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package bigfft
import (
"math/big"
)
// Arithmetic modulo 2^n+1.
// A fermat of length w+1 represents a number modulo 2^(w*_W) + 1. The last
// word is zero or one. A number has at most two representatives satisfying the
// 0-1 last word constraint.
type fermat nat
func (n fermat) String() string { return nat(n).String() }
func (z fermat) norm() {
n := len(z) - 1
c := z[n]
if c == 0 {
return
}
if z[0] >= c {
z[n] = 0
z[0] -= c
return
}
// z[0] < z[n].
subVW(z, z, c) // Substract c
if c > 1 {
z[n] -= c - 1
c = 1
}
// Add back c.
if z[n] == 1 {
z[n] = 0
return
} else {
addVW(z, z, 1)
}
}
// Shift computes (x << k) mod (2^n+1).
func (z fermat) Shift(x fermat, k int) {
if len(z) != len(x) {
panic("len(z) != len(x) in Shift")
}
n := len(x) - 1
// Shift by n*_W is taking the opposite.
k %= 2 * n * _W
if k < 0 {
k += 2 * n * _W
}
neg := false
if k >= n*_W {
k -= n * _W
neg = true
}
kw, kb := k/_W, k%_W
z[n] = 1 // Add (-1)
if !neg {
for i := 0; i < kw; i++ {
z[i] = 0
}
// Shift left by kw words.
// x = a·2^(n-k) + b
// x<<k = (b<<k) - a
copy(z[kw:], x[:n-kw])
b := subVV(z[:kw+1], z[:kw+1], x[n-kw:])
if z[kw+1] > 0 {
z[kw+1] -= b
} else {
subVW(z[kw+1:], z[kw+1:], b)
}
} else {
for i := kw + 1; i < n; i++ {
z[i] = 0
}
// Shift left and negate, by kw words.
copy(z[:kw+1], x[n-kw:n+1]) // z_low = x_high
b := subVV(z[kw:n], z[kw:n], x[:n-kw]) // z_high -= x_low
z[n] -= b
}
// Add back 1.
if z[n] > 0 {
z[n]--
} else if z[0] < ^big.Word(0) {
z[0]++
} else {
addVW(z, z, 1)
}
// Shift left by kb bits
shlVU(z, z, uint(kb))
z.norm()
}
// ShiftHalf shifts x by k/2 bits the left. Shifting by 1/2 bit
// is multiplication by sqrt(2) mod 2^n+1 which is 2^(3n/4) - 2^(n/4).
// A temporary buffer must be provided in tmp.
func (z fermat) ShiftHalf(x fermat, k int, tmp fermat) {
n := len(z) - 1
if k%2 == 0 {
z.Shift(x, k/2)
return
}
u := (k - 1) / 2
a := u + (3*_W/4)*n
b := u + (_W/4)*n
z.Shift(x, a)
tmp.Shift(x, b)
z.Sub(z, tmp)
}
// Add computes addition mod 2^n+1.
func (z fermat) Add(x, y fermat) fermat {
if len(z) != len(x) {
panic("Add: len(z) != len(x)")
}
addVV(z, x, y) // there cannot be a carry here.
z.norm()
return z
}
// Sub computes substraction mod 2^n+1.
func (z fermat) Sub(x, y fermat) fermat {
if len(z) != len(x) {
panic("Add: len(z) != len(x)")
}
n := len(y) - 1
b := subVV(z[:n], x[:n], y[:n])
b += y[n]
// If b > 0, we need to subtract b<<n, which is the same as adding b.
z[n] = x[n]
if z[0] <= ^big.Word(0)-b {
z[0] += b
} else {
addVW(z, z, b)
}
z.norm()
return z
}
func (z fermat) Mul(x, y fermat) fermat {
if len(x) != len(y) {
panic("Mul: len(x) != len(y)")
}
n := len(x) - 1
if n < 30 {
z = z[:2*n+2]
basicMul(z, x, y)
z = z[:2*n+1]
} else {
var xi, yi, zi big.Int
xi.SetBits(x)
yi.SetBits(y)
zi.SetBits(z)
zb := zi.Mul(&xi, &yi).Bits()
if len(zb) <= n {
// Short product.
copy(z, zb)
for i := len(zb); i < len(z); i++ {
z[i] = 0
}
return z
}
z = zb
}
// len(z) is at most 2n+1.
if len(z) > 2*n+1 {
panic("len(z) > 2n+1")
}
// We now have
// z = z[:n] + 1<<(n*W) * z[n:2n+1]
// which normalizes to:
// z = z[:n] - z[n:2n] + z[2n]
c1 := big.Word(0)
if len(z) > 2*n {
c1 = addVW(z[:n], z[:n], z[2*n])
}
c2 := big.Word(0)
if len(z) >= 2*n {
c2 = subVV(z[:n], z[:n], z[n:2*n])
} else {
m := len(z) - n
c2 = subVV(z[:m], z[:m], z[n:])
c2 = subVW(z[m:n], z[m:n], c2)
}
// Restore carries.
// Substracting z[n] -= c2 is the same
// as z[0] += c2
z = z[:n+1]
z[n] = c1
c := addVW(z, z, c2)
if c != 0 {
panic("impossible")
}
z.norm()
return z
}
// copied from math/big
//
// 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 fermat) {
// initialize z
for i := 0; i < len(z); i++ {
z[i] = 0
}
for i, d := range y {
if d != 0 {
z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
}
}
}
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