big: completed set of Int division routines & cleanups

- renamed Len -> BitLen, simplified implementation
- renamed old Div, Mod, DivMod -> Que, Rem, QuoRem
- implemented Div, Mod, DivMod (Euclidian definition, more
  useful in a mathematical context)
- fixed a bug in Exp (-0 was possible)
- added extra tests to check normalized results everywhere
- uniformly set Int.neg flag at the end of computations
- minor cosmetic cleanups
- ran all tests

R=rsc
CC=golang-dev
https://golang.org/cl/1091041

8 files changed, 332 insertions, 172 deletions

diff --git a/src/pkg/big/int.go b/src/pkg/big/int.go
index 4d1be5db69..2382924787 100755
--- a/src/pkg/big/int.go
+++ b/src/pkg/big/int.go
@@ -14,8 +14,11 @@ type Int struct {
 }
 
 
-// New allocates and returns a new Int set to x.
-func (z *Int) New(x int64) *Int {
+var intOne = &Int{false, natOne}
+
+
+// SetInt64 sets z to x and returns z.
+func (z *Int) SetInt64(x int64) *Int {
 	z.neg = false
 	if x < 0 {
 		z.neg = true
@@ -27,7 +30,9 @@ func (z *Int) New(x int64) *Int {
 
 
 // NewInt allocates and returns a new Int set to x.
-func NewInt(x int64) *Int { return new(Int).New(x) }
+func NewInt(x int64) *Int {
+	return new(Int).SetInt64(x)
+}
 
 
 // Set sets z to x.
@@ -38,57 +43,51 @@ func (z *Int) Set(x *Int) *Int {
 }
 
 
-// Add computes z = x+y.
+// 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.neg = x.neg
 		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.neg = x.neg
 			z.abs = z.abs.sub(x.abs, y.abs)
 		} else {
-			z.neg = !x.neg
+			neg = !neg
 			z.abs = z.abs.sub(y.abs, x.abs)
 		}
 	}
-	if len(z.abs) == 0 {
-		z.neg = false // 0 has no sign
-	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
 	return z
 }
 
 
-// Sub computes z = x-y.
+// 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.neg = x.neg
 		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.neg = x.neg
 			z.abs = z.abs.sub(x.abs, y.abs)
 		} else {
-			z.neg = !x.neg
+			neg = !neg
 			z.abs = z.abs.sub(y.abs, x.abs)
 		}
 	}
-	if len(z.abs) == 0 {
-		z.neg = false // 0 has no sign
-	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
 	return z
 }
 
 
-// Mul computes z = x*y.
+// 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)
@@ -100,38 +99,117 @@ func (z *Int) Mul(x, y *Int) *Int {
 }
 
 
-// Div calculates q = (x-r)/y and sets z = q.
-func (z *Int) Div(x, y *Int) *Int {
-	r := new(Int)
-	div(z, r, x, y)
+// 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
 }
 
 
-// Mod calculates q = (x-r)/y and sets z = r.
-func (z *Int) Mod(x, y *Int) *Int {
-	q := new(Int)
-	div(q, z, x, y)
+// 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
 }
 
 
-// DivMod calculates q = (x-r)/y and sets z = q.  (It returns z, r.)
-func (z *Int) DivMod(x, y, r *Int) (*Int, *Int) {
-	div(z, r, x, y)
+// 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
 }
 
 
-func div(q, r, x, y *Int) {
-	q.neg = x.neg != y.neg
-	r.neg = x.neg
-	q.abs, r.abs = q.abs.div(r.abs, x.abs, y.abs)
-	return
+// 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 Euclidian division and modulus (unlike Go):
+//
+//	q = x div y  such that
+//      m = x - y*q  with 0 <= m < |q|
+//
+// (See Raymond T. Boute, ``The Euclidian 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
 }
 
 
-// Neg computes z = -x.
+// Neg computes the negation z = -x.
 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
@@ -139,7 +217,7 @@ func (z *Int) Neg(x *Int) *Int {
 }
 
 
-// Cmp compares x and y. The result is
+// Cmp compares x and y and returns:
 //
 //   -1 if x <  y
 //    0 if x == y
@@ -205,26 +283,23 @@ func (z *Int) SetString(s string, base int) (*Int, bool) {
 		goto Error
 	}
 
