2Order(k) = 0 or n > probablySafe repeat
currPrime := nextPrime currPrime
probablySafe:=probablySafe*(100::I)
rabinProvesComposite(currPrime,n,nm1,q,k) => return false
true
nextPrime n ==
-- computes the first prime after n
n < two => two
if odd? n then n := n + two else n := n + 1
while not prime? n repeat n := n + two
n
prevPrime n ==
-- computes the first prime before n
n < 3::I => error "no primes less than 2"
n = 3::I => two
if odd? n then n := n - two else n := n - 1
while not prime? n repeat n := n - two
n
@
\section{package IROOT IntegerRoots}
<<package IROOT IntegerRoots>>=
)abbrev package IROOT IntegerRoots
++ Author: Michael Monagan
++ Date Created: November 1987
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: integer roots
++ Examples:
++ References:
++ Description: The \spadtype{IntegerRoots} package computes square roots and
++ nth roots of integers efficiently.
IntegerRoots(I:IntegerNumberSystem): Exports == Implementation where
NNI ==> NonNegativeInteger
Exports ==> with
perfectNthPower?: (I, NNI) -> Boolean
++ \spad{perfectNthPower?(n,r)} returns true if n is an \spad{r}th
++ power and false otherwise
perfectNthRoot: (I,NNI) -> Union(I,"failed")
++ \spad{perfectNthRoot(n,r)} returns the \spad{r}th root of n if n
++ is an \spad{r}th power and returns "failed" otherwise
perfectNthRoot: I -> Record(base:I, exponent:NNI)
++ \spad{perfectNthRoot(n)} returns \spad{[x,r]}, where \spad{n = x\^r}
++ and r is the largest integer such that n is a perfect \spad{r}th power
approxNthRoot: (I,NNI) -> I
++ \spad{approxRoot(n,r)} returns an approximation x
++ to \spad{n**(1/r)} such that \spad{-1 < x - n**(1/r) < 1}
perfectSquare?: I -> Boolean
++ \spad{perfectSquare?(n)} returns true if n is a perfect square
++ and false otherwise
perfectSqrt: I -> Union(I,"failed")
++ \spad{perfectSqrt(n)} returns the square root of n if n is a
++ perfect square and returns "failed" otherwise
approxSqrt: I -> I
++ \spad{approxSqrt(n)} returns an approximation x
++ to \spad{sqrt(n)} such that \spad{-1 < x - sqrt(n) < 1}.
++ Compute an approximation s to \spad{sqrt(n)} such that
++ \spad{-1 < s - sqrt(n) < 1}
++ A variable precision Newton iteration is used.
++ The running time is \spad{O( log(n)**2 )}.
Implementation ==> add
import IntegerPrimesPackage(I)
resMod144: List I := [0::I,1::I,4::I,9::I,16::I,25::I,36::I,49::I,_
52::I,64::I,73::I,81::I,97::I,100::I,112::I,121::I]
two := 2::I
perfectSquare? a == (perfectSqrt a) case I
perfectNthPower?(b, n) == perfectNthRoot(b, n) case I
perfectNthRoot n == -- complexity (log log n)**2 (log n)**2
one? n or zero? n or n = -1 => [n, 1]
e:NNI := 1
p:NNI := 2
while p::I <= length(n) + 1 repeat
m: NNI := 0
while (r := perfectNthRoot(n, p)) case I repeat
n := r::I
m := m + 1
e := e * p ** m
p := convert(nextPrime(p::I))@Integer :: NNI
[n, e]
approxNthRoot(a, n) == -- complexity (log log n) (log n)**2
zero? n => error "invalid arguments"
one? n => a
n=2 => approxSqrt a
negative? a =>
odd? n => - approxNthRoot(-a, n)
0
zero? a => 0
one? a => 1
-- quick check for case of large n
((3*n) quo 2)::I >= (l := length a) => two
-- the initial approximation must be >= the root
y := max(two, shift(1, (n::I+l-1) quo (n::I)))
z:I := 1
n1:= (n-1)::NNI
x: I
while z > 0 repeat
x := y
xn:= x**n1
y := (n1*x*xn+a) quo (n*xn)
z := x-y
x
perfectNthRoot(b, n) ==
(r := approxNthRoot(b, n)) ** n = b => r
"failed"
perfectSqrt a ==
negative? a or not member?(a rem (144::I), resMod144) => "failed"
(s := approxSqrt a) * s = a => s
"failed"
approxSqrt a ==
a < 1 => 0
if (n := length a) > (100::I) then
-- variable precision newton iteration
n := n quo (4::I)
s := approxSqrt shift(a, -2 * n)
s := shift(s, n)
return ((1 + s + a quo s) quo two)
-- initial approximation for the root is within a factor of 2
(new, old) := (shift(1, n quo two), 1)
while new ~= old repeat
(new, old) := ((1 + new + a quo new) quo two, new)
new
@
\section{package INTFACT IntegerFactorizationPackage}
<<package INTFACT IntegerFactorizationPackage>>=
)abbrev package INTFACT IntegerFactorizationPackage
++ This Package contains basic methods for integer factorization.
++ The factor operation employs trial division up to 10,000. It
++ then tests to see if n is a perfect power before using Pollards
++ rho method. Because Pollards method may fail, the result
++ of factor may contain composite factors. We should also employ
++ Lenstra's eliptic curve method.
