crypto/internal/fips140/bigmod: add Nat.InverseVarTime

Will be needed for RSA key generation.

We now require Modulus to be > 1 because we don't want to worry about 1
being out of range. There is no use for a Modulus of 1 anyway, and we
already return an error from NewModulus.

Ported from https://cs.opensource.google/boringssl/boringssl/+/master:crypto/fipsmodule/bn/gcd_extra.cc.inc;drc=5813c2c10c73d800f1b0d890a7d74ff973abbffc.

Updates #69799
For #69536

Change-Id: I9850bcc461565b23fa7186a09c65355f7da3e5ba
Reviewed-on: https://go-review.googlesource.com/c/go/+/632415
Auto-Submit: Filippo Valsorda <[email protected]>
Reviewed-by: Daniel McCarney <[email protected]>
LUCI-TryBot-Result: Go LUCI <[email protected]>
Reviewed-by: Roland Shoemaker <[email protected]>
Reviewed-by: Russ Cox <[email protected]>

Author: FiloSottile

src/crypto/internal/fips140/bigmod/nat.go

@@ -473,15 +473,16 @@ func minusInverseModW(x uint) uint {
 	return -y
 }
 
-// NewModulus creates a new Modulus from a slice of big-endian bytes.
+// NewModulus creates a new Modulus from a slice of big-endian bytes. The
+// modulus must be greater than one.
 //
 // The number of significant bits and whether the modulus is even is leaked
 // through timing side-channels.
 func NewModulus(b []byte) (*Modulus, error) {
 	m := &Modulus{}
 	m.nat = NewNat().resetToBytes(b)
-	if len(m.nat.limbs) == 0 {
-		return nil, errors.New("modulus must be > 0")
+	if m.nat.IsZero() == yes || m.nat.IsOne() == yes {
+		return nil, errors.New("modulus must be > 1")
 	}
 	m.leading = _W - bitLen(m.nat.limbs[len(m.nat.limbs)-1])
 	if m.nat.IsOdd() == 1 {
@@ -963,3 +964,129 @@ func (out *Nat) ExpShortVarTime(x *Nat, e uint, m *Modulus) *Nat {
 	}
 	return out.montgomeryReduction(m)
 }
+
+// InverseVarTime calculates x = a⁻¹ mod m and returns (x, true) if a is
+// invertible. Otherwise, InverseVarTime returns (x, false) and x is not
+// modified.
+//
+// a must be reduced modulo m, but doesn't need to have the same size. The
+// output will be resized to the size of m and overwritten.
+func (x *Nat) InverseVarTime(a *Nat, m *Modulus) (*Nat, bool) {
+	// This is the extended binary GCD algorithm described in the Handbook of
+	// Applied Cryptography, Algorithm 14.61, adapted by BoringSSL to bound
+	// coefficients and avoid negative numbers. For more details and proof of
+	// correctness, see https://github.com/mit-plv/fiat-crypto/pull/333/files.
+	//
+	// Following the proof linked in the PR above, the changes are:
+	//
+	// 1. Negate [B] and [C] so they are positive. The invariant now involves a
+	//    subtraction.
+	// 2. If step 2 (both [x] and [y] are even) runs, abort immediately. This
+	//    algorithm only cares about [x] and [y] relatively prime.
+	// 3. Subtract copies of [x] and [y] as needed in step 6 (both [u] and [v]
+	//    are odd) so coefficients stay in bounds.
+	// 4. Replace the [u >= v] check with [u > v]. This changes the end
+	//    condition to [v = 0] rather than [u = 0]. This saves an extra
+	//    subtraction due to which coefficients were negated.
+	// 5. Rename x and y to a and n, to capture that one is a modulus.
+	// 6. Rearrange steps 4 through 6 slightly. Merge the loops in steps 4 and
+	//    5 into the main loop (step 7's goto), and move step 6 to the start of
+	//    the loop iteration, ensuring each loop iteration halves at least one
+	//    value.
+	//
+	// Note this algorithm does not handle either input being zero.
+
+	if a.IsZero() == yes {
+		return x, false
+	}
+	if a.IsOdd() == no && !m.odd {
+		// a and m are not coprime, as they are both even.
+		return x, false
+	}
+
+	u := NewNat().set(a).ExpandFor(m)
+	v := m.Nat()
+
+	A := NewNat().reset(len(m.nat.limbs))
+	A.limbs[0] = 1
+	B := NewNat().reset(len(a.limbs))
+	C := NewNat().reset(len(m.nat.limbs))
+	D := NewNat().reset(len(a.limbs))
+	D.limbs[0] = 1
+
+	// Before and after each loop iteration, the following hold:
+	//
+	//   u = A*a - B*m
+	//   v = D*m - C*a
+	//   0 < u <= a
+	//   0 <= v <= m
+	//   0 <= A < m
+	//   0 <= B <= a
+	//   0 <= C < m
+	//   0 <= D <= a
+	//
+	// After each loop iteration, u and v only get smaller, and at least one of
+	// them shrinks by at least a factor of two.
+	for {
+		// If both u and v are odd, subtract the smaller from the larger.
+		// If u = v, we need to subtract from v to hit the modified exit condition.
+		if u.IsOdd() == yes && v.IsOdd() == yes {
+			if v.cmpGeq(u) == no {
+				u.sub(v)
+				A.Add(C, m)
+				B.Add(D, &Modulus{nat: a})
+			} else {
+				v.sub(u)
+				C.Add(A, m)
+				D.Add(B, &Modulus{nat: a})
+			}
+		}
+
+		// Exactly one of u and v is now even.
+		if u.IsOdd() == v.IsOdd() {
+			panic("bigmod: internal error: u and v are not in the expected state")
+		}
+
+		// Halve the even one and adjust the corresponding coefficient.
+		if u.IsOdd() == no {
+			rshift1(u, 0)
+			if A.IsOdd() == yes || B.IsOdd() == yes {
+				rshift1(A, A.add(m.nat))
+				rshift1(B, B.add(a))
+			} else {
+				rshift1(A, 0)
+				rshift1(B, 0)
+			}
+		} else { // v.IsOdd() == no
+			rshift1(v, 0)
+			if C.IsOdd() == yes || D.IsOdd() == yes {
+				rshift1(C, C.add(m.nat))
+				rshift1(D, D.add(a))
+			} else {
+				rshift1(C, 0)
+				rshift1(D, 0)
+			}
+		}
+
+		if v.IsZero() == yes {
+			if u.IsOne() == no {
+				return x, false
+			}
+			return x.set(A), true
+		}
+	}
+}
+
+func rshift1(a *Nat, carry uint) {
+	size := len(a.limbs)
+	aLimbs := a.limbs[:size]
+
+	for i := range size {
+		aLimbs[i] >>= 1
+		if i+1 < size {
+			aLimbs[i] |= aLimbs[i+1] << (_W - 1)
+		} else {
+			aLimbs[i] |= carry << (_W - 1)
+		}
+	}
+}