+	neg := false
 	if s[0] == '-' {
-		z.neg = true
+		neg = true
 		s = s[1:]
-	} else {
-		z.neg = false
 	}
 
 	z.abs, _, scanned = z.abs.scan(s, base)
 	if scanned != len(s) {
 		goto Error
 	}
-	if len(z.abs) == 0 {
-		z.neg = false // 0 has no sign
-	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
 
 	return z, true
 
 Error:
-	z.neg = false
 	z.abs = nil
+	z.neg = false
 	return z, false
 }
 
@@ -234,7 +309,6 @@ Error:
 func (z *Int) SetBytes(b []byte) *Int {
 	s := int(_S)
 	z.abs = z.abs.make((len(b) + s - 1) / s)
-	z.neg = false
 
 	j := 0
 	for len(b) >= s {
@@ -262,7 +336,7 @@ func (z *Int) SetBytes(b []byte) *Int {
 	}
 
 	z.abs = z.abs.norm()
-
+	z.neg = false
 	return z
 }
 
@@ -289,14 +363,10 @@ func (z *Int) Bytes() []byte {
 }
 
 
-// Len returns the length of the absolute value of z in bits. Zero is
-// considered to have a length of zero.
-func (z *Int) Len() int {
-	if len(z.abs) == 0 {
-		return 0
-	}
-
-	return len(z.abs)*_W - int(leadingZeros(z.abs[len(z.abs)-1]))
+// 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()
 }
 
 
@@ -304,7 +374,7 @@ func (z *Int) Len() int {
 // See Knuth, volume 2, section 4.6.3.
 func (z *Int) Exp(x, y, m *Int) *Int {
 	if y.neg || len(y.abs) == 0 {
-		z.New(1)
+		z.SetInt64(1)
 		z.neg = x.neg
 		return z
 	}
@@ -315,7 +385,7 @@ func (z *Int) Exp(x, y, m *Int) *Int {
 	}
 
 	z.abs = z.abs.expNN(x.abs, y.abs, mWords)
-	z.neg = x.neg && y.abs[0]&1 == 1
+	z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
 	return z
 }
 
@@ -326,12 +396,12 @@ func (z *Int) Exp(x, y, m *Int) *Int {
 // 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.New(0)
+		d.SetInt64(0)
 		if x != nil {
-			x.New(0)
+			x.SetInt64(0)
 		}
 		if y != nil {
-			y.New(0)
+			y.SetInt64(0)
 		}
 		return
 	}
@@ -340,9 +410,9 @@ func GcdInt(d, x, y, a, b *Int) {
 	B := new(Int).Set(b)
 
 	X := new(Int)
-	Y := new(Int).New(1)
+	Y := new(Int).SetInt64(1)
 
-	lastX := new(Int).New(1)
+	lastX := new(Int).SetInt64(1)
 	lastY := new(Int)
 
 	q := new(Int)
@@ -350,7 +420,7 @@ func GcdInt(d, x, y, a, b *Int) {
 
 	for len(B.abs) > 0 {
 		r := new(Int)
-		q, r = q.DivMod(A, B, r)
+		q, r = q.QuoRem(A, B, r)
 
 		A, B = B, r
 
@@ -382,13 +452,15 @@ func GcdInt(d, x, y, a, b *Int) {
 // 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) }
+func ProbablyPrime(z *Int, n int) bool {
+	return !z.neg && z.abs.probablyPrime(n)
+}
 
 
 // Lsh sets z = x << n and returns z.
 func (z *Int) Lsh(x *Int, n uint) *Int {
-	z.neg = x.neg
 	z.abs = z.abs.shl(x.abs, n)
+	z.neg = x.neg
 	return z
 }
 
@@ -397,18 +469,19 @@ func (z *Int) Lsh(x *Int, n uint) *Int {
 func (z *Int) Rsh(x *Int, n uint) *Int {
 	if x.neg {
 		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
-		z.neg = true
 		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.neg = false
 	z.abs = z.abs.shr(x.abs, n)
+	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 {
@@ -416,14 +489,14 @@ func (z *Int) And(x, y *Int) *Int {
 			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
 			x1 := nat{}.sub(x.abs, natOne)
 			y1 := z.abs.sub(y.abs, natOne)
-			z.neg = true
 			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.neg = false
 		z.abs = z.abs.and(x.abs, y.abs)
+		z.neg = false
 		return z
 	}
 
@@ -434,8 +507,8 @@ func (z *Int) And(x, y *Int) *Int {
 
 	// x & (-y) == x & ^(y-1) == x &^ (y-1)
 	y1 := z.abs.sub(y.abs, natOne)
-	z.neg = false
 	z.abs = z.abs.andNot(x.abs, y1)
+	z.neg = false
 	return z
 }
 
