math/big: additional Montgomery cleanup

Also fix bug reported in CL 17510.

Found during fix of #13515 in CL 17672, but separate from the fix.

Change-Id: I4b1024569a98f5cfd2ebb442ec3d64356164d284
Reviewed-on: https://go-review.googlesource.com/17673Reviewed-by: Robert Griesemer <gri@golang.org>

src/math/big/int_test.go

@@ -544,6 +544,9 @@ var expTests = []struct {
 	{"0x8000000000000000", "1000", "6719", "1603"},
 	{"0x8000000000000000", "1000000", "6719", "3199"},
 	{"0x8000000000000000", "-1000000", "6719", "1"},
+
+	{"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
+
 	{
 		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
 		"298472983472983471903246121093472394872319615612417471234712061",
@@ -551,6 +554,12 @@ var expTests = []struct {
 		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
 	},
 	// test case for issue 8822
+	{
+		"11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
+		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+		"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
+		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+	},
 	{
 		"-0x1BCE04427D8032319A89E5C4136456671AC620883F2C4139E57F91307C485AD2D6204F4F87A58262652DB5DBBAC72B0613E51B835E7153BEC6068F5C8D696B74DBD18FEC316AEF73985CF0475663208EB46B4F17DD9DA55367B03323E5491A70997B90C059FB34809E6EE55BCFBD5F2F52233BFE62E6AA9E4E26A1D4C2439883D14F2633D55D8AA66A1ACD5595E778AC3A280517F1157989E70C1A437B849F1877B779CC3CDDEDE2DAA6594A6C66D181A00A5F777EE60596D8773998F6E988DEAE4CCA60E4DDCF9590543C89F74F603259FCAD71660D30294FBBE6490300F78A9D63FA660DC9417B8B9DDA28BEB3977B621B988E23D4D954F322C3540541BC649ABD504C50FADFD9F0987D58A2BF689313A285E773FF02899A6EF887D1D4A0D2",
 		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",

src/math/big/nat.go

@@ -213,25 +213,25 @@ func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
 	if len(x) != n || len(y) != n || len(m) != n {
 		panic("math/big: mismatched montgomery number lengths")
 	}
-	var c1, c2, c3 Word
 	z = z.make(n)
 	z.clear()
+	var c Word
 	for i := 0; i < n; i++ {
 		d := y[i]
-		c2 = addMulVVW(z, x, d)
+		c2 := addMulVVW(z, x, d)
 		t := z[0] * k
-		c3 = addMulVVW(z, m, t)
+		c3 := addMulVVW(z, m, t)
 		copy(z, z[1:])
-		cx := c1 + c2
+		cx := c + c2
 		cy := cx + c3
 		z[n-1] = cy
 		if cx < c2 || cy < c3 {
-			c1 = 1
+			c = 1
 		} else {
-			c1 = 0
+			c = 0
 		}
 	}
-	if c1 != 0 {
+	if c != 0 {
 		subVV(z, z, m)
 	}
 	return z
@@ -1056,23 +1056,19 @@ func (z nat) expNNWindowed(x, y, m nat) nat {
 // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
 // Uses Montgomery representation.
 func (z nat) expNNMontgomery(x, y, m nat) nat {
-	var zz, one, rr, RR nat
-
 	numWords := len(m)
 
 	// We want the lengths of x and m to be equal.
+	// It is OK if x >= m as long as len(x) == len(m).
 	if len(x) > numWords {
-		_, rr = rr.div(rr, x, m)
-	} else if len(x) < numWords {
-		rr = rr.make(numWords)
-		rr.clear()
-		for i := range x {
-			rr[i] = x[i]
-		}
-	} else {
-		rr = x
+		_, x = nat(nil).div(nil, x, m)
+		// Note: now len(x) <= numWords, not guaranteed ==.
+	}
+	if len(x) < numWords {
+		rr := make(nat, numWords)
+		copy(rr, x)
+		x = rr
 	}
-	x = rr
 
 	// Ideally the precomputations would be performed outside, and reused
 	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
@@ -1086,8 +1082,8 @@ func (z nat) expNNMontgomery(x, y, m nat) nat {
 	k0 = -k0
 
 	// RR = 2**(2*_W*len(m)) mod m
-	RR = RR.setWord(1)
-	zz = zz.shl(RR, uint(2*numWords*_W))
+	RR := nat(nil).setWord(1)
+	zz := nat(nil).shl(RR, uint(2*numWords*_W))
 	_, RR = RR.div(RR, zz, m)
 	if len(RR) < numWords {
 		zz = zz.make(numWords)
@@ -1095,8 +1091,7 @@ func (z nat) expNNMontgomery(x, y, m nat) nat {
 		RR = zz
 	}
 	// one = 1, with equal length to that of m
-	one = one.make(numWords)
-	one.clear()
+	one := make(nat, numWords)
 	one[0] = 1
 
 	const n = 4

src/math/big/nat_test.go

@@ -483,6 +483,12 @@ var expNNTests = []struct {
 		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
 		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
 	},
+	{
+		"11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
+		"426343618817810911523",
+		"444747819283133684179",
+		"42",
+	},
 }
 
 func TestExpNN(t *testing.T) {