IntegerFactorizationPackage(I): Exports == Implementation where
I: IntegerNumberSystem
B ==> Boolean
FF ==> Factored I
NNI ==> NonNegativeInteger
LMI ==> ListMultiDictionary I
FFE ==> Record(flg:Union("nil","sqfr","irred","prime"),
fctr:I, xpnt:Integer)
Exports ==> with
factor : I -> FF
++ factor(n) returns the full factorization of integer n
squareFree : I -> FF
++ squareFree(n) returns the square free factorization of integer n
BasicMethod : I -> FF
++ BasicMethod(n) returns the factorization
++ of integer n by trial division
PollardSmallFactor: I -> Union(I,"failed")
++ PollardSmallFactor(n) returns a factor
++ of n or "failed" if no one is found
Implementation ==> add
import IntegerRoots(I)
BasicSieve: (I, I) -> FF
squareFree(n:I):FF ==
u:I
if negative? n then (m := -n; u := -1)
else (m := n; u := 1)
(m > 1) and ((v := perfectSqrt m) case I) =>
sv : FF
l : List FFE
for rec in (l := factorList(sv := squareFree(v::I))) repeat
rec.xpnt := 2 * rec.xpnt
makeFR(u * unit sv, l)
-- avoid using basic sieve when the lim is too big
lim := 1 + approxNthRoot(m,3)
lim > (100000::I) => makeFR(u, factorList factor m)
x := BasicSieve(m, lim)
y :=
one?(m:= unit x) => factorList x
(v := perfectSqrt m) case I =>
concat!(factorList x, ["sqfr",v,2]$FFE)
concat!(factorList x, ["sqfr",m,1]$FFE)
makeFR(u, y)
-- Pfun(y: I,n: I): I == (y**2 + 5) rem n
PollardSmallFactor(n:I):Union(I,"failed") ==
-- Use the Brent variation
x0 := random()$I
m := 100::I
y := x0 rem n
r:I := 1
q:I := 1
G:I := 1
ys: I
x: I
l: I
k: I
until G > 1 repeat
x := y
ys := y
for i in 1..convert(r)@Integer repeat
y := (y*y+5::I) rem n
q := (q*abs(x-y)) rem n
k := 0::I
G := gcd(q,n)
until (k>=r) or (G>1) repeat
ys := y
for i in 1..convert(min(m,r-k))@Integer repeat
y := (y*y+5::I) rem n
q := (q*abs(x-y)) rem n
G := gcd(q,n)
k := k+m
k := k + r
r := 2*r
if G=n then
l := k - m
G := 1::I
until G>1 repeat
ys := (ys*ys+5::I) rem n
G := gcd(abs(x-ys),n)
l := l + 1
if G = n then
y := x0
x := x0
for i in 1..convert(l)@Integer repeat
y := (y*y + 5::I) rem n
G := gcd(abs(x-y),n)
until G > 1 repeat
y := (y*y + 5::I) rem n
x := (x*x + 5::I) rem n
G := gcd(abs(x-y),n)
G=n => "failed"
G
PollardSmallFactor20(n: I): Union(I,"failed") ==
r: Union(I,"failed")
for i in 1..20 repeat
r := PollardSmallFactor n
r case I => return r
r
BasicSieve(r, lim) ==
l:List(I) :=
[1::I,2::I,2::I,4::I,2::I,4::I,2::I,4::I,6::I,2::I,6::I]
concat!(l, rest(l, 3))
d := 2::I
n := r
ls := empty()$List(FFE)
for s in l repeat
d > lim => return makeFR(n, ls)
if n<d*d then
if n>1 then ls := concat!(ls, ["prime",n,1]$FFE)
return makeFR(1, ls)
m : Integer
for m in 0.. while zero?(n rem d) repeat n := n quo d
if m>0 then ls := concat!(ls, ["prime",d,convert m]$FFE)
d := d+s
BasicMethod n ==
u:I
if negative? n then (m := -n; u := -1)
else (m := n; u := 1)
x := BasicSieve(m, 1 + approxSqrt m)
makeFR(u, factorList x)
factor m ==
u:I
zero? m => 0
if negative? m then (n := -m; u := -1)
else (n := m; u := 1)
b := BasicSieve(n, 10000::I)
flb := factorList b
one?(n := unit b) => makeFR(u, flb)
a:LMI := dictionary() -- numbers yet to be factored
b:LMI := dictionary() -- prime factors found
f:LMI := dictionary() -- number which could not be factored
insert!(n, a)
while not empty? a repeat
n := inspect a;
c := count(n, a);
remove!(n, a)
prime?(n)$IntegerPrimesPackage(I) => insert!(n, b, c)
-- test for a perfect power
(s := perfectNthRoot n).exponent > 1 =>
insert!(s.base, a, c * s.exponent)
-- test for a difference of square
x:=approxSqrt n;
if (x**2<n) then x:=x+1
(y:=perfectSqrt (x**2-n)) case I =>
insert!(x+y,a,c)
insert!(x-y,a,c)
(d := PollardSmallFactor20 n) case I =>
m' : NonNegativeInteger
for m' in 0.. while zero?(n rem d) repeat n := n quo d
insert!(d, a, m' * c)
if n > 1 then insert!(n, a, c)
-- an elliptic curve factorization attempt should be made here
insert!(n, f, c)
-- insert prime factors found
while not empty? b repeat
n := inspect b; c := count(n, b); remove!(n, b)
flb := concat!(flb, ["prime",n,convert c]$FFE)
-- insert non-prime factors found
while not empty? f repeat
n := inspect f; c := count(n, f); remove!(n, f)
flb := concat!(flb, ["nil",n,convert c]$FFE)
makeFR(u, flb)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package PRIMES IntegerPrimesPackage>>
<<package IROOT IntegerRoots>>
<<package INTFACT IntegerFactorizationPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|