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author | Ian Lance Taylor <ian@gcc.gnu.org> | 2011-12-07 01:11:29 +0000 |
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committer | Ian Lance Taylor <ian@gcc.gnu.org> | 2011-12-07 01:11:29 +0000 |
commit | 9c63abc9a1d127f95162756467284cf76b47aff8 (patch) | |
tree | 84f27a6ab44d932e4b0455f18390b070b4de626e /libgo/go/big/nat.go | |
parent | 374280238f934fa851273e2ee16ba53be890c6b8 (diff) | |
download | gcc-9c63abc9a1d127f95162756467284cf76b47aff8.zip gcc-9c63abc9a1d127f95162756467284cf76b47aff8.tar.gz gcc-9c63abc9a1d127f95162756467284cf76b47aff8.tar.bz2 |
libgo: Update to weekly 2011-11-09.
From-SVN: r182073
Diffstat (limited to 'libgo/go/big/nat.go')
-rw-r--r-- | libgo/go/big/nat.go | 1271 |
1 files changed, 0 insertions, 1271 deletions
diff --git a/libgo/go/big/nat.go b/libgo/go/big/nat.go deleted file mode 100644 index a46f782..0000000 --- a/libgo/go/big/nat.go +++ /dev/null @@ -1,1271 +0,0 @@ -// Copyright 2009 The Go 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 big implements multi-precision arithmetic (big numbers). -// The following numeric types are supported: -// -// - Int signed integers -// - Rat rational numbers -// -// All methods on Int take the result as the receiver; if it is one -// of the operands it may be overwritten (and its memory reused). -// To enable chaining of operations, the result is also returned. -// -package big - -// This file contains operations on unsigned multi-precision integers. -// These are the building blocks for the operations on signed integers -// and rationals. - -import ( - "errors" - "io" - "rand" -) - -// An unsigned integer x of the form -// -// x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0] -// -// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n, -// with the digits x[i] as the slice elements. -// -// A number is normalized if the slice contains no leading 0 digits. -// During arithmetic operations, denormalized values may occur but are -// always normalized before returning the final result. The normalized -// representation of 0 is the empty or nil slice (length = 0). -// -type nat []Word - -var ( - natOne = nat{1} - natTwo = nat{2} - natTen = nat{10} -) - -func (z nat) clear() { - for i := range z { - z[i] = 0 - } -} - -func (z nat) norm() nat { - i := len(z) - for i > 0 && z[i-1] == 0 { - i-- - } - return z[0:i] -} - -func (z nat) make(n int) nat { - if n <= cap(z) { - return z[0:n] // reuse z - } - // Choosing a good value for e has significant performance impact - // because it increases the chance that a value can be reused. - const e = 4 // extra capacity - return make(nat, n, n+e) -} - -func (z nat) setWord(x Word) nat { - if x == 0 { - return z.make(0) - } - z = z.make(1) - z[0] = x - return z -} - -func (z nat) setUint64(x uint64) nat { - // single-digit values - if w := Word(x); uint64(w) == x { - return z.setWord(w) - } - - // compute number of words n required to represent x - n := 0 - for t := x; t > 0; t >>= _W { - n++ - } - - // split x into n words - z = z.make(n) - for i := range z { - z[i] = Word(x & _M) - x >>= _W - } - - return z -} - -func (z nat) set(x nat) nat { - z = z.make(len(x)) - copy(z, x) - return z -} - -func (z nat) add(x, y nat) nat { - m := len(x) - n := len(y) - - switch { - case m < n: - return z.add(y, x) - case m == 0: - // n == 0 because m >= n; result is 0 - return z.make(0) - case n == 0: - // result is x - return z.set(x) - } - // m > 0 - - z = z.make(m + 1) - c := addVV(z[0:n], x, y) - if m > n { - c = addVW(z[n:m], x[n:], c) - } - z[m] = c - - return z.norm() -} - -func (z nat) sub(x, y nat) nat { - m := len(x) - n := len(y) - - switch { - case m < n: - panic("underflow") - case m == 0: - // n == 0 because m >= n; result is 0 - return z.make(0) - case n == 0: - // result is x - return z.set(x) - } - // m > 0 - - z = z.make(m) - c := subVV(z[0:n], x, y) - if m > n { - c = subVW(z[n:], x[n:], c) - } - if c != 0 { - panic("underflow") - } - - return z.norm() -} - -func (x nat) cmp(y nat) (r int) { - m := len(x) - n := len(y) - if m != n || m == 0 { - switch { - case m < n: - r = -1 - case m > n: - r = 1 - } - return - } - - i := m - 1 - for i > 0 && x[i] == y[i] { - i-- - } - - switch { - case x[i] < y[i]: - r = -1 - case x[i] > y[i]: - r = 1 - } - return -} - -func (z nat) mulAddWW(x nat, y, r Word) nat { - m := len(x) - if m == 0 || y == 0 { - return z.setWord(r) // result is r - } - // m > 0 - - z = z.make(m + 1) - z[m] = mulAddVWW(z[0:m], x, y, r) - - return z.norm() -} - -// basicMul multiplies x and y and leaves the result in z. -// The (non-normalized) result is placed in z[0 : len(x) + len(y)]. -func basicMul(z, x, y nat) { - z[0 : len(x)+len(y)].clear() // initialize z - for i, d := range y { - if d != 0 { - z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d) - } - } -} - -// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks. -// Factored out for readability - do not use outside karatsuba. -func karatsubaAdd(z, x nat, n int) { - if c := addVV(z[0:n], z, x); c != 0 { - addVW(z[n:n+n>>1], z[n:], c) - } -} - -// Like karatsubaAdd, but does subtract. -func karatsubaSub(z, x nat, n int) { - if c := subVV(z[0:n], z, x); c != 0 { - subVW(z[n:n+n>>1], z[n:], c) - } -} - -// Operands that are shorter than karatsubaThreshold are multiplied using -// "grade school" multiplication; for longer operands the Karatsuba algorithm -// is used. -var karatsubaThreshold int = 32 // computed by calibrate.go - -// karatsuba multiplies x and y and leaves the result in z. -// Both x and y must have the same length n and n must be a -// power of 2. The result vector z must have len(z) >= 6*n. -// The (non-normalized) result is placed in z[0 : 2*n]. -func karatsuba(z, x, y nat) { - n := len(y) - - // Switch to basic multiplication if numbers are odd or small. - // (n is always even if karatsubaThreshold is even, but be - // conservative) - if n&1 != 0 || n < karatsubaThreshold || n < 2 { - basicMul(z, x, y) - return - } - // n&1 == 0 && n >= karatsubaThreshold && n >= 2 - - // Karatsuba multiplication is based on the observation that - // for two numbers x and y with: - // - // x = x1*b + x0 - // y = y1*b + y0 - // - // the product x*y can be obtained with 3 products z2, z1, z0 - // instead of 4: - // - // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0 - // = z2*b*b + z1*b + z0 - // - // with: - // - // xd = x1 - x0 - // yd = y0 - y1 - // - // z1 = xd*yd + z1 + z0 - // = (x1-x0)*(y0 - y1) + z1 + z0 - // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z1 + z0 - // = x1*y0 - z1 - z0 + x0*y1 + z1 + z0 - // = x1*y0 + x0*y1 - - // split x, y into "digits" - n2 := n >> 1 // n2 >= 1 - x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0 - y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0 - - // z is used for the result and temporary storage: - // - // 6*n 5*n 4*n 3*n 2*n 1*n 0*n - // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ] - // - // For each recursive call of karatsuba, an unused slice of - // z is passed in that has (at least) half the length of the - // caller's z. - - // compute z0 