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// 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.
// This file implements signed multi-precision integers.
package big
import (
"fmt"
"io"
"os"
"rand"
"strings"
)
// An Int represents a signed multi-precision integer.
// The zero value for an Int represents the value 0.
type Int struct {
neg bool // sign
abs nat // absolute value of the integer
}
var intOne = &Int{false, natOne}
// Sign returns:
//
// -1 if x < 0
// 0 if x == 0
// +1 if x > 0
//
func (x *Int) Sign() int {
if len(x.abs) == 0 {
return 0
}
if x.neg {
return -1
}
return 1
}
// SetInt64 sets z to x and returns z.
func (z *Int) SetInt64(x int64) *Int {
neg := false
if x < 0 {
neg = true
x = -x
}
z.abs = z.abs.setUint64(uint64(x))
z.neg = neg
return z
}
// NewInt allocates and returns a new Int set to x.
func NewInt(x int64) *Int {
return new(Int).SetInt64(x)
}
// Set sets z to x and returns z.
func (z *Int) Set(x *Int) *Int {
z.abs = z.abs.set(x.abs)
z.neg = x.neg
return z
}
// Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Int) Abs(x *Int) *Int {
z.abs = z.abs.set(x.abs)
z.neg = false
return z
}
// Neg sets z to -x and returns z.
func (z *Int) Neg(x *Int) *Int {
z.abs = z.abs.set(x.abs)
z.neg = len(z.abs) > 0 && !x.neg // 0 has no sign
return z
}
// Add sets z to the sum x+y and returns z.
func (z *Int) Add(x, y *Int) *Int {
neg := x.neg
if x.neg == y.neg {
// x + y == x + y
// (-x) + (-y) == -(x + y)
z.abs = z.abs.add(x.abs, y.abs)
} else {
// x + (-y) == x - y == -(y - x)
// (-x) + y == y - x == -(x - y)
if x.abs.cmp(y.abs) >= 0 {
z.abs = z.abs.sub(x.abs, y.abs)
} else {
neg = !neg
z.abs = z.abs.sub(y.abs, x.abs)
}
}
z.neg = len(z.abs) > 0 && neg // 0 has no sign
return z
}
// Sub sets z to the difference x-y and returns z.
func (z *Int) Sub(x, y *Int) *Int {
neg := x.neg
if x.neg != y.neg {
// x - (-y) == x + y
// (-x) - y == -(x + y)
z.abs = z.abs.add(x.abs, y.abs)
} else {
// x - y == x - y == -(y - x)
// (-x) - (-y) == y - x == -(x - y)
if x.abs.cmp(y.abs) >= 0 {
z.abs = z.abs.sub(x.abs, y.abs)
} else {
neg = !neg
z.abs = z.abs.sub(y.abs, x.abs)
}
}
z.neg = len(z.abs) > 0 && neg // 0 has no sign
return z
}
// Mul sets z to the product x*y and returns z.
func (z *Int) Mul(x, y *Int) *Int {
// x * y == x * y
// x * (-y) == -(x * y)
// (-x) * y == -(x * y)
// (-x) * (-y) == x * y
z.abs = z.abs.mul(x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
return z
}
// MulRange sets z to the product of all integers
// in the range [a, b] inclusively and returns z.
// If a > b (empty range), the result is 1.
func (z *Int) MulRange(a, b int64) *Int {
switch {
case a > b:
return z.SetInt64(1) // empty range
case a <= 0 && b >= 0:
return z.SetInt64(0) // range includes 0
}
// a <= b && (b < 0 || a > 0)
neg := false
if a < 0 {
neg = (b-a)&1 == 0
a, b = -b, -a
}
z.abs = z.abs.mulRange(uint64(a), uint64(b))
z.neg = neg
return z
}
// Binomial sets z to the binomial coefficient of (n, k) and returns z.
func (z *Int) Binomial(n, k int64) *Int {
var a, b Int
a.MulRange(n-k+1, n)
b.MulRange(1, k)
return z.Quo(&a, &b)
}
// Quo sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// See QuoRem for more details.