@@ -447,29 +520,29 @@ func (z *Int) AndNot(x, y *Int) *Int {
 			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
 			x1 := nat{}.sub(x.abs, natOne)
 			y1 := z.abs.sub(y.abs, natOne)
-			z.neg = false
 			z.abs = z.abs.andNot(y1, x1)
+			z.neg = false
 			return z
 		}
 
 		// x &^ y == x &^ y
-		z.neg = false
 		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 := z.abs.sub(x.abs, natOne)
-		z.neg = true
 		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 := z.abs.add(y.abs, natOne)
-	z.neg = false
 	z.abs = z.abs.and(x.abs, y1)
+	z.neg = false
 	return z
 }
 
@@ -481,14 +554,14 @@ func (z *Int) Or(x, y *Int) *Int {
 			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
 			x1 := nat{}.sub(x.abs, natOne)
 			y1 := z.abs.sub(y.abs, natOne)
-			z.neg = true
 			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.neg = false
 		z.abs = z.abs.or(x.abs, y.abs)
+		z.neg = false
 		return z
 	}
 
@@ -499,8 +572,8 @@ func (z *Int) Or(x, y *Int) *Int {
 
 	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
 	y1 := z.abs.sub(y.abs, natOne)
-	z.neg = true
 	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
 }
 
@@ -512,26 +585,26 @@ func (z *Int) Xor(x, y *Int) *Int {
 			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
 			x1 := nat{}.sub(x.abs, natOne)
 			y1 := z.abs.sub(y.abs, natOne)
-			z.neg = false
 			z.abs = z.abs.xor(x1, y1)
+			z.neg = false
 			return z
 		}
 
 		// x ^ y == x ^ y
-		z.neg = false
 		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 = y, x // ^ is symmetric
 	}
 
 	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
 	y1 := z.abs.sub(y.abs, natOne)
-	z.neg = true
 	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
 }
 
@@ -540,13 +613,13 @@ func (z *Int) Xor(x, y *Int) *Int {
 func (z *Int) Not(x *Int) *Int {
 	if x.neg {
 		// ^(-x) == ^(^(x-1)) == x-1
-		z.neg = false
 		z.abs = z.abs.sub(x.abs, natOne)
+		z.neg = false
 		return z
 	}
 
 	// ^x == -x-1 == -(x+1)
-	z.neg = true
 	z.abs = z.abs.add(x.abs, natOne)
+	z.neg = true // z cannot be zero if x is positive
 	return z
 }
diff --git a/src/pkg/big/int_test.go b/src/pkg/big/int_test.go
index deacdfac4f..fe2974308d 100755
--- a/src/pkg/big/int_test.go
+++ b/src/pkg/big/int_test.go
@@ -11,9 +11,13 @@ import (
 	"testing/quick"
 )
 
-func newZ(x int64) *Int {
-	var z Int
-	return z.New(x)
+
+func isNormalized(x *Int) bool {
+	if len(x.abs) == 0 {
+		return !x.neg
+	}
+	// len(x.abs) > 0
+	return x.abs[len(x.abs)-1] != 0
 }
 
 
@@ -24,20 +28,20 @@ type argZZ struct {
 
 
 var sumZZ = []argZZ{
-	argZZ{newZ(0), newZ(0), newZ(0)},
-	argZZ{newZ(1), newZ(1), newZ(0)},
-	argZZ{newZ(1111111110), newZ(123456789), newZ(987654321)},
-	argZZ{newZ(-1), newZ(-1), newZ(0)},
-	argZZ{newZ(864197532), newZ(-123456789), newZ(987654321)},
-	argZZ{newZ(-1111111110), newZ(-123456789), newZ(-987654321)},
+	argZZ{NewInt(0), NewInt(0), NewInt(0)},
+	argZZ{NewInt(1), NewInt(1), NewInt(0)},
+	argZZ{NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+	argZZ{NewInt(-1), NewInt(-1), NewInt(0)},
+	argZZ{NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+	argZZ{NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
 }
 
 
 var prodZZ = []argZZ{
-	argZZ{newZ(0), newZ(0), newZ(0)},
-	argZZ{newZ(0), newZ(1), newZ(0)},
-	argZZ{newZ(1), newZ(1), newZ(1)},
-	argZZ{newZ(-991 * 991), newZ(991), newZ(-991)},
+	argZZ{NewInt(0), NewInt(0), NewInt(0)},
+	argZZ{NewInt(0), NewInt(1), NewInt(0)},
+	argZZ{NewInt(1), NewInt(1), NewInt(1)},
+	argZZ{NewInt(-991 * 991), NewInt(991), NewInt(-991)},
 	// TODO(gri) add larger products
 }
 