and z2 with the result "in place" in z - karatsuba(z, x0, y0) // z0 = x0*y0 - karatsuba(z[n:], x1, y1) // z2 = x1*y1 - - // compute xd (or the negative value if underflow occurs) - s := 1 // sign of product xd*yd - xd := z[2*n : 2*n+n2] - if subVV(xd, x1, x0) != 0 { // x1-x0 - s = -s - subVV(xd, x0, x1) // x0-x1 - } - - // compute yd (or the negative value if underflow occurs) - yd := z[2*n+n2 : 3*n] - if subVV(yd, y0, y1) != 0 { // y0-y1 - s = -s - subVV(yd, y1, y0) // y1-y0 - } - - // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0 - // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0 - p := z[n*3:] - karatsuba(p, xd, yd) - - // save original z2:z0 - // (ok to use upper half of z since we're done recursing) - r := z[n*4:] - copy(r, z) - - // add up all partial products - // - // 2*n n 0 - // z = [ z2 | z0 ] - // + [ z0 ] - // + [ z2 ] - // + [ p ] - // - karatsubaAdd(z[n2:], r, n) - karatsubaAdd(z[n2:], r[n:], n) - if s > 0 { - karatsubaAdd(z[n2:], p, n) - } else { - karatsubaSub(z[n2:], p, n) - } -} - -// alias returns true if x and y share the same base array. -func alias(x, y nat) bool { - return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1] -} - -// addAt implements z += x*(1<<(_W*i)); z must be long enough. -// (we don't use nat.add because we need z to stay the same -// slice, and we don't need to normalize z after each addition) -func addAt(z, x nat, i int) { - if n := len(x); n > 0 { - if c := addVV(z[i:i+n], z[i:], x); c != 0 { - j := i + n - if j < len(z) { - addVW(z[j:], z[j:], c) - } - } - } -} - -func max(x, y int) int { - if x > y { - return x - } - return y -} - -// karatsubaLen computes an approximation to the maximum k <= n such that -// k = p<<i for a number p <= karatsubaThreshold and an i >= 0. Thus, the -// result is the largest number that can be divided repeatedly by 2 before -// becoming about the value of karatsubaThreshold. -func karatsubaLen(n int) int { - i := uint(0) - for n > karatsubaThreshold { - n >>= 1 - i++ - } - return n << i -} - -func (z nat) mul(x, y nat) nat { - m := len(x) - n := len(y) - - switch { - case m < n: - return z.mul(y, x) - case m == 0 || n == 0: - return z.make(0) - case n == 1: - return z.mulAddWW(x, y[0], 0) - } - // m >= n > 1 - - // determine if z can be reused - if alias(z, x) || alias(z, y) { - z = nil // z is an alias for x or y - cannot reuse - } - - // use basic multiplication if the numbers are small - if n < karatsubaThreshold || n < 2 { - z = z.make(m + n) - basicMul(z, x, y) - return z.norm() - } - // m >= n && n >= karatsubaThreshold && n >= 2 - - // determine Karatsuba length k such that - // - // x = x1*b + x0 - // y = y1*b + y0 (and k <= len(y), which implies k <= len(x)) - // b = 1<<(_W*k) ("base" of digits xi, yi) - // - k := karatsubaLen(n) - // k <= n - - // multiply x0 and y0 via Karatsuba - x0 := x[0:k] // x0 is not normalized - y0 := y[0:k] // y0 is not normalized - z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y - karatsuba(z, x0, y0) - z = z[0 : m+n] // z has final length but may be incomplete, upper portion is garbage - - // If x1 and/or y1 are not 0, add missing terms to z explicitly: - // - // m+n 2*k 0 - // z = [ ... | x0*y0 ] - // + [ x1*y1 ] - // + [ x1*y0 ] - // + [ x0*y1 ] - // - if k < n || m != n { - x1 := x[k:] // x1 is normalized because x is - y1 := y[k:] // y1 is normalized because y is - var t nat - t = t.mul(x1, y1) - copy(z[2*k:], t) - z[2*k+len(t):].clear() // upper portion of z is garbage - t = t.mul(x1, y0.norm()) - addAt(z, t, k) - t = t.mul(x0.norm(), y1) - addAt(z, t, k) - } - - return