crypto/rsa: support > 3 primes.

With full multi-prime support we can support version 1 PKCS#1 private
keys. This means exporting all the members of rsa.PrivateKey, thus
making the API a little messy. However there has already been another
request to export this so it seems to be something that's needed.

Over time, rsa.GenerateMultiPrimeKey will replace rsa.GenerateKey, but
I need to work on the prime balance first because we're no longer
generating primes which are a multiples of 8 bits.

Fixes #987.

R=rsc
CC=golang-dev
https://golang.org/cl/4378046

src/pkg/crypto/openpgp/packet/private_key.go

@@ -164,8 +164,10 @@ func (pk *PrivateKey) parseRSAPrivateKey(data []byte) (err os.Error) {
 	}
 
 	rsaPriv.D = new(big.Int).SetBytes(d)
-	rsaPriv.P = new(big.Int).SetBytes(p)
-	rsaPriv.Q = new(big.Int).SetBytes(q)
+	rsaPriv.Primes = make([]*big.Int, 2)
+	rsaPriv.Primes[0] = new(big.Int).SetBytes(p)
+	rsaPriv.Primes[1] = new(big.Int).SetBytes(q)
+	rsaPriv.Precompute()
 	pk.PrivateKey = rsaPriv
 	pk.Encrypted = false
 	pk.encryptedData = nil

src/pkg/crypto/rsa/rsa.go

@@ -13,7 +13,6 @@ import (
 	"hash"
 	"io"
 	"os"
-	"sync"
 )
 
 var bigZero = big.NewInt(0)
@@ -92,48 +91,58 @@ type PublicKey struct {
 type PrivateKey struct {
 	PublicKey            // public part.
 	D         *big.Int   // private exponent
-	P, Q, R   *big.Int // prime factors of N (R may be nil)
+	Primes    []*big.Int // prime factors of N, has >= 2 elements.
 
-	rwMutex    sync.RWMutex // protects the following
-	dP, dQ, dR *big.Int     // D mod (P-1) (or mod Q-1 etc) 
-	qInv       *big.Int     // q^-1 mod p
-	pq         *big.Int     // P*Q
-	tr         *big.Int     // pq·tr ≡ 1 mod r
+	// Precomputed contains precomputed values that speed up private
+	// operations, if availible.
+	Precomputed PrecomputedValues
+}
+
+type PrecomputedValues struct {
+	Dp, Dq *big.Int // D mod (P-1) (or mod Q-1) 
+	Qinv   *big.Int // Q^-1 mod Q
+
+	// CRTValues is used for the 3rd and subsequent primes. Due to a
+	// historical accident, the CRT for the first two primes is handled
+	// differently in PKCS#1 and interoperability is sufficiently
+	// important that we mirror this.
+	CRTValues []CRTValue
+}
+
+// CRTValue contains the precomputed chinese remainder theorem values.
+type CRTValue struct {
+	Exp   *big.Int // D mod (prime-1).
+	Coeff *big.Int // R·Coeff ≡ 1 mod Prime.
+	R     *big.Int // product of primes prior to this (inc p and q).
 }
 
 // Validate performs basic sanity checks on the key.
 // It returns nil if the key is valid, or else an os.Error describing a problem.
 
 func (priv *PrivateKey) Validate() os.Error {
-	// Check that p, q and, maybe, r are prime. Note that this is just a
-	// sanity check. Since the random witnesses chosen by ProbablyPrime are
-	// deterministic, given the candidate number, it's easy for an attack
-	// to generate composites that pass this test.
-	if !big.ProbablyPrime(priv.P, 20) {
-		return os.ErrorString("P is composite")
+	// Check that the prime factors are actually prime. Note that this is
+	// just a sanity check. Since the random witnesses chosen by
+	// ProbablyPrime are deterministic, given the candidate number, it's
+	// easy for an attack to generate composites that pass this test.
+	for _, prime := range priv.Primes {
+		if !big.ProbablyPrime(prime, 20) {
+			return os.ErrorString("Prime factor is composite")
 		}
-	if !big.ProbablyPrime(priv.Q, 20) {
-		return os.ErrorString("Q is composite")
-	}
-	if priv.R != nil && !big.ProbablyPrime(priv.R, 20) {
-		return os.ErrorString("R is composite")
 	}
 