func (z *Int) Quo(x, y *Int) *Int {
z.abs, _ = z.abs.div(nil, x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
return z
}
// Rem sets z to the remainder x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// See QuoRem for more details.
func (z *Int) Rem(x, y *Int) *Int {
_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
return z
}
// QuoRem sets z to the quotient x/y and r to the remainder x%y
// and returns the pair (z, r) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// QuoRem implements T-division and modulus (like Go):
//
// q = x/y with the result truncated to zero
// r = x - y*q
//
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
//
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
return z, r
}
// Div sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// See DivMod for more details.
func (z *Int) Div(x, y *Int) *Int {
y_neg := y.neg // z may be an alias for y
var r Int
z.QuoRem(x, y, &r)
if r.neg {
if y_neg {
z.Add(z, intOne)
} else {
z.Sub(z, intOne)
}
}
return z
}
// Mod sets z to the modulus x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// See DivMod for more details.
func (z *Int) Mod(x, y *Int) *Int {
y0 := y // save y
if z == y || alias(z.abs, y.abs) {
y0 = new(Int).Set(y)
}
var q Int
q.QuoRem(x, y, z)
if z.neg {
if y0.neg {
z.Sub(z, y0)
} else {
z.Add(z, y0)
}
}
return z
}
// DivMod sets z to the quotient x div y and m to the modulus x mod y
// and returns the pair (z, m) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// DivMod implements Euclidean division and modulus (unlike Go):
//
// q = x div y such that
// m = x - y*q with 0 <= m < |q|
//
// (See Raymond T. Boute, ``The Euclidean definition of the functions
// div and mod''. ACM Transactions on Programming Languages and
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
// ACM press.)
//
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
y0 := y // save y
if z == y || alias(z.abs, y.abs) {
y0 = new(Int).Set(y)
}
z.QuoRem(x, y, m)
if m.neg {
if y0.neg {
z.Add(z, intOne)
m.Sub(m, y0)
} else {
z.Sub(z, intOne)
m.Add(m, y0)
}
}
return z, m
}
// Cmp compares x and y and returns:
//
// -1 if x < y
// 0 if x == y
// +1 if x > y
//
func (x *Int) Cmp(y *Int) (r int) {
// x cmp y == x cmp y
// x cmp (-y) == x
// (-x) cmp y == y
// (-x) cmp (-y) == -(x cmp y)
switch {
case x.neg == y.neg:
r = x.abs.cmp(y.abs)
if x.neg {
r = -r
}
case x.neg:
r = -1
default:
r = 1
}
return
}
func (x *Int) String() string {
switch {
case x == nil:
return "<nil>"
case x.neg:
return "-" + x.abs.decimalString()
}
return x.abs.decimalString()
}
func charset(ch int) string {
switch ch {
case 'b':
return lowercaseDigits[0:2]
case 'o':
return lowercaseDigits[0:8]
case 'd', 's', 'v':
return lowercaseDigits[0:10]
case 'x':
return lowercaseDigits[0:16]
case 'X':
return uppercaseDigits[0:16]
}
return "" // unknown format
}
// write count copies of text to s
func writeMultiple(s fmt.State, text string, count int) {
if len(text) > 0 {
b := []byte(text)
for ; count > 0; count-- {
s.Write(b)
}
}
}
// Format is a support routine for fmt.Formatter. It accepts
// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
// Also supported are the full suite of package fmt's format
// verbs for integral types, including '+', '-', and ' '
// for sign control, '#' for leading zero in octal and for
// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
// respectively, specification of minimum digits precision,
// output field width, space or zero padding, and left or
// right justification.