z.norm() -} - -// mulRange computes the product of all the unsigned integers in the -// range [a, b] inclusively. If a > b (empty range), the result is 1. -func (z nat) mulRange(a, b uint64) nat { - switch { - case a == 0: - // cut long ranges short (optimization) - return z.setUint64(0) - case a > b: - return z.setUint64(1) - case a == b: - return z.setUint64(a) - case a+1 == b: - return z.mul(nat{}.setUint64(a), nat{}.setUint64(b)) - } - m := (a + b) / 2 - return z.mul(nat{}.mulRange(a, m), nat{}.mulRange(m+1, b)) -} - -// q = (x-r)/y, with 0 <= r < y -func (z nat) divW(x nat, y Word) (q nat, r Word) { - m := len(x) - switch { - case y == 0: - panic("division by zero") - case y == 1: - q = z.set(x) // result is x - return - case m == 0: - q = z.make(0) // result is 0 - return - } - // m > 0 - z = z.make(m) - r = divWVW(z, 0, x, y) - q = z.norm() - return -} - -func (z nat) div(z2, u, v nat) (q, r nat) { - if len(v) == 0 { - panic("division by zero") - } - - if u.cmp(v) < 0 { - q = z.make(0) - r = z2.set(u) - return - } - - if len(v) == 1 { - var rprime Word - q, rprime = z.divW(u, v[0]) - if rprime > 0 { - r = z2.make(1) - r[0] = rprime - } else { - r = z2.make(0) - } - return - } - - q, r = z.divLarge(z2, u, v) - return -} - -// q = (uIn-r)/v, with 0 <= r < y -// Uses z as storage for q, and u as storage for r if possible. -// See Knuth, Volume 2, section 4.3.1, Algorithm D. -// Preconditions: -// len(v) >= 2 -// len(uIn) >= len(v) -func (z nat) divLarge(u, uIn, v nat) (q, r nat) { - n := len(v) - m := len(uIn) - n - - // determine if z can be reused - // TODO(gri) should find a better solution - this if statement - // is very costly (see e.g. time pidigits -s -n 10000) - if alias(z, uIn) || alias(z, v) { - z = nil // z is an alias for uIn or v - cannot reuse - } - q = z.make(m + 1) - - qhatv := make(nat, n+1) - if alias(u, uIn) || alias(u, v) { - u = nil // u is an alias for uIn or v - cannot reuse - } - u = u.make(len(uIn) + 1) - u.clear() - - // D1. - shift := leadingZeros(v[n-1]) - if shift > 0 { - // do not modify v, it may be used by another goroutine simultaneously - v1 := make(nat, n) - shlVU(v1, v, shift) - v = v1 - } - u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift) - - // D2. - for j := m; j >= 0; j-- { - // D3. - qhat := Word(_M) - if u[j+n] != v[n-1] { - var rhat Word - qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1]) - - // x1 | x2 = q̂v_{n-2} - x1, x2 := mulWW(qhat, v[n-2]) - // test if q̂v_{n-2} > br̂ + u_{j+n-2} - for greaterThan(x1, x2, rhat, u[j+n-2]) { - qhat-- - prevRhat := rhat - rhat += v[n-1] - // v[n-1] >= 0, so this tests for overflow. - if rhat < prevRhat { - break - } - x1, x2 = mulWW(qhat, v[n-2]) - } - } - - // D4. - qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0) - - c := subVV(u[j:j+len(qhatv)], u[j:], qhatv) - if c != 0 { - c := addVV(u[j:j+n], u[j:], v) - u[j+n] += c - qhat-- - } - - q[j] = qhat - } - - q = q.norm() - shrVU(u, u, shift) - r = u.norm() - - return q, r -} - -// Length of x in bits. x must be normalized. -func (x nat) bitLen() int { - if i := len(x) - 1; i >= 0 { - return i*_W + bitLen(x[i]) - } - return 0 -} - -// MaxBase is the largest number base accepted for string conversions. -const MaxBase = 'z' - 'a' + 10 + 1 // = hexValue('z') + 1 - -func hexValue(ch rune) Word { - d := MaxBase + 1 // illegal base - switch { - case '0' <= ch && ch <= '9': - d = int(ch - '0') - case 'a' <= ch && ch <= 'z': - d = int(ch - 'a' + 10) - case 'A' <= ch && ch <= 'Z': - d = int(ch - 'A' + 10) - } - return