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math/big: speed up GCD x, y calculation

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The current implementation of the extended Euclidean GCD algorithm
calculates both cosequences x and y inside the division loop. This
is unneccessary since the second Bezout coefficient can be obtained
at the end of calculation via a multiplication, subtraction and a
division.  In case only one coefficient is needed, e.g. ModInverse
this calculation can be skipped entirely.  This is a standard
optimization, see e.g.

"Handbook of Elliptic and Hyperelliptic Curve Cryptography"
Cohen et al pp 191
Available at:
http://cs.ucsb.edu/~koc/ccs130h/2013/EllipticHyperelliptic-CohenFrey.pdf

Updates #15833

Change-Id: I1e0d2e63567cfed97fd955048fe6373d36f22757
Reviewed-on: https://go-review.googlesource.com/50530
Reviewed-by: Robert Griesemer <gri@golang.org>
This commit is contained in:
Brian Kessler 2017-07-20 23:31:51 -07:00 committed by Robert Griesemer
parent 1246566142
commit 53836a74f8
1 changed files with 6 additions and 13 deletions
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19
src/math/big/int.go
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@ -447,7 +447,7 @@ func (z *Int) Exp(x, y, m *Int) *Int {
// GCD sets z to the greatest common divisor of a and b, which both must
// be > 0, and returns z.
// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
// If x or y are not nil, GCD sets their value such that z = a*x + b*y.
// If either a or b is <= 0, GCD sets z = x = y = 0.
func (z *Int) GCD(x, y, a, b *Int) *Int {
if a.Sign() <= 0 || b.Sign() <= 0 {
@ -468,10 +468,7 @@ func (z *Int) GCD(x, y, a, b *Int) *Int {
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)
@ -484,15 +481,8 @@ func (z *Int) GCD(x, y, a, b *Int) *Int {
temp.Set(X)
X.Mul(X, q)
X.neg = !X.neg
X.Add(X, lastX)
X.Sub(lastX, X)
lastX.Set(temp)
temp.Set(Y)
Y.Mul(Y, q)
Y.neg = !Y.neg
Y.Add(Y, lastY)
lastY.Set(temp)
}
if x != nil {
@ -500,7 +490,10 @@ func (z *Int) GCD(x, y, a, b *Int) *Int {
}
if y != nil {
*y = *lastY
// y = (z - a*x)/b
y.Mul(a, lastX)
y.Sub(A, y)
y.Div(y, b)
}
*z = *A