src/crypto/internal/fips140/bigmod/nat_test.go

@@ -5,12 +5,15 @@
 package bigmod
 
 import (
+	"bufio"
 	"bytes"
 	cryptorand "crypto/rand"
+	"encoding/hex"
 	"fmt"
 	"math/big"
 	"math/bits"
 	"math/rand"
+	"os"
 	"reflect"
 	"slices"
 	"strings"
@@ -632,7 +635,7 @@ func BenchmarkExp(b *testing.B) {
 }
 
 func TestNewModulus(t *testing.T) {
-	expected := "modulus must be > 0"
+	expected := "modulus must be > 1"
 	_, err := NewModulus([]byte{})
 	if err == nil || err.Error() != expected {
 		t.Errorf("NewModulus(0) got %q, want %q", err, expected)
@@ -645,6 +648,14 @@ func TestNewModulus(t *testing.T) {
 	if err == nil || err.Error() != expected {
 		t.Errorf("NewModulus(0) got %q, want %q", err, expected)
 	}
+	_, err = NewModulus([]byte{1})
+	if err == nil || err.Error() != expected {
+		t.Errorf("NewModulus(1) got %q, want %q", err, expected)
+	}
+	_, err = NewModulus([]byte{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1})
+	if err == nil || err.Error() != expected {
+		t.Errorf("NewModulus(1) got %q, want %q", err, expected)
+	}
 }
 
 func makeTestValue(nbits int) []uint {
@@ -683,3 +694,71 @@ func TestAddMulVVWSized(t *testing.T) {
 		})
 	}
 }
+
+func TestInverse(t *testing.T) {
+	f, err := os.Open("testdata/mod_inv_tests.txt")
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	var ModInv, A, M string
+	var lineNum int
+	scanner := bufio.NewScanner(f)
+	for scanner.Scan() {
+		lineNum++
+		line := scanner.Text()
+		if len(line) == 0 || line[0] == '#' {
+			continue
+		}
+
+		k, v, _ := strings.Cut(line, " = ")
+		switch k {
+		case "ModInv":
+			ModInv = v
+		case "A":
+			A = v
+		case "M":
+			M = v
+
+			t.Run(fmt.Sprintf("line %d", lineNum), func(t *testing.T) {
+				m, err := NewModulus(decodeHex(t, M))
+				if err != nil {
+					t.Skip("modulus <= 1")
+				}
+				a, err := NewNat().SetBytes(decodeHex(t, A), m)
+				if err != nil {
+					t.Fatal(err)
+				}
+
+				got, ok := NewNat().InverseVarTime(a, m)
+				if !ok {
+					t.Fatal("not invertible")
+				}
+				exp, err := NewNat().SetBytes(decodeHex(t, ModInv), m)
+				if err != nil {
+					t.Fatal(err)
+				}
+				if got.Equal(exp) != 1 {
+					t.Errorf("%v != %v", got, exp)
+				}
+			})
+		default:
+			t.Fatalf("unknown key %q on line %d", k, lineNum)
+		}
+	}
+	if err := scanner.Err(); err != nil {
+		t.Fatal(err)
+	}
+}
+
+func decodeHex(t *testing.T, s string) []byte {
+	t.Helper()
+	if len(s)%2 != 0 {
+		s = "0" + s
+	}
+	b, err := hex.DecodeString(s)
+	if err != nil {
+		t.Fatalf("failed to decode hex %q: %v", s, err)
+	}
+	return b
+}

