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mersenne.go
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mersenne.go
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// Copyright (c) 2014 The mersenne Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
/*
Package mersenne collects utilities related to Mersenne numbers[1] and/or some
of their properties.
Exponent
In this documentation the term 'exponent' refers to 'n' of a Mersenne number Mn
equal to 2^n-1. This package supports only uint32 sized exponents. New()
currently supports exponents only up to math.MaxInt32 (31 bits, up to 256 MB
required to represent such Mn in memory as a big.Int).
Links
Referenced from above:
[1] http://en.wikipedia.org/wiki/Mersenne_number
*/
package mersenne
import (
"math"
"math/big"
"github.com/cznic/mathutil"
"github.com/remyoudompheng/bigfft"
)
var (
_0 = big.NewInt(0)
_1 = big.NewInt(1)
_2 = big.NewInt(2)
)
// Knowns list the exponent of currently (March 2012) known Mersenne primes
// exponents in order. See also: http://oeis.org/A000043 for a partial list.
var Knowns = []uint32{
2, // #1
3, // #2
5, // #3
7, // #4
13, // #5
17, // #6
19, // #7
31, // #8
61, // #9
89, // #10
107, // #11
127, // #12
521, // #13
607, // #14
1279, // #15
2203, // #16
2281, // #17
3217, // #18
4253, // #19
4423, // #20
9689, // #21
9941, // #22
11213, // #23
19937, // #24
21701, // #25
23209, // #26
44497, // #27
86243, // #28
110503, // #29
132049, // #30
216091, // #31
756839, // #32
859433, // #33
1257787, // #34
1398269, // #35
2976221, // #36
3021377, // #37
6972593, // #38
13466917, // #39
20996011, // #40
24036583, // #41
25964951, // #42
30402457, // #43
32582657, // #44
37156667, // #45
42643801, // #46
43112609, // #47
57885161, // #48
}
// Known maps the exponent of known Mersenne primes its ordinal number/rank.
// Ranks > 41 are currently provisional.
var Known map[uint32]int
func init() {
Known = map[uint32]int{}
for i, v := range Knowns {
Known[v] = i + 1
}
}
// New returns Mn == 2^n-1 for n <= math.MaxInt32 or nil otherwise.
func New(n uint32) (m *big.Int) {
if n > math.MaxInt32 {
return
}
m = big.NewInt(0)
return m.Sub(m.SetBit(m, int(n), 1), _1)
}
// HasFactorUint32 returns true if d | Mn. Typical run time for a 32 bit factor
// and a 32 bit exponent is < 1 µs.
func HasFactorUint32(d, n uint32) bool {
return d == 1 || d&1 != 0 && mathutil.ModPowUint32(2, n, d) == 1
}
// HasFactorUint64 returns true if d | Mn. Typical run time for a 64 bit factor
// and a 32 bit exponent is < 30 µs.
func HasFactorUint64(d uint64, n uint32) bool {
return d == 1 || d&1 != 0 && mathutil.ModPowUint64(2, uint64(n), d) == 1
}
// HasFactorBigInt returns true if d | Mn, d > 0. Typical run time for a 128
// bit factor and a 32 bit exponent is < 75 µs.
func HasFactorBigInt(d *big.Int, n uint32) bool {
return d.Cmp(_1) == 0 || d.Sign() > 0 && d.Bit(0) == 1 &&
mathutil.ModPowBigInt(_2, big.NewInt(int64(n)), d).Cmp(_1) == 0
}
// HasFactorBigInt2 returns true if d | Mn, d > 0
func HasFactorBigInt2(d, n *big.Int) bool {
return d.Cmp(_1) == 0 || d.Sign() > 0 && d.Bit(0) == 1 &&
mathutil.ModPowBigInt(_2, n, d).Cmp(_1) == 0
}
/*
FromFactorBigInt returns n such that d | Mn if n <= max and d is odd. In other
cases zero is returned.
It is conjectured that every odd d ∊ N divides infinitely many Mersenne numbers.
The returned n should be the exponent of smallest such Mn.
NOTE: The computation of n from a given d performs roughly in O(n). It is
thus highly recomended to use the 'max' argument to limit the "searched"
exponent upper bound as appropriate. Otherwise the computation can take a long
time as a large factor can be a divisor of a Mn with exponent above the uint32
limits.