-	// Check that p*q*r == n.
-	modulus := new(big.Int).Mul(priv.P, priv.Q)
-	if priv.R != nil {
-		modulus.Mul(modulus, priv.R)
+	// Check that Πprimes == n.
+	modulus := new(big.Int).Set(bigOne)
+	for _, prime := range priv.Primes {
+		modulus.Mul(modulus, prime)
 	}
 	if modulus.Cmp(priv.N) != 0 {
 		return os.ErrorString("invalid modulus")
 	}
-	// Check that e and totient(p, q, r) are coprime.
-	pminus1 := new(big.Int).Sub(priv.P, bigOne)
-	qminus1 := new(big.Int).Sub(priv.Q, bigOne)
-	totient := new(big.Int).Mul(pminus1, qminus1)
-	if priv.R != nil {
-		rminus1 := new(big.Int).Sub(priv.R, bigOne)
-		totient.Mul(totient, rminus1)
+	// Check that e and totient(Πprimes) are coprime.
+	totient := new(big.Int).Set(bigOne)
+	for _, prime := range priv.Primes {
+		pminus1 := new(big.Int).Sub(prime, bigOne)
+		totient.Mul(totient, pminus1)
 	}
 	e := big.NewInt(int64(priv.E))
 	gcd := new(big.Int)
@@ -143,7 +152,7 @@ func (priv *PrivateKey) Validate() os.Error {
 	if gcd.Cmp(bigOne) != 0 {
 		return os.ErrorString("invalid public exponent E")
 	}
-	// Check that de ≡ 1 (mod totient(p, q, r))
+	// Check that de ≡ 1 (mod totient(Πprimes))
 	de := new(big.Int).Mul(priv.D, e)
 	de.Mod(de, totient)
 	if de.Cmp(bigOne) != 0 {
@@ -154,6 +163,20 @@ func (priv *PrivateKey) Validate() os.Error {
 
 // GenerateKey generates an RSA keypair of the given bit size.
 func GenerateKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
+	return GenerateMultiPrimeKey(rand, 2, bits)
+}
+
+// GenerateMultiPrimeKey generates a multi-prime RSA keypair of the given bit
+// size, as suggested in [1]. Although the public keys are compatible
+// (actually, indistinguishable) from the 2-prime case, the private keys are
+// not. Thus it may not be possible to export multi-prime private keys in
+// certain formats or to subsequently import them into other code.
+//
+// Table 1 in [2] suggests maximum numbers of primes for a given size.
+//
+// [1] US patent 4405829 (1972, expired)
+// [2] http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-16.pdf
+func GenerateMultiPrimeKey(rand io.Reader, nprimes int, bits int) (priv *PrivateKey, err os.Error) {
 	priv = new(PrivateKey)
 	// Smaller public exponents lead to faster public key
 	// operations. Since the exponent must be coprime to
@@ -165,100 +188,41 @@ func GenerateKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
 	// [1] http://marc.info/?l=cryptography&m=115694833312008&w=2
 	priv.E = 3
 
-	pminus1 := new(big.Int)
-	qminus1 := new(big.Int)
-	totient := new(big.Int)
+	if nprimes < 2 {
+		return nil, os.ErrorString("rsa.GenerateMultiPrimeKey: nprimes must be >= 2")
+	}
+
+	primes := make([]*big.Int, nprimes)
 
+NextSetOfPrimes:
 	for {
-		p, err := randomPrime(rand, bits/2)
+		todo := bits
+		for i := 0; i < nprimes; i++ {
+			primes[i], err = randomPrime(rand, todo/(nprimes-i))
 			if err != nil {
 				return nil, err
 			}
-
-		q, err := randomPrime(rand, bits/2)
-		if err != nil {
-			return nil, err
+			todo -= primes[i].BitLen()
 		}
 
-		if p.Cmp(q) == 0 {
-			continue
+		// Make sure that primes is pairwise unequal.
+		for i, prime := range primes {
+			for j := 0; j < i; j++ {
+				if prime.Cmp(primes[j]) == 0 {
+					continue NextSetOfPrimes
 				}
-
-		n := new(big.Int).Mul(p, q)
-		pminus1.Sub(p, bigOne)
-		qminus1.Sub(q, bigOne)
-		totient.Mul(pminus1, qminus1)
-
-		g := new(big.Int)
-		priv.D = new(big.Int)
-		y := new(big.Int)
-		e := big.NewInt(int64(priv.E))
-		big.GcdInt(g, priv.D, y, e, totient)
-
-		if g.Cmp(bigOne) == 0 {
-			priv.D.Add(priv.D, totient)
-			priv.P = p
-			priv.Q = q
-			priv.N = n
-
-			break
 			}
 		}
 