Word(d) -} - -// scan sets z to the natural number corresponding to the longest possible prefix -// read from r representing an unsigned integer in a given conversion base. -// It returns z, the actual conversion base used, and an error, if any. In the -// error case, the value of z is undefined. The syntax follows the syntax of -// unsigned integer literals in Go. -// -// The base argument must be 0 or a value from 2 through MaxBase. If the base -// is 0, the string prefix determines the actual conversion base. A prefix of -// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a -// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10. -// -func (z nat) scan(r io.RuneScanner, base int) (nat, int, error) { - // reject illegal bases - if base < 0 || base == 1 || MaxBase < base { - return z, 0, errors.New("illegal number base") - } - - // one char look-ahead - ch, _, err := r.ReadRune() - if err != nil { - return z, 0, err - } - - // determine base if necessary - b := Word(base) - if base == 0 { - b = 10 - if ch == '0' { - switch ch, _, err = r.ReadRune(); err { - case nil: - b = 8 - switch ch { - case 'x', 'X': - b = 16 - case 'b', 'B': - b = 2 - } - if b == 2 || b == 16 { - if ch, _, err = r.ReadRune(); err != nil { - return z, 0, err - } - } - case io.EOF: - return z.make(0), 10, nil - default: - return z, 10, err - } - } - } - - // convert string - // - group as many digits d as possible together into a "super-digit" dd with "super-base" bb - // - only when bb does not fit into a word anymore, do a full number mulAddWW using bb and dd - z = z.make(0) - bb := Word(1) - dd := Word(0) - for max := _M / b; ; { - d := hexValue(ch) - if d >= b { - r.UnreadRune() // ch does not belong to number anymore - break - } - - if bb <= max { - bb *= b - dd = dd*b + d - } else { - // bb * b would overflow - z = z.mulAddWW(z, bb, dd) - bb = b - dd = d - } - - if ch, _, err = r.ReadRune(); err != nil { - if err != io.EOF { - return z, int(b), err - } - break - } - } - - switch { - case bb > 1: - // there was at least one mantissa digit - z = z.mulAddWW(z, bb, dd) - case base == 0 && b == 8: - // there was only the octal prefix 0 (possibly followed by digits > 7); - // return base 10, not 8 - return z, 10, nil - case base != 0 || b != 8: - // there was neither a mantissa digit nor the octal prefix 0 - return z, int(b), errors.New("syntax error scanning number") - } - - return z.norm(), int(b), nil -} - -// Character sets for string conversion. -const ( - lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz" - uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ" -) - -// decimalString returns a decimal representation of x. -// It calls x.string with the charset "0123456789". -func (x nat) decimalString() string { - return x.string(lowercaseDigits[0:10]) -} - -// string converts x to a string using digits from a charset; a digit with -// value d is represented by charset[d]. The conversion base is determined -// by len(charset), which must be >= 2. -func (x nat) string(charset string) string { - b := Word(len(charset)) - - // special cases - switch { - case b < 2 || b > 256: - panic("illegal base") - case len(x) == 0: - return string(charset[0]) - } - - // allocate buffer for conversion - i := x.bitLen()/log2(b) + 1 // +1: round up - s := make([]byte, i) - - // special case: power of two bases can avoid divisions completely - if b == b&-b { - // shift is base-b digit size in bits - shift := uint(trailingZeroBits(b)) // shift > 0 because b >= 2 - mask := Word(1)<<shift - 1 - w := x[0] - nbits := uint(_W) // number of unprocessed bits in w - - // convert less-significant words - for k := 1; k < len(x); k++ { - // convert full digits - for nbits >= shift { - i-- - s[i] = charset[w&mask] - w >>= shift - nbits -= shift - } - - // convert any partial leading digit and advance to next word - if nbits == 0 { - // no partial digit remaining, just advance - w = x[k] - nbits = _W - } else { - // partial digit in current (k-1) and next (k) word - w |= x[k] << nbits - i-- - s[i] = charset[w&mask] - - // advance - w = x[k] >> (shift - nbits) - nbits = _W - (shift - nbits) - } - } - - // convert digits of most-significant word (omit leading zeros) - for nbits >= 0 && w != 0 { - i-- - s[i] = charset[w&mask] - w >>= shift - nbits -= shift - } - - return string(s[i:]) - } - - // general case: extract groups of digits by multiprecision division - - // maximize ndigits where b**ndigits < 2^_W; bb (big base) is b**ndigits - bb := Word(1) - ndigits := 0 - for max := Word(_M / b); bb <= max; bb *= b { - ndigits++ - } - - // preserve x, create local copy for use in repeated divisions - q := nat{}.set(x) - var r Word - - // convert - if b == 10 { // hard-coding for 10 here speeds this up by 1.25x - for len(q) > 0 { - // extract least significant, base bb "digit" - q, r = q.divW(q, bb) // N.B. >82% of time is here. Optimize divW - if len(q) == 0 { - // skip leading zeros in most-significant group of digits - for j := 0; j < ndigits && r != 0; j++ { - i-- - s[i] = charset[r%10] - r /= 10 - } - } else { - for j := 0; j < ndigits; j++ { - i-- - s[i] = charset[r%10] - r /= 10 - } - } - } - } else { - for len(q) > 0 { - // extract least significant group of digits - q, r = q.divW(q, bb) // N.B. >82% of time is here. Optimize divW - if len(q) == 0 { - // skip leading zeros in most-significant group of digits - for j := 0; j < ndigits && r != 0; j++ { - i-- - s[i] = charset[r%b] - r /= b - } - } else { - for j := 0; j < ndigits; j++ { - i-- - s[i] = charset[r%b] - r /= b - } - } - } - } - - return string(s[i:]) -} - -const deBruijn32 = 0x077CB531 - -var deBruijn32Lookup = []byte{ - 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8, - 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9, -} - -const deBruijn64 = 0x03f79d71b4ca8b09 - -var deBruijn64Lookup = []byte{ - 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4, - 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5, - 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11, - 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6, -} - -// trailingZeroBits returns the number of consecutive zero bits on the right -// side of the given Word. -// See Knuth, volume 4, section 7.3.1 -func trailingZeroBits(x Word) int { - // x & -x leaves only the right-most bit set in the word. Let k be the - // index of that bit. Since only a single bit is set, the value is two - // to the power of k. Multiplying by a power of two is equivalent to - // left shifting, in this case by k bits. The de Bruijn constant is - // such that all six bit, consecutive substrings are distinct. - // Therefore, if we have a left shifted version of this constant we can - // find by how many bits it was shifted by looking at which six bit - // substring ended up at the top of the word. - switch _W { - case 32: - return int(deBruijn32Lookup[((x&-x)*deBruijn32)>>27]) - case 64: - return int(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58]) - default: - panic("Unknown word size") - } - - return 0 -} - -// z = x << s -func (z nat) shl(x nat, s uint) nat { - m := len(x) - if m == 0 { - return z.make(0) - } - // m > 0 - - n := m + int(s/_W) - z = z.make(n + 1) - z[n] = shlVU(z[n-m:n], x, s%_W) - z[0 : n-m].clear() - - return z.norm() -} - -// z = x >> s -func (z nat) shr(x nat, s uint) nat { - m := len(x) - n := m - int(s/_W) - if n <= 0 { - return z.make(0) - } - // n > 0 - - z = z.make(n) - shrVU(z, x[m-n:], s%_W) - - return z.norm() -} - -func (z nat) setBit(x nat, i uint, b uint) nat { - j := int(i / _W) - m := Word(1) << (i % _W) - n := len(x) - switch b { - case 0: - z = z.make(n) - copy(z, x) - if j >= n { - // no need to grow - return z - } - z[j] &^= m - return z.norm() - case 1: - if j >= n { - n = j + 1 - } - z = z.make(n) - copy(z, x) - z[j] |= m - // no need to normalize - return z - } - panic("set bit is not 0 or 1") -} - -func (z nat) bit(i uint) uint { - j := int(i / _W) - if j >= len(z) { - return 0 - } - return uint(z[j] >> (i % _W) & 1) -} - -func (z nat) and(x, y nat) nat { - m := len(x) - n := len(y) - if m > n { - m = n - } - // m <= n - - z = z.make(m) - for i := 0; i < m; i++ { - z[i] = x[i] & y[i] - } - - return z.norm() -} - -func (z nat) andNot(x, y nat) nat { - m := len(x) - n := len(y) - if n > m { - n = m - } - // m >= n - - z = z.make(m) - for i := 0; i < n; i++ { - z[i] = x[i] &^ y[i] - } - copy(z[n:m], x[n:m]) - - return z.norm() -} - -func (z nat) or(x, y nat) nat { - m := len(x) - n := len(y) - s := x - if m < n { - n, m = m, n - s = y - } - // m >= n - - z = z.make(m) - for i := 0; i < n; i++ { - z[i] = x[i] | y[i] - } - copy(z[n:m], s[n:m]) - - return z.norm() -} - -func (z nat) xor(x, y nat) nat { - m := len(x) - n := len(y) - s := x - if m < n { - n, m = m, n - s = y - } - // m >= n - - z = z.make(m) - for i := 0; i < n; i++ { - z[i] = x[i] ^ y[i] - } - copy(z[n:m], s[n:m]) - - return z.norm() -} - -// greaterThan returns true iff (x1<<_W + x2) > (y1<<_W + y2) -func greaterThan(x1, x2, y1, y2 Word) bool { - return x1 > y1 || x1 == y1 && x2 > y2 -} - -// modW returns x % d. -func (x nat) modW(d Word) (r Word) { - // TODO(agl): we don't actually need to store the q value. - var q nat - q = q.make(len(x)) - return divWVW(q, 0, x, d) -} - -// powersOfTwoDecompose finds q and k with x = q * 1<<k and q is odd, or q and k are 0. -func (x nat) powersOfTwoDecompose() (q nat, k int) { - if len(x) == 0 { - return x, 0 - } - - // One of the words must be non-zero by definition, - // so this loop will terminate with i < len(x), and - // i is the number of 0 words. - i := 0 - for x[i] == 0 { - i++ - } - n := trailingZeroBits(x[i]) // x[i] != 0 - - q = make(nat, len(x)-i) - shrVU(q, x[i:], uint(n)) - - q = q.norm() - k = i*_W + n - return -} - -// random creates a random integer in [0..limit), using the space in z if -// possible. n is the bit length of limit. -func (z nat) random(rand *rand.Rand, limit nat, n int) nat { - bitLengthOfMSW := uint(n % _W) - if bitLengthOfMSW == 0 { - bitLengthOfMSW = _W - } - mask := Word((1 << bitLengthOfMSW) - 1) - z = z.make(len(limit)) - - for { - for i := range z { - switch _W { - case 32: - z[i] = Word(rand.Uint32()) - case 64: - z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32 - } - } - - z[len(limit)-1] &= mask - - if z.cmp(limit) < 0 { - break - } - } - - return z.norm() -} - -// If m != nil, expNN calculates x**y mod m. Otherwise it calculates x**y. It -// reuses the storage of z if possible. -func (z nat) expNN(x, y, m nat) nat { - if alias(z, x) || alias(z, y) { - // We cannot allow in place modification of x or y. - z = nil - } - - if len(y) == 0 { - z = z.make(1) - z[0] = 1 - return z - } - - if