src/crypto/internal/fips140/bigmod/testdata/mod_inv_tests.txt

@@ -0,0 +1,115 @@
+# ModInv tests.
+#
+# These test vectors satisfy ModInv * A = 1 (mod M) and 0 <= ModInv < M.
+
+ModInv = 00
+A = 00
+M = 01
+
+ModInv = 00
+A = 01
+M = 01
+
+ModInv = 00
+A = 02
+M = 01
+
+ModInv = 00
+A = 03
+M = 01
+
+ModInv = 64
+A = 54
+M = e3
+
+ModInv = 13
+A = 2b
+M = 30
+
+ModInv = 2f
+A = 30
+M = 37
+
+ModInv = 4
+A = 13
+M = 4b
+
+ModInv = 1c47
+A = cd4
+M = 6a21
+
+ModInv = 2b97
+A = 8e7
+M = 49c0
+
+ModInv = 29b9
+A = fcb
+M = 3092
+
+ModInv = a83
+A = 14bf
+M = 41ae
+
+ModInv = 18f15fe1
+A = 11b5d53e
+M = 322e92a1
+
+ModInv = 32f9453b
+A = 8af6df6
+M = 33d45eb7
+
+ModInv = d696369
+A = c5f89dd5
+M = fc09c17c
+
+ModInv = 622839d8
+A = 60c2526
+M = 74200493
+
+ModInv = fb5a8aee7bbc4ef
+A = 24ebd835a70be4e2
+M = 9c7256574e0c5e93
+
+ModInv = 846bc225402419c
+A = 23026003ab1fbdb
+M = 1683cbe32779c59b
+
+ModInv = 5ff84f63a78982f9
+A = 4a2420dc733e1a0f
+M = a73c6bfabefa09e6
+
+ModInv = 133e74d28ef42b43
+A = 2e9511ae29cdd41
+M = 15234df99f19fcda
+
+ModInv = 46ae1fabe9521e4b99b198fc8439609023aa69be2247c0d1e27c2a0ea332f9c5
+A = 6331fec5f01014046788c919ed50dc86ac7a80c085f1b6f645dd179c0f0dc9cd
+M = 8ef409de82318259a8655a39293b1e762fa2cc7e0aeb4c59713a1e1fff6af640
+
+ModInv = 444ccea3a7b21677dd294d34de53cc8a5b51e69b37782310a00fc6bcc975709b
+A = 679280bd880994c08322143a4ea8a0825d0466fda1bb6b3eb86fc8e90747512b
+M = e4fecab84b365c63a0dab4244ce3f921a9c87ec64d69a2031939f55782e99a2e
+
+ModInv = 1ac7d7a03ceec5f690f567c9d61bf3469c078285bcc5cf00ac944596e887ca17
+A = 1593ef32d9c784f5091bdff952f5c5f592a3aed6ba8ea865efa6d7df87be1805
+M = 1e276882f90c95e0c1976eb079f97af075445b1361c02018d6bd7191162e67b2
+
+ModInv = 639108b90dfe946f498be21303058413bbb0e59d0bd6a6115788705abd0666d6
+A = 9258d6238e4923d120b2d1033573ffcac691526ad0842a3b174dccdbb79887bd
+M = ce62909c39371d463aaba3d4b72ea6da49cb9b529e39e1972ef3ccd9a66fe08f
+
+ModInv = aebde7654cb17833a106231c4b9e2f519140e85faee1bfb4192830f03f385e773c0f4767e93e874ffdc3b7a6b7e6a710e5619901c739ee8760a26128e8c91ef8cf761d0e505d8b28ae078d17e6071c372893bb7b72538e518ebc57efa70b7615e406756c49729b7c6e74f84aed7a316b6fa748ff4b9f143129d29dad1bff98bb
+A = a29dacaf5487d354280fdd2745b9ace4cd50f2bde41d0ee529bf26a1913244f708085452ff32feab19a7418897990da46a0633f7c8375d583367319091bbbe069b0052c5e48a7daac9fb650db5af768cd2508ec3e2cda7456d4b9ce1c39459627a8b77e038b826cd7e326d0685b0cd0cb50f026f18300dae9f5fd42aa150ee8b