//
func (x *Int) Format(s fmt.State, ch int) {
cs := charset(ch)
// special cases
switch {
case cs == "":
// unknown format
fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
return
case x == nil:
fmt.Fprint(s, "<nil>")
return
}
// determine sign character
sign := ""
switch {
case x.neg:
sign = "-"
case s.Flag('+'): // supersedes ' ' when both specified
sign = "+"
case s.Flag(' '):
sign = " "
}
// determine prefix characters for indicating output base
prefix := ""
if s.Flag('#') {
switch ch {
case 'o': // octal
prefix = "0"
case 'x': // hexadecimal
prefix = "0x"
case 'X':
prefix = "0X"
}
}
// determine digits with base set by len(cs) and digit characters from cs
digits := x.abs.string(cs)
// number of characters for the three classes of number padding
var left int // space characters to left of digits for right justification ("%8d")
var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
var right int // space characters to right of digits for left justification ("%-8d")
// determine number padding from precision: the least number of digits to output
precision, precisionSet := s.Precision()
if precisionSet {
switch {
case len(digits) < precision:
zeroes = precision - len(digits) // count of zero padding
case digits == "0" && precision == 0:
return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
}
}
// determine field pad from width: the least number of characters to output
length := len(sign) + len(prefix) + zeroes + len(digits)
if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
switch d := width - length; {
case s.Flag('-'):
// pad on the right with spaces; supersedes '0' when both specified
right = d
case s.Flag('0') && !precisionSet:
// pad with zeroes unless precision also specified
zeroes = d
default:
// pad on the left with spaces
left = d
}
}
// print number as [left pad][sign][prefix][zero pad][digits][right pad]
writeMultiple(s, " ", left)
writeMultiple(s, sign, 1)
writeMultiple(s, prefix, 1)
writeMultiple(s, "0", zeroes)
writeMultiple(s, digits, 1)
writeMultiple(s, " ", right)
}
// scan sets z to the integer value corresponding to the longest possible prefix
// read from r representing a signed integer number 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
// 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 *Int) scan(r io.RuneScanner, base int) (*Int, int, os.Error) {
// determine sign
ch, _, err := r.ReadRune()
if err != nil {
return z, 0, err
}
neg := false
switch ch {
case '-':
neg = true
case '+': // nothing to do
default:
r.UnreadRune()
}
// determine mantissa
z.abs, base, err = z.abs.scan(r, base)
if err != nil {
return z, base, err
}
z.neg = len(z.abs) > 0 && neg // 0 has no sign
return z, base, nil
}
// Scan is a support routine for fmt.Scanner; it sets z to the value of
// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
func (z *Int) Scan(s fmt.ScanState, ch int) os.Error {
s.SkipSpace() // skip leading space characters
base := 0
switch ch {
case 'b':
base = 2
case 'o':
base = 8
case 'd':
base = 10
case 'x', 'X':
base = 16
case 's', 'v':
// let scan determine the base
default:
return os.NewError("Int.Scan: invalid verb")
}
_, _, err := z.scan(s, base)
return err
}
// Int64 returns the int64 representation of x.
// If x cannot be represented in an int64, the result is undefined.
func (x *Int) Int64() int64 {
if len(x.abs) == 0 {
return 0
}
v := int64(x.abs[0])
if _W == 32 && len(x.abs) > 1 {
v |= int64(x.abs[1]) << 32
}
if x.neg {
v = -v
}
return v
}
// SetString sets z to the value of s, interpreted in the given base,
// and returns z and a boolean indicating success. If SetString fails,
// the value of z is undefined.
//
// 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 *Int) SetString(s string, base int) (*Int, bool) {
r := strings.NewReader(s)
_, _, err := z.scan(r, base)
if err != nil {
return z, false
}
_, _, err = r.ReadRune()
return z, err == os.EOF // err == os.EOF => scan consumed all of s
}
// SetBytes interprets buf as the bytes of a big-endian unsigned
// integer, sets z to that value, and returns z.
func (z *Int) SetBytes(buf []byte) *Int {
z.abs = z.abs.setBytes(buf)
z.neg = false
return z
}
// Bytes returns the absolute value of z as a big-endian byte slice.
func (z *Int) Bytes() []byte {
buf := make([]byte, len(z.abs)*_S)
return buf[z.abs.bytes(buf):]
}
// BitLen returns the length of the absolute value of z in bits.