The FromFactorBigInt function is a modification of the original Will
Edgington's "reverse method", discussed here:
http://tech.groups.yahoo.com/group/primenumbers/message/15061
*/
func FromFactorBigInt(d *big.Int, max uint32) (n uint32) {
if d.Bit(0) == 0 {
return
}
var m big.Int
for n < max {
m.Add(&m, d)
i := 0
for ; m.Bit(i) == 1; i++ {
if n == math.MaxUint32 {
return 0
}
n++
}
m.Rsh(&m, uint(i))
if m.Sign() == 0 {
if n > max {
n = 0
}
return
}
}
return 0
}
// Mod sets mod to n % Mexp and returns mod. It panics for exp == 0 || exp >=
// math.MaxInt32 || n < 0.
func Mod(mod, n *big.Int, exp uint32) *big.Int {
if exp == 0 || exp >= math.MaxInt32 || n.Sign() < 0 {
panic(0)
}
m := New(exp)
mod.Set(n)
var x big.Int
for mod.BitLen() > int(exp) {
x.Set(mod)
x.Rsh(&x, uint(exp))
mod.And(mod, m)
mod.Add(mod, &x)
}
if mod.BitLen() == int(exp) && mod.Cmp(m) == 0 {
mod.SetInt64(0)
}
return mod
}
// ModPow2 returns x such that 2^Me % Mm == 2^x. It panics for m < 2. Typical
// run time is < 1 µs. Use instead of ModPow(2, e, m) wherever possible.
func ModPow2(e, m uint32) (x uint32) {
/*
m < 2 -> panic
e == 0 -> x == 0
e == 1 -> x == 1
2^M1 % M2 == 2^1 % 3 == 2^1 10 // 2^1, 3, 5, 7 ... +2k
2^M1 % M3 == 2^1 % 7 == 2^1 010 // 2^1, 4, 7, ... +3k
2^M1 % M4 == 2^1 % 15 == 2^1 0010 // 2^1, 5, 9, 13... +4k
2^M1 % M5 == 2^1 % 31 == 2^1 00010 // 2^1, 6, 11, 16... +5k
2^M2 % M2 == 2^3 % 3 == 2^1 10.. // 2^3, 5, 7, 9, 11, ... +2k
2^M2 % M3 == 2^3 % 7 == 2^0 001... // 2^3, 6, 9, 12, 15, ... +3k
2^M2 % M4 == 2^3 % 15 == 2^3 1000 // 2^3, 7, 11, 15, 19, ... +4k
2^M2 % M5 == 2^3 % 31 == 2^3 01000 // 2^3, 8, 13, 18, 23, ... +5k
2^M3 % M2 == 2^7 % 3 == 2^1 10..--.. // 2^3, 5, 7... +2k
2^M3 % M3 == 2^7 % 7 == 2^1 010...--- // 2^1, 4, 7... +3k
2^M3 % M4 == 2^7 % 15 == 2^3 1000.... // +4k
2^M3 % M5 == 2^7 % 31 == 2^2 00100..... // +5k
2^M3 % M6 == 2^7 % 63 == 2^1 000010...... // +6k
2^M3 % M7 == 2^7 % 127 == 2^0 0000001.......
2^M3 % M8 == 2^7 % 255 == 2^7 10000000
2^M3 % M9 == 2^7 % 511 == 2^7 010000000
2^M4 % M2 == 2^15 % 3 == 2^1 10..--..--..--..
2^M4 % M3 == 2^15 % 7 == 2^0 1...---...---...
2^M4 % M4 == 2^15 % 15 == 2^3 1000....----....
2^M4 % M5 == 2^15 % 31 == 2^0 1.....-----.....
2^M4 % M6 == 2^15 % 63 == 2^3 1000......------
2^M4 % M7 == 2^15 % 127 == 2^1 10.......-------
2^M4 % M8 == 2^15 % 255 == 2^7 10000000........
2^M4 % M9 == 2^15 % 511 == 2^6 1000000.........
*/
switch {
case m < 2:
panic(0)
case e < 2:
return e
}
if x = mathutil.ModPowUint32(2, e, m); x == 0 {
return m - 1
}
return x - 1
}
// ModPow returns b^Me % Mm. Run time grows quickly with 'e' and/or 'm' when b
// != 2 (then ModPow2 is used).
func ModPow(b, e, m uint32) (r *big.Int) {
if m == 1 {
return big.NewInt(0)
}
if b == 2 {
x := ModPow2(e, m)
r = big.NewInt(0)
r.SetBit(r, int(x), 1)
return
}
bb := big.NewInt(int64(b))
r = big.NewInt(1)
for ; e != 0; e-- {
r = bigfft.Mul(r, bb)
Mod(r, r, m)
bb = bigfft.Mul(bb, bb)
Mod(bb, bb, m)
}
return
}
// ProbablyPrime returns true if Mn is prime or is a pseudoprime to base a.
// Note: Every Mp, prime p, is a prime or is a pseudoprime to base 2, actually
// to every base 2^i, i ∊ [1, p). In contrast - it is conjectured (w/o any
// known counterexamples) that no composite Mp, prime p, is a pseudoprime to
// base 3.
func ProbablyPrime(n, a uint32) bool {
//TODO +test, +bench
if a == 2 {
return ModPow2(n-1, n) == 0
}
nMinus1 := New(n)
nMinus1.Sub(nMinus1, _1)
x := ModPow(a, n-1, n)
return x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0
}