@@ -46,6 +50,9 @@ func TestSetZ(t *testing.T) {
 	for _, a := range sumZZ {
 		var z Int
 		z.Set(a.z)
+		if !isNormalized(&z) {
+			t.Errorf("%v is not normalized", z)
+		}
 		if (&z).Cmp(a.z) != 0 {
 			t.Errorf("got z = %v; want %v", z, a.z)
 		}
@@ -56,6 +63,9 @@ func TestSetZ(t *testing.T) {
 func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
 	var z Int
 	f(&z, a.x, a.y)
+	if !isNormalized(&z) {
+		t.Errorf("msg: %v is not normalized", z, msg)
+	}
 	if (&z).Cmp(a.z) != 0 {
 		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
 	}
@@ -186,7 +196,7 @@ func TestSetString(t *testing.T) {
 	for i, test := range fromStringTests {
 		n1, ok1 := new(Int).SetString(test.in, test.base)
 		n2, ok2 := n2.SetString(test.in, test.base)
-		expected := new(Int).New(test.out)
+		expected := NewInt(test.out)
 		if ok1 != test.ok || ok2 != test.ok {
 			t.Errorf("#%d (input '%s') ok incorrect (should be %t)", i, test.in, test.ok)
 			continue
@@ -195,6 +205,13 @@ func TestSetString(t *testing.T) {
 			continue
 		}
 
+		if ok1 && !isNormalized(n1) {
+			t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n1)
+		}
+		if ok2 && !isNormalized(n2) {
+			t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n2)
+		}
+
 		if n1.Cmp(expected) != 0 {
 			t.Errorf("#%d (input '%s') got: %s want: %d\n", i, test.in, n1, test.out)
 		}
@@ -205,33 +222,77 @@ func TestSetString(t *testing.T) {
 }
 
 
-type divSignsTest struct {
+type divisionSignsTest struct {
 	x, y int64
-	q, r int64
+	q, r int64 // T-division
+	d, m int64 // Euclidian division
 }
 
 
-// These examples taken from the Go Language Spec, section "Arithmetic operators"
-var divSignsTests = []divSignsTest{
-	divSignsTest{5, 3, 1, 2},
-	divSignsTest{-5, 3, -1, -2},
-	divSignsTest{5, -3, -1, 2},
-	divSignsTest{-5, -3, 1, -2},
-	divSignsTest{1, 2, 0, 1},
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []divisionSignsTest{
+	divisionSignsTest{5, 3, 1, 2, 1, 2},
+	divisionSignsTest{-5, 3, -1, -2, -2, 1},
+	divisionSignsTest{5, -3, -1, 2, -1, 2},
+	divisionSignsTest{-5, -3, 1, -2, 2, 1},
+	divisionSignsTest{1, 2, 0, 1, 0, 1},
+	divisionSignsTest{8, 4, 2, 0, 2, 0},
 }
 
 
-func TestDivSigns(t *testing.T) {
-	for i, test := range divSignsTests {
-		x := new(Int).New(test.x)
-		y := new(Int).New(test.y)
-		r := new(Int)
-		q, r := new(Int).DivMod(x, y, r)
-		expectedQ := new(Int).New(test.q)
-		expectedR := new(Int).New(test.r)
+func TestDivisionSigns(t *testing.T) {
+	for i, test := range divisionSignsTests {
+		x := NewInt(test.x)
+		y := NewInt(test.y)
+		q := NewInt(test.q)
+		r := NewInt(test.r)
+		d := NewInt(test.d)
+		m := NewInt(test.m)
 
-		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
-			t.Errorf("#%d: got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+		q1 := new(Int).Quo(x, y)
+		r1 := new(Int).Rem(x, y)
+		if !isNormalized(q1) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+		}
+		if !isNormalized(r1) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+		}
+		if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+		}
+
+		q2, r2 := new(Int).QuoRem(x, y, new(Int))
+		if !isNormalized(q2) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+		}
+		if !isNormalized(r2) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+		}
+		if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+		}
+
+		d1 := new(Int).Div(x, y)
+		m1 := new(Int).Mod(x, y)
+		if !isNormalized(d1) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d1)
+		}
+		if !isNormalized(m1) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+		}
+		if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+		}
+
+		d2, m2 := new(Int).DivMod(x, y, new(Int))
+		if !isNormalized(d2) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d2)
+		}
+		if !isNormalized(m2) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+		}
+		if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
 		}
 	}
 }
@@ -273,7 +334,7 @@ func TestBytes(t *testing.T) {
 }
 