// The bit length of 0 is 0.
func (z *Int) BitLen() int {
return z.abs.bitLen()
}
// Exp sets z = x**y mod m. If m is nil, z = x**y.
// See Knuth, volume 2, section 4.6.3.
func (z *Int) Exp(x, y, m *Int) *Int {
if y.neg || len(y.abs) == 0 {
neg := x.neg
z.SetInt64(1)
z.neg = neg
return z
}
var mWords nat
if m != nil {
mWords = m.abs
}
z.abs = z.abs.expNN(x.abs, y.abs, mWords)
z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
return z
}
// GcdInt sets d to the greatest common divisor of a and b, which must be
// positive numbers.
// If x and y are not nil, GcdInt sets x and y such that d = a*x + b*y.
// If either a or b is not positive, GcdInt sets d = x = y = 0.
func GcdInt(d, x, y, a, b *Int) {
if a.neg || b.neg {
d.SetInt64(0)
if x != nil {
x.SetInt64(0)
}
if y != nil {
y.SetInt64(0)
}
return
}
A := new(Int).Set(a)
B := new(Int).Set(b)
X := new(Int)
Y := new(Int).SetInt64(1)
lastX := new(Int).SetInt64(1)
lastY := new(Int)
q := new(Int)
temp := new(Int)
for len(B.abs) > 0 {
r := new(Int)
q, r = q.QuoRem(A, B, r)
A, B = B, r
temp.Set(X)
X.Mul(X, q)
X.neg = !X.neg
X.Add(X, lastX)
lastX.Set(temp)
temp.Set(Y)
Y.Mul(Y, q)
Y.neg = !Y.neg
Y.Add(Y, lastY)
lastY.Set(temp)
}
if x != nil {
*x = *lastX
}
if y != nil {
*y = *lastY
}
*d = *A
}
// ProbablyPrime performs n Miller-Rabin tests to check whether z is prime.
// If it returns true, z is prime with probability 1 - 1/4^n.
// If it returns false, z is not prime.
func ProbablyPrime(z *Int, n int) bool {
return !z.neg && z.abs.probablyPrime(n)
}
// Rand sets z to a pseudo-random number in [0, n) and returns z.
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
z.neg = false
if n.neg == true || len(n.abs) == 0 {
z.abs = nil
return z
}
z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
return z
}
// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
// p is a prime) and returns z.
func (z *Int) ModInverse(g, p *Int) *Int {
var d Int
GcdInt(&d, z, nil, g, p)
// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
// that modulo p results in g*x = 1, therefore x is the inverse element.
if z.neg {
z.Add(z, p)
}
return z
}
// Lsh sets z = x << n and returns z.
func (z *Int) Lsh(x *Int, n uint) *Int {
z.abs = z.abs.shl(x.abs, n)
z.neg = x.neg
return z
}
// Rsh sets z = x >> n and returns z.
func (z *Int) Rsh(x *Int, n uint) *Int {
if x.neg {
// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
t = t.shr(t, n)
z.abs = t.add(t, natOne)
z.neg = true // z cannot be zero if x is negative
return z
}
z.abs = z.abs.shr(x.abs, n)
z.neg = false
return z
}
// Bit returns the value of the i'th bit of z. That is, it
// returns (z>>i)&1. The bit index i must be >= 0.
func (z *Int) Bit(i int) uint {
if i < 0 {
panic("negative bit index")
}
if z.neg {
t := nat{}.sub(z.abs, natOne)
return t.bit(uint(i)) ^ 1
}
return z.abs.bit(uint(i))
}
// SetBit sets the i'th bit of z to bit and returns z.
