/usr/share/gocode/src/github.com/restic/chunker/polynomials.go is in golang-github-restic-chunker-dev 0.2.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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import (
"crypto/rand"
"encoding/binary"
"errors"
"fmt"
"io"
"strconv"
)
// Pol is a polynomial from F_2[X].
type Pol uint64
// Add returns x+y.
func (x Pol) Add(y Pol) Pol {
r := Pol(uint64(x) ^ uint64(y))
return r
}
// mulOverflows returns true if the multiplication would overflow uint64.
// Code by Rob Pike, see
// https://groups.google.com/d/msg/golang-nuts/h5oSN5t3Au4/KaNQREhZh0QJ
func mulOverflows(a, b Pol) bool {
if a <= 1 || b <= 1 {
return false
}
c := a.mul(b)
d := c.Div(b)
if d != a {
return true
}
return false
}
func (x Pol) mul(y Pol) Pol {
if x == 0 || y == 0 {
return 0
}
var res Pol
for i := 0; i <= y.Deg(); i++ {
if (y & (1 << uint(i))) > 0 {
res = res.Add(x << uint(i))
}
}
return res
}
// Mul returns x*y. When an overflow occurs, Mul panics.
func (x Pol) Mul(y Pol) Pol {
if mulOverflows(x, y) {
panic("multiplication would overflow uint64")
}
return x.mul(y)
}
// Deg returns the degree of the polynomial x. If x is zero, -1 is returned.
func (x Pol) Deg() int {
// the degree of 0 is -1
if x == 0 {
return -1
}
// see https://graphics.stanford.edu/~seander/bithacks.html#IntegerLog
r := 0
if uint64(x)&0xffffffff00000000 > 0 {
x >>= 32
r |= 32
}
if uint64(x)&0xffff0000 > 0 {
x >>= 16
r |= 16
}
if uint64(x)&0xff00 > 0 {
x >>= 8
r |= 8
}
if uint64(x)&0xf0 > 0 {
x >>= 4
r |= 4
}
if uint64(x)&0xc > 0 {
x >>= 2
r |= 2
}
if uint64(x)&0x2 > 0 {
x >>= 1
r |= 1
}
return r
}
// String returns the coefficients in hex.
func (x Pol) String() string {
return "0x" + strconv.FormatUint(uint64(x), 16)
}
// Expand returns the string representation of the polynomial x.
func (x Pol) Expand() string {
if x == 0 {
return "0"
}
s := ""
for i := x.Deg(); i > 1; i-- {
if x&(1<<uint(i)) > 0 {
s += fmt.Sprintf("+x^%d", i)
}
}
if x&2 > 0 {
s += "+x"
}
if x&1 > 0 {
s += "+1"
}
return s[1:]
}
// DivMod returns x / d = q, and remainder r,
// see https://en.wikipedia.org/wiki/Division_algorithm
func (x Pol) DivMod(d Pol) (Pol, Pol) {
if x == 0 {
return 0, 0
}
if d == 0 {
panic("division by zero")
}
D := d.Deg()
diff := x.Deg() - D
if diff < 0 {
return 0, x
}
var q Pol
for diff >= 0 {
m := d << uint(diff)
q |= (1 << uint(diff))
x = x.Add(m)
diff = x.Deg() - D
}
return q, x
}
// Div returns the integer division result x / d.
func (x Pol) Div(d Pol) Pol {
q, _ := x.DivMod(d)
return q
}
// Mod returns the remainder of x / d
func (x Pol) Mod(d Pol) Pol {
_, r := x.DivMod(d)
return r
}
// I really dislike having a function that does not terminate, so specify a
// really large upper bound for finding a new irreducible polynomial, and
// return an error when no irreducible polynomial has been found within
// randPolMaxTries.
const randPolMaxTries = 1e6
// RandomPolynomial returns a new random irreducible polynomial
// of degree 53 using the default System CSPRNG as source.
// It is equivalent to calling DerivePolynomial(rand.Reader).
func RandomPolynomial() (Pol, error) {
return DerivePolynomial(rand.Reader)
}
// DerivePolynomial returns an irreducible polynomial of degree 53
// (largest prime number below 64-8) by reading bytes from source.
// There are (2^53-2/53) irreducible polynomials of degree 53 in
// F_2[X], c.f. Michael O. Rabin (1981): "Fingerprinting by Random
// Polynomials", page 4. If no polynomial could be found in one
// million tries, an error is returned.
func DerivePolynomial(source io.Reader) (Pol, error) {
for i := 0; i < randPolMaxTries; i++ {
var f Pol
// choose polynomial at (pseudo)random
err := binary.Read(source, binary.LittleEndian, &f)
if err != nil {
return 0, err
}
// mask away bits above bit 53
f &= Pol((1 << 54) - 1)
// set highest and lowest bit so that the degree is 53 and the
// polynomial is not trivially reducible
f |= (1 << 53) | 1
// test if f is irreducible
if f.Irreducible() {
return f, nil
}
}
// If this is reached, we haven't found an irreducible polynomial in
// randPolMaxTries. This error is very unlikely to occur.
return 0, errors.New("unable to find new random irreducible polynomial")
}
// GCD computes the Greatest Common Divisor x and f.
func (x Pol) GCD(f Pol) Pol {
if f == 0 {
return x
}
if x == 0 {
return f
}
if x.Deg() < f.Deg() {
x, f = f, x
}
return f.GCD(x.Mod(f))
}
// Irreducible returns true iff x is irreducible over F_2. This function
// uses Ben Or's reducibility test.
//
// For details see "Tests and Constructions of Irreducible Polynomials over
// Finite Fields".
func (x Pol) Irreducible() bool {
for i := 1; i <= x.Deg()/2; i++ {
if x.GCD(qp(uint(i), x)) != 1 {
return false
}
}
return true
}
// MulMod computes x*f mod g
func (x Pol) MulMod(f, g Pol) Pol {
if x == 0 || f == 0 {
return 0
}
var res Pol
for i := 0; i <= f.Deg(); i++ {
if (f & (1 << uint(i))) > 0 {
a := x
for j := 0; j < i; j++ {
a = a.Mul(2).Mod(g)
}
res = res.Add(a).Mod(g)
}
}
return res
}
// qp computes the polynomial (x^(2^p)-x) mod g. This is needed for the
// reducibility test.
func qp(p uint, g Pol) Pol {
num := (1 << p)
i := 1
// start with x
res := Pol(2)
for i < num {
// repeatedly square res
res = res.MulMod(res, g)
i *= 2
}
// add x
return res.Add(2).Mod(g)
}
// MarshalJSON returns the JSON representation of the Pol.
func (x Pol) MarshalJSON() ([]byte, error) {
buf := strconv.AppendUint([]byte{'"'}, uint64(x), 16)
buf = append(buf, '"')
return buf, nil
}
// UnmarshalJSON parses a Pol from the JSON data.
func (x *Pol) UnmarshalJSON(data []byte) error {
if len(data) < 2 {
return errors.New("invalid string for polynomial")
}
n, err := strconv.ParseUint(string(data[1:len(data)-1]), 16, 64)
if err != nil {
return err
}
*x = Pol(n)
return nil
}
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