 
-func checkDiv(x, y []byte) bool {
+func checkQuo(x, y []byte) bool {
 	u := new(Int).SetBytes(x)
 	v := new(Int).SetBytes(y)
 
@@ -282,7 +343,7 @@ func checkDiv(x, y []byte) bool {
 	}
 
 	r := new(Int)
-	q, r := new(Int).DivMod(u, v, r)
+	q, r := new(Int).QuoRem(u, v, r)
 
 	if r.Cmp(v) >= 0 {
 		return false
@@ -296,20 +357,20 @@ func checkDiv(x, y []byte) bool {
 }
 
 
-type divTest struct {
+type quoTest struct {
 	x, y string
 	q, r string
 }
 
 
-var divTests = []divTest{
-	divTest{
+var quoTests = []quoTest{
+	quoTest{
 		"476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
 		"9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
 		"50911",
 		"1",
 	},
-	divTest{
+	quoTest{
 		"11510768301994997771168",
 		"1328165573307167369775",
 		"8",
@@ -318,19 +379,19 @@ var divTests = []divTest{
 }
 
 
-func TestDiv(t *testing.T) {
-	if err := quick.Check(checkDiv, nil); err != nil {
+func TestQuo(t *testing.T) {
+	if err := quick.Check(checkQuo, nil); err != nil {
 		t.Error(err)
 	}
 
-	for i, test := range divTests {
+	for i, test := range quoTests {
 		x, _ := new(Int).SetString(test.x, 10)
 		y, _ := new(Int).SetString(test.y, 10)
 		expectedQ, _ := new(Int).SetString(test.q, 10)
 		expectedR, _ := new(Int).SetString(test.r, 10)
 
 		r := new(Int)
-		q, r := new(Int).DivMod(x, y, r)
+		q, r := new(Int).QuoRem(x, y, r)
 
 		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
 			t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
@@ -339,7 +400,7 @@ func TestDiv(t *testing.T) {
 }
 
 
-func TestDivStepD6(t *testing.T) {
+func TestQuoStepD6(t *testing.T) {
 	// See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
 	// a code path which only triggers 1 in 10^{-19} cases.
 
@@ -347,7 +408,7 @@ func TestDivStepD6(t *testing.T) {
 	v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
 
 	r := new(Int)
-	q, r := new(Int).DivMod(u, v, r)
+	q, r := new(Int).QuoRem(u, v, r)
 	const expectedQ64 = "18446744073709551613"
 	const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
 	const expectedQ32 = "4294967293"
@@ -359,35 +420,38 @@ func TestDivStepD6(t *testing.T) {
 }
 
 
-type lenTest struct {
+type bitLenTest struct {
 	in  string
 	out int
 }
 
 
-var lenTests = []lenTest{
-	lenTest{"0", 0},
-	lenTest{"1", 1},
-	lenTest{"2", 2},
-	lenTest{"4", 3},
-	lenTest{"0x8000", 16},
-	lenTest{"0x80000000", 32},
-	lenTest{"0x800000000000", 48},
-	lenTest{"0x8000000000000000", 64},
-	lenTest{"0x80000000000000000000", 80},
+var bitLenTests = []bitLenTest{
+	bitLenTest{"-1", 1},
+	bitLenTest{"0", 0},
+	bitLenTest{"1", 1},
+	bitLenTest{"2", 2},
+	bitLenTest{"4", 3},
+	bitLenTest{"0xabc", 12},
+	bitLenTest{"0x8000", 16},
+	bitLenTest{"0x80000000", 32},
+	bitLenTest{"0x800000000000", 48},
+	bitLenTest{"0x8000000000000000", 64},
+	bitLenTest{"0x80000000000000000000", 80},
+	bitLenTest{"-0x4000000000000000000000", 87},
 }
 
 
-func TestLen(t *testing.T) {
-	for i, test := range lenTests {
-		n, ok := new(Int).SetString(test.in, 0)
+func TestBitLen(t *testing.T) {
+	for i, test := range bitLenTests {
+		x, ok := new(Int).SetString(test.in, 0)
 		if !ok {
 			t.Errorf("#%d test input invalid: %s", i, test.in)
 			continue
 		}
 