// That is, if bit is 1 SetBit sets z = x | (1 << i);
// if bit is 0 it sets z = x &^ (1 << i). If bit is not 0 or 1,
// SetBit will panic.
func (z *Int) SetBit(x *Int, i int, b uint) *Int {
if i < 0 {
panic("negative bit index")
}
if x.neg {
t := z.abs.sub(x.abs, natOne)
t = t.setBit(t, uint(i), b^1)
z.abs = t.add(t, natOne)
z.neg = len(z.abs) > 0
return z
}
z.abs = z.abs.setBit(x.abs, uint(i), b)
z.neg = false
return z
}
// And sets z = x & y and returns z.
func (z *Int) And(x, y *Int) *Int {
if x.neg == y.neg {
if x.neg {
// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
x1 := nat{}.sub(x.abs, natOne)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
z.neg = true // z cannot be zero if x and y are negative
return z
}
// x & y == x & y
z.abs = z.abs.and(x.abs, y.abs)
z.neg = false
return z
}
// x.neg != y.neg
if x.neg {
x, y = y, x // & is symmetric
}
// x & (-y) == x & ^(y-1) == x &^ (y-1)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.andNot(x.abs, y1)
z.neg = false
return z
}
// AndNot sets z = x &^ y and returns z.
func (z *Int) AndNot(x, y *Int) *Int {
if x.neg == y.neg {
if x.neg {
// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
x1 := nat{}.sub(x.abs, natOne)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.andNot(y1, x1)
z.neg = false
return z
}
// x &^ y == x &^ y
z.abs = z.abs.andNot(x.abs, y.abs)
z.neg = false
return z
}
if x.neg {
// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
x1 := nat{}.sub(x.abs, natOne)
z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
z.neg = true // z cannot be zero if x is negative and y is positive
return z
}
// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
y1 := nat{}.add(y.abs, natOne)
z.abs = z.abs.and(x.abs, y1)
z.neg = false
return z
}
// Or sets z = x | y and returns z.
func (z *Int) Or(x, y *Int) *Int {
if x.neg == y.neg {
if x.neg {
// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
x1 := nat{}.sub(x.abs, natOne)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
z.neg = true // z cannot be zero if x and y are negative
return z
}
// x | y == x | y
z.abs = z.abs.or(x.abs, y.abs)
z.neg = false
return z
}
// x.neg != y.neg
if x.neg {
x, y = y, x // | is symmetric
}
// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
z.neg = true // z cannot be zero if one of x or y is negative
return z
}
// Xor sets z = x ^ y and returns z.
func (z *Int) Xor(x, y *Int) *Int {
if x.neg == y.neg {
if x.neg {
// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
x1 := nat{}.sub(x.abs, natOne)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.xor(x1, y1)
z.neg = false
return z
}
// x ^ y == x ^ y
z.abs = z.abs.xor(x.abs, y.abs)
z.neg = false
return z
}
// x.neg != y.neg
if x.neg {
x, y = y, x // ^ is symmetric
}
// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
y1 := nat{}.sub(y.abs, natOne)
z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
z.neg = true // z cannot be zero if only one of x or y is negative
return z
}
// Not sets z = ^x and returns z.
func (z *Int) Not(x *Int) *Int {
if x.neg {
// ^(-x) == ^(^(x-1)) == x-1
z.abs = z.abs.sub(x.abs, natOne)
z.neg = false
return z
}
// ^x == -x-1 == -(x+1)
z.abs = z.abs.add(x.abs, natOne)
z.neg = true // z cannot be zero if x is positive
return z
}
// Gob codec version. Permits backward-compatible changes to the encoding.
const intGobVersion byte = 1
// GobEncode implements the gob.GobEncoder interface.
func (z *Int) GobEncode() ([]byte, os.Error) {
buf := make([]byte, 1+len(z.abs)*_S) // extra byte for version and sign bit
i := z.abs.bytes(buf) - 1 // i >= 0
b := intGobVersion << 1 // make space for sign bit
if z.neg {
b |= 1
}
buf[i] = b
return buf[i:], nil
}
// GobDecode implements the gob.GobDecoder interface.
func (z *Int) GobDecode(buf []byte) os.Error {
if len(buf) == 0 {
return os.NewError("Int.GobDecode: no data")
}
b := buf[0]
if b>>1 != intGobVersion {
return os.NewError(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
}
z.neg = b&1 != 0
z.abs = z.abs.setBytes(buf[1:])
return nil
}