math/big: fix big.Exp and document better

- always return 1 for y <= 0
- document that the sign of m is ignored
- protect against div-0 panics by treating
  m == 0 the same way as m == nil
- added extra tests

Fixes #4239.

R=agl, remyoudompheng, agl
CC=golang-dev
https://golang.org/cl/6724046

src/pkg/math/big/int.go

@@ -561,19 +561,18 @@ func (x *Int) BitLen() int {
 	return x.abs.bitLen()
 }
 
-// Exp sets z = x**y mod m and returns z. If m is nil, z = x**y.
+// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
+// If y <= 0, the result is 1; if m == nil or m == 0, 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
+		return z.SetInt64(1)
 	}
+	// y > 0
 
 	var mWords nat
 	if m != nil {
-		mWords = m.abs
+		mWords = m.abs // m.abs may be nil for m == 0
 	}
 
 	z.abs = z.abs.expNN(x.abs, y.abs, mWords)

src/pkg/math/big/int_test.go

@@ -767,8 +767,10 @@ var expTests = []struct {
 	x, y, m string
 	out     string
 }{
+	{"5", "-7", "", "1"},
+	{"-5", "-7", "", "1"},
 	{"5", "0", "", "1"},
-	{"-5", "0", "", "-1"},
+	{"-5", "0", "", "1"},
 	{"5", "1", "", "5"},
 	{"-5", "1", "", "-5"},
 	{"-2", "3", "2", "0"},
@@ -779,6 +781,7 @@ var expTests = []struct {
 	{"0x8000000000000000", "3", "6719", "5447"},
 	{"0x8000000000000000", "1000", "6719", "1603"},
 	{"0x8000000000000000", "1000000", "6719", "3199"},
+	{"0x8000000000000000", "-1000000", "6719", "1"},
 	{
 		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
 		"298472983472983471903246121093472394872319615612417471234712061",
@@ -807,12 +810,22 @@ func TestExp(t *testing.T) {
 			continue
 		}
 
-		z := y.Exp(x, y, m)
-		if !isNormalized(z) {
-			t.Errorf("#%d: %v is not normalized", i, *z)
+		z1 := new(Int).Exp(x, y, m)
+		if !isNormalized(z1) {
+			t.Errorf("#%d: %v is not normalized", i, *z1)
 		}
-		if z.Cmp(out) != 0 {
-			t.Errorf("#%d: got %s want %s", i, z, out)
+		if z1.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z1, out)
+		}
+
+		if m == nil {
+			// the result should be the same as for m == 0;
+			// specifically, there should be no div-zero panic
+			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+			z2 := new(Int).Exp(x, y, m)
+			if z2.Cmp(z1) != 0 {
+				t.Errorf("#%d: got %s want %s", i, z1, z2)
+			}
 		}
 	}
 }

src/pkg/math/big/nat.go

@@ -1227,8 +1227,8 @@ func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
 	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.
+// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
+// otherwise it sets z to x**y. The result is the value of z.
 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.
@@ -1240,15 +1240,15 @@ func (z nat) expNN(x, y, m nat) nat {
 		z[0] = 1
 		return z
 	}
+	// y > 0
 
-	if m != nil {
+	if len(m) != 0 {
 		// 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.
+
+	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
 	shift := leadingZeros(v) + 1
 	v <<= shift
 	var q nat
@@ -1272,7 +1272,7 @@ func (z nat) expNN(x, y, m nat) nat {
 			zz, z = z, zz
 		}
 
-		if m != nil {
+		if len(m) != 0 {
 			zz, r = zz.div(r, z, m)
 			zz, r, q, z = q, z, zz, r
 		}
@@ -1292,7 +1292,7 @@ func (z nat) expNN(x, y, m nat) nat {
 				zz, z = z, zz
 			}
 
-			if m != nil {
+			if len(m) != 0 {
 				zz, r = zz.div(r, z, m)
 				zz, r, q, z = q, z, zz, r
 			}