-		if n.Len() != test.out {
-			t.Errorf("#%d got %d want %d\n", i, n.Len(), test.out)
+		if n := x.BitLen(); n != test.out {
+			t.Errorf("#%d got %d want %d\n", i, n, test.out)
 		}
 	}
 }
@@ -404,6 +468,7 @@ var expTests = []expTest{
 	expTest{"-5", "0", "", "-1"},
 	expTest{"5", "1", "", "5"},
 	expTest{"-5", "1", "", "-5"},
+	expTest{"-2", "3", "2", "0"},
 	expTest{"5", "2", "", "25"},
 	expTest{"1", "65537", "2", "1"},
 	expTest{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
@@ -436,13 +501,16 @@ func TestExp(t *testing.T) {
 		}
 
 		if !ok1 || !ok2 || !ok3 || !ok4 {
-			t.Errorf("#%d error in input", i)
+			t.Errorf("#%d: error in input", i)
 			continue
 		}
 
 		z := new(Int).Exp(x, y, m)
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
 		if z.Cmp(out) != 0 {
-			t.Errorf("#%d got %s want %s", i, z, out)
+			t.Errorf("#%d: got %s want %s", i, z, out)
 		}
 	}
 }
@@ -478,16 +546,16 @@ var gcdTests = []gcdTest{
 
 func TestGcd(t *testing.T) {
 	for i, test := range gcdTests {
-		a := new(Int).New(test.a)
-		b := new(Int).New(test.b)
+		a := NewInt(test.a)
+		b := NewInt(test.b)
 
 		x := new(Int)
 		y := new(Int)
 		d := new(Int)
 
-		expectedX := new(Int).New(test.x)
-		expectedY := new(Int).New(test.y)
-		expectedD := new(Int).New(test.d)
+		expectedX := NewInt(test.x)
+		expectedY := NewInt(test.y)
+		expectedD := NewInt(test.d)
 
 		GcdInt(d, x, y, a, b)
 
@@ -594,8 +662,11 @@ func TestRsh(t *testing.T) {
 		expected, _ := new(Int).SetString(test.out, 10)
 		out := new(Int).Rsh(in, test.shift)
 
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
 		if out.Cmp(expected) != 0 {
-			t.Errorf("#%d got %s want %s", i, out, expected)
+			t.Errorf("#%d: got %s want %s", i, out, expected)
 		}
 	}
 }
@@ -607,8 +678,11 @@ func TestRshSelf(t *testing.T) {
 		expected, _ := new(Int).SetString(test.out, 10)
 		z.Rsh(z, test.shift)
 
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
 		if z.Cmp(expected) != 0 {
-			t.Errorf("#%d got %s want %s", i, z, expected)
+			t.Errorf("#%d: got %s want %s", i, z, expected)
 		}
 	}
 }
@@ -643,8 +717,11 @@ func TestLsh(t *testing.T) {
 		expected, _ := new(Int).SetString(test.out, 10)
 		out := new(Int).Lsh(in, test.shift)
 
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
 		if out.Cmp(expected) != 0 {
-			t.Errorf("#%d got %s want %s", i, out, expected)
+			t.Errorf("#%d: got %s want %s", i, out, expected)
 		}
 	}
 }
@@ -656,8 +733,11 @@ func TestLshSelf(t *testing.T) {
 		expected, _ := new(Int).SetString(test.out, 10)
 		z.Lsh(z, test.shift)
 
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
 		if z.Cmp(expected) != 0 {
-			t.Errorf("#%d got %s want %s", i, z, expected)
+			t.Errorf("#%d: got %s want %s", i, z, expected)
 		}
 	}
 }
@@ -669,8 +749,11 @@ func TestLshRsh(t *testing.T) {
 		out := new(Int).Lsh(in, test.shift)
 		out = out.Rsh(out, test.shift)
 
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
 		if in.Cmp(out) != 0 {
-			t.Errorf("#%d got %s want %s", i, out, in)
+			t.Errorf("#%d: got %s want %s", i, out, in)
 		}
 	}
 	for i, test := range lshTests {
@@ -678,8 +761,11 @@ func TestLshRsh(t *testing.T) {
 		out := new(Int).Lsh(in, test.shift)
 		out.Rsh(out, test.shift)
 