m != nil { - // We likely end up being as long as the modulus. - z = z.make(len(m)) - } - z = z.set(x) - v := y[len(y)-1] - // It's invalid for the most significant word to be zero, therefore we - // will find a one bit. - shift := leadingZeros(v) + 1 - v <<= shift - var q nat - - const mask = 1 << (_W - 1) - - // We walk through the bits of the exponent one by one. Each time we - // see a bit, we square, thus doubling the power. If the bit is a one, - // we also multiply by x, thus adding one to the power. - - w := _W - int(shift) - for j := 0; j < w; j++ { - z = z.mul(z, z) - - if v&mask != 0 { - z = z.mul(z, x) - } - - if m != nil { - q, z = q.div(z, z, m) - } - - v <<= 1 - } - - for i := len(y) - 2; i >= 0; i-- { - v = y[i] - - for j := 0; j < _W; j++ { - z = z.mul(z, z) - - if v&mask != 0 { - z = z.mul(z, x) - } - - if m != nil { - q, z = q.div(z, z, m) - } - - v <<= 1 - } - } - - return z -} - -// probablyPrime performs reps Miller-Rabin tests to check whether n is prime. -// If it returns true, n is prime with probability 1 - 1/4^reps. -// If it returns false, n is not prime. -func (n nat) probablyPrime(reps int) bool { - if len(n) == 0 { - return false - } - - if len(n) == 1 { - if n[0] < 2 { - return false - } - - if n[0]%2 == 0 { - return n[0] == 2 - } - - // We have to exclude these cases because we reject all - // multiples of these numbers below. - switch n[0] { - case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53: - return true - } - } - - const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29} - const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53} - - var r Word - switch _W { - case 32: - r = n.modW(primesProduct32) - case 64: - r = n.modW(primesProduct64 & _M) - default: - panic("Unknown word size") - } - - if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 || - r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 { - return false - } - - if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 || - r%43 == 0 || r%47 == 0 || r%53 == 0) { - return false - } - - nm1 := nat{}.sub(n, natOne) - // 1<<k * q = nm1; - q, k := nm1.powersOfTwoDecompose() - - nm3 := nat{}.sub(nm1, natTwo) - rand := rand.New(rand.NewSource(int64(n[0]))) - - var x, y, quotient nat - nm3Len := nm3.bitLen() - -NextRandom: - for i := 0; i < reps; i++ { - x = x.random(rand, nm3, nm3Len) - x = x.add(x, natTwo) - y = y.expNN(x, q, n) - if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 { - continue - } - for j := 1; j < k; j++ { - y = y.mul(y, y) - quotient, y = quotient.div(y, y, n) - if y.cmp(nm1) == 0 { - continue NextRandom - } - if y.cmp(natOne) == 0 { - return false - } - } - return false - } - - return true -} - -// bytes writes the value of z into buf using big-endian encoding. -// len(buf) must be >= len(z)*_S. The value of z is encoded in the -// slice buf[i:]. The number i of unused bytes at the beginning of -// buf is returned as result. -func (z nat) bytes(buf []byte) (i int) { - i = len(buf) - for _, d := range z { - for j := 0; j < _S; j++ { - i-- - buf[i] = byte(d) - d >>= 8 - } - } - - for i < len(buf) && buf[i] == 0 { - i++ - } - - return -} - -// setBytes interprets buf as the bytes of a big-endian unsigned -// integer, sets z to that value, and returns z. -func (z nat) setBytes(buf []byte) nat { - z = z.make((len(buf) + _S - 1) / _S) - - k := 0 - s := uint(0) - var d Word - for i := len(buf); i > 0; i-- { - d |= Word(buf[i-1]) << s - if s += 8; s == _S*8 { - z[k] = d - k++ - s = 0 - d = 0 - } - } - if k < len(z) { - z[k] = d - } - - return z.norm() -} |