+M = d686f9b86697313251685e995c09b9f1e337ddfaa050bd2df15bf4ca1dc46c5565021314765299c434ea1a6ec42bf92a29a7d1ffff599f4e50b79a82243fb24813060580c770d4c1140aeb2ab2685007e948b6f1f62e8001a0545619477d498132c907774479f6d95899e6251e7136f79ab6d3b7c82e4aca421e7d22fe7db19c
+
+ModInv = 1ec872f4f20439e203597ca4de9d1296743f95781b2fe85d5def808558bbadef02a46b8955f47c83e1625f8bb40228eab09cad2a35c9ad62ab77a30e3932872959c5898674162da244a0ec1f68c0ed89f4b0f3572bfdc658ad15bf1b1c6e1176b0784c9935bd3ff1f49bb43753eacee1d8ca1c0b652d39ec727da83984fe3a0f
+A = 2e527b0a1dc32460b2dd94ec446c692989f7b3c7451a5cbeebf69fc0ea9c4871fbe78682d5dc5b66689f7ed889b52161cd9830b589a93d21ab26dbede6c33959f5a0f0d107169e2daaac78bac8cf2d41a1eb1369cb6dc9e865e73bb2e51b886f4e896082db199175e3dde0c4ed826468f238a77bd894245d0918efc9ca84f945
+M = b13133a9ebe0645f987d170c077eea2aa44e85c9ab10386d02867419a590cb182d9826a882306c212dbe75225adde23f80f5b37ca75ed09df20fc277cc7fbbfac8d9ef37a50f6b68ea158f5447283618e64e1426406d26ea85232afb22bf546c75018c1c55cb84c374d58d9d44c0a13ba88ac2e387765cb4c3269e3a983250fa
+
+ModInv = 30ffa1876313a69de1e4e6ee132ea1d3a3da32f3b56f5cfb11402b0ad517dce605cf8e91d69fa375dd887fa8507bd8a28b2d5ce745799126e86f416047709f93f07fbd88918a047f13100ea71b1d48f6fc6d12e5c917646df3041b302187af641eaedf4908abc36f12c204e1526a7d80e96e302fb0779c28d7da607243732f26
+A = 31157208bde6b85ebecaa63735947b3b36fa351b5c47e9e1c40c947339b78bf96066e5dbe21bb42629e6fcdb81f5f88db590bfdd5f4c0a6a0c3fc6377e5c1fd8235e46e291c688b6d6ecfb36604891c2a7c9cbcc58c26e44b43beecb9c5044b58bb58e35de3cf1128f3c116534fe4e421a33f83603c3df1ae36ec88092f67f2a
+M = 53408b23d6cb733e6c9bc3d1e2ea2286a5c83cc4e3e7470f8af3a1d9f28727f5b1f8ae348c1678f5d1105dc3edf2de64e65b9c99545c47e64b770b17c8b4ef5cf194b43a0538053e87a6b95ade1439cebf3d34c6aa72a11c1497f58f76011e16c5be087936d88aba7a740113120e939e27bd3ddcb6580c2841aa406566e33c35
+
+ModInv = 87355002f305c81ba0dc97ca2234a2bc02528cefde38b94ac5bd95efc7bf4c140899107fff47f0df9e3c6aa70017ebc90610a750f112cd4f475b9c76b204a953444b4e7196ccf17e93fdaed160b7345ca9b397eddf9446e8ea8ee3676102ce70eaafbe9038a34639789e6f2f1e3f352638f2e8a8f5fc56aaea7ec705ee068dd5
+A = 42a25d0bc96f71750f5ac8a51a1605a41b506cca51c9a7ecf80cad713e56f70f1b4b6fa51cbb101f55fd74f318adefb3af04e0c8a7e281055d5a40dd40913c0e1211767c5be915972c73886106dc49325df6c2df49e9eea4536f0343a8e7d332c6159e4f5bdb20d89f90e67597c4a2a632c31b2ef2534080a9ac61f52303990d
+M = d3d3f95d50570351528a76ab1e806bae1968bd420899bdb3d87c823fac439a4354c31f6c888c939784f18fe10a95e6d203b1901caa18937ba6f8be033af10c35fc869cf3d16bef479f280f53b3499e645d0387554623207ca4989e5de00bfeaa5e9ab56474fc60dd4967b100e0832eaaf2fcb2ef82a181567057b880b3afef62