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
 		if in.Cmp(out) != 0 {
-			t.Errorf("#%d got %s want %s", i, out, in)
+			t.Errorf("#%d: got %s want %s", i, out, in)
 		}
 	}
 }
@@ -721,6 +807,7 @@ var bitwiseTests = []bitwiseTest{
 	bitwiseTest{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
 	bitwiseTest{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
 	bitwiseTest{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+	bitwiseTest{"-0xAF", "-0x50", "0x00", "-0xFF", "-0x01", "-0x01"},
 	bitwiseTest{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
 	bitwiseTest{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
 	bitwiseTest{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
diff --git a/src/pkg/big/nat.go b/src/pkg/big/nat.go
index fd4c49f5cf..1cad23777b 100755
--- a/src/pkg/big/nat.go
+++ b/src/pkg/big/nat.go
@@ -356,7 +356,7 @@ func karatsuba(z, x, y nat) {
 
 // alias returns true if x and y share the same base array.
 func alias(x, y nat) bool {
-	return &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
 }
 
 
@@ -412,7 +412,7 @@ func (z nat) mul(x, y nat) nat {
 	// m >= n > 1
 
 	// determine if z can be reused
-	if len(z) > 0 && (alias(z, x) || alias(z, y)) {
+	if alias(z, x) || alias(z, y) {
 		z = nil // z is an alias for x or y - cannot reuse
 	}
 
@@ -757,7 +757,7 @@ func (z nat) shl(x nat, s uint) nat {
 
 	// determine if z can be reused
 	// TODO(gri) change shlVW so we don't need this
-	if len(z) > 0 && alias(z, x) {
+	if alias(z, x) {
 		z = nil // z is an alias for x - cannot reuse
 	}
 
@@ -780,7 +780,7 @@ func (z nat) shr(x nat, s uint) nat {
 
 	// determine if z can be reused
 	// TODO(gri) change shrVW so we don't need this
-	if len(z) > 0 && alias(z, x) {
+	if alias(z, x) {
 		z = nil // z is an alias for x - cannot reuse
 	}
 
diff --git a/src/pkg/bignum/integer.go b/src/pkg/bignum/integer.go
index 873b2664a7..a8d26829d1 100644
--- a/src/pkg/bignum/integer.go
+++ b/src/pkg/bignum/integer.go
@@ -253,7 +253,7 @@ func (x *Integer) QuoRem(y *Integer) (*Integer, *Integer) {
 // Div and Mod implement Euclidian division and modulus:
 //
 //   q = x.Div(y)
-//   r = x.Mod(y) with: 0 <= r < |q| and: y = x*q + r
+//   r = x.Mod(y) with: 0 <= r < |q| and: x = y*q + r
 //
 // (Raymond T. Boute, ``The Euclidian definition of the functions
 // div and mod''. ACM Transactions on Programming Languages and
diff --git a/src/pkg/crypto/rsa/pkcs1v15.go b/src/pkg/crypto/rsa/pkcs1v15.go
index cfad954544..5fd25d58c7 100644
--- a/src/pkg/crypto/rsa/pkcs1v15.go
+++ b/src/pkg/crypto/rsa/pkcs1v15.go
@@ -18,7 +18,7 @@ import (
 // WARNING: use of this function to encrypt plaintexts other than session keys
 // is dangerous. Use RSA OAEP in new protocols.
 func EncryptPKCS1v15(rand io.Reader, pub *PublicKey, msg []byte) (out []byte, err os.Error) {
-	k := (pub.N.Len() + 7) / 8
+	k := (pub.N.BitLen() + 7) / 8
 	if len(msg) > k-11 {
 		err = MessageTooLongError{}
 		return
@@ -66,7 +66,7 @@ func DecryptPKCS1v15(rand io.Reader, priv *PrivateKey, ciphertext []byte) (out [
 // Encryption Standard PKCS #1'', Daniel Bleichenbacher, Advances in Cryptology
 // (Crypto '98),
 func DecryptPKCS1v15SessionKey(rand io.Reader, priv *PrivateKey, ciphertext []byte, key []byte) (err os.Error) {
-	k := (priv.N.Len() + 7) / 8
+	k := (priv.N.BitLen() + 7) / 8
 	if k-(len(key)+3+8) < 0 {
 		err = DecryptionError{}
 		return
@@ -83,7 +83,7 @@ func DecryptPKCS1v15SessionKey(rand io.Reader, priv *PrivateKey, ciphertext []by
 }
 
 func decryptPKCS1v15(rand io.Reader, priv *PrivateKey, ciphertext []byte) (valid int, msg []byte, err os.Error) {
-	k := (priv.N.Len() + 7) / 8
+	k := (priv.N.BitLen() + 7) / 8
 	if k < 11 {
 		err = DecryptionError{}
 		return
@@ -179,7 +179,7 @@ func SignPKCS1v15(rand io.Reader, priv *PrivateKey, hash PKCS1v15Hash, hashed []
 	}
 
 	tLen := len(prefix) + hashLen
-	k := (priv.N.Len() + 7) / 8
+	k := (priv.N.BitLen() + 7) / 8
 	if k < tLen+11 {
 		return nil, MessageTooLongError{}
 	}
@@ -212,7 +212,7 @@ func VerifyPKCS1v15(pub *PublicKey, hash PKCS1v15Hash, hashed []byte, sig []byte
 	}
 
 	tLen := len(prefix) + hashLen
-	k := (pub.N.Len() + 7) / 8
+	k := (pub.N.BitLen() + 7) / 8
 	if k < tLen+11 {
 		err = VerificationError{}
 		return
diff --git a/src/pkg/crypto/rsa/pkcs1v15_test.go b/src/pkg/crypto/rsa/pkcs1v15_test.go
index 69edeaa2ee..bfc12be285 100644
--- a/src/pkg/crypto/rsa/pkcs1v15_test.go
+++ b/src/pkg/crypto/rsa/pkcs1v15_test.go
@@ -67,7 +67,7 @@ func TestEncryptPKCS1v15(t *testing.T) {
 	if err != nil {
 		t.Errorf("Failed to open /dev/urandom")
 	}
-	k := (rsaPrivateKey.N.Len() + 7) / 8
+	k := (rsaPrivateKey.N.BitLen() + 7) / 8
 
 	tryEncryptDecrypt := func(in []byte, blind bool) bool {
 		if len(in) > k-11 {
diff --git a/src/pkg/crypto/rsa/rsa.go b/src/pkg/crypto/rsa/rsa.go
index 941b061b5f..c7a8d2053d 100644
--- a/src/pkg/crypto/rsa/rsa.go
+++ b/src/pkg/crypto/rsa/rsa.go
@@ -50,11 +50,11 @@ func randomPrime(rand io.Reader, bits int) (p *big.Int, err os.Error) {
 
 // randomNumber returns a uniform random value in [0, max).
 func randomNumber(rand io.Reader, max *big.Int) (n *big.Int, err os.Error) {
-	k := (max.Len() + 7) / 8
+	k := (max.BitLen() + 7) / 8
 
 	// r is the number of bits in the used in the most significant byte of
 	// max.
-	r := uint(max.Len() % 8)
+	r := uint(max.BitLen() % 8)
 	if r == 0 {
 		r = 8
 	}
@@ -244,7 +244,7 @@ func encrypt(c *big.Int, pub *PublicKey, m *big.Int) *big.Int {
 // twice the hash length plus 2.
 func EncryptOAEP(hash hash.Hash, rand io.Reader, pub *PublicKey, msg []byte, label []byte) (out []byte, err os.Error) {
 	hash.Reset()
-	k := (pub.N.Len() + 7) / 8
+	k := (pub.N.BitLen() + 7) / 8
 	if len(msg) > k-2*hash.Size()-2 {
 		err = MessageTooLongError{}
 		return
@@ -365,7 +365,7 @@ func decrypt(rand io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err os.E
 // DecryptOAEP decrypts ciphertext using RSA-OAEP.
 // If rand != nil, DecryptOAEP uses RSA blinding to avoid timing side-channel attacks.
 func DecryptOAEP(hash hash.Hash, rand io.Reader, priv *PrivateKey, ciphertext []byte, label []byte) (msg []byte, err os.Error) {
-	k := (priv.N.Len() + 7) / 8
+	k := (priv.N.BitLen() + 7) / 8
 	if len(ciphertext) > k ||
 		k < hash.Size()*2+2 {
 		err = DecryptionError{}
diff --git a/test/bench/pidigits.go b/test/bench/pidigits.go
index a05515028a..dcfb502ce2 100644
--- a/test/bench/pidigits.go
+++ b/test/bench/pidigits.go
@@ -81,8 +81,8 @@ func extract_digit() int64 {
 
 func next_term(k int64) {
 	// TODO(eds) If big.Int ever gets a Scale method, y2 and bigk could be int64
-	y2.New(k*2 + 1)
-	bigk.New(k)
+	y2.SetInt64(k*2 + 1)
+	bigk.SetInt64(k)
 
 	tmp1.Lsh(numer, 1)
 	accum.Add(accum, tmp1)