-	return
-}
-
-// Generate3PrimeKey generates a 3-prime RSA keypair of the given bit size, as
-// suggested in [1]. Although the public keys are compatible (actually,
-// indistinguishable) from the 2-prime case, the private keys are not. Thus it
-// may not be possible to export 3-prime private keys in certain formats or to
-// subsequently import them into other code.
-//
-// Table 1 in [2] suggests that size should be >= 1024 when using 3 primes.
-//
-// [1] US patent 4405829 (1972, expired)
-// [2] http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-16.pdf
-func Generate3PrimeKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
-	priv = new(PrivateKey)
-	priv.E = 3
-
+		n := new(big.Int).Set(bigOne)
+		totient := new(big.Int).Set(bigOne)
 		pminus1 := new(big.Int)
-	qminus1 := new(big.Int)
-	rminus1 := new(big.Int)
-	totient := new(big.Int)
-
-	for {
-		p, err := randomPrime(rand, bits/3)
-		if err != nil {
-			return nil, err
-		}
-
-		todo := bits - p.BitLen()
-		q, err := randomPrime(rand, todo/2)
-		if err != nil {
-			return nil, err
+		for _, prime := range primes {
+			n.Mul(n, prime)
+			pminus1.Sub(prime, bigOne)
+			totient.Mul(totient, pminus1)
 		}
 
-		todo -= q.BitLen()
-		r, err := randomPrime(rand, todo)
-		if err != nil {
-			return nil, err
-		}
-
-		if p.Cmp(q) == 0 ||
-			q.Cmp(r) == 0 ||
-			r.Cmp(p) == 0 {
-			continue
-		}
-
-		n := new(big.Int).Mul(p, q)
-		n.Mul(n, r)
-		pminus1.Sub(p, bigOne)
-		qminus1.Sub(q, bigOne)
-		rminus1.Sub(r, bigOne)
-		totient.Mul(pminus1, qminus1)
-		totient.Mul(totient, rminus1)
-
 		g := new(big.Int)
 		priv.D = new(big.Int)
 		y := new(big.Int)
@@ -267,15 +231,14 @@ func Generate3PrimeKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error
 
 		if g.Cmp(bigOne) == 0 {
 			priv.D.Add(priv.D, totient)
-			priv.P = p
-			priv.Q = q
-			priv.R = r
+			priv.Primes = primes
 			priv.N = n
 
 			break
 		}
 	}
 
+	priv.Precompute()
 	return
 }
 
@@ -409,23 +372,34 @@ func modInverse(a, n *big.Int) (ia *big.Int, ok bool) {
 	return x, true
 }
 
-// precompute performs some calculations that speed up private key operations
+// Precompute performs some calculations that speed up private key operations
 // in the future.
-func (priv *PrivateKey) precompute() {
-	priv.dP = new(big.Int).Sub(priv.P, bigOne)
-	priv.dP.Mod(priv.D, priv.dP)
+func (priv *PrivateKey) Precompute() {
+	if priv.Precomputed.Dp != nil {
+		return
+	}
 
-	priv.dQ = new(big.Int).Sub(priv.Q, bigOne)
-	priv.dQ.Mod(priv.D, priv.dQ)
+	priv.Precomputed.Dp = new(big.Int).Sub(priv.Primes[0], bigOne)
+	priv.Precomputed.Dp.Mod(priv.D, priv.Precomputed.Dp)
 
-	priv.qInv = new(big.Int).ModInverse(priv.Q, priv.P)
+	priv.Precomputed.Dq = new(big.Int).Sub(priv.Primes[1], bigOne)
+	priv.Precomputed.Dq.Mod(priv.D, priv.Precomputed.Dq)
 
-	if priv.R != nil {
-		priv.dR = new(big.Int).Sub(priv.R, bigOne)
-		priv.dR.Mod(priv.D, priv.dR)
+	priv.Precomputed.Qinv = new(big.Int).ModInverse(priv.Primes[1], priv.Primes[0])
 
-		priv.pq = new(big.Int).Mul(priv.P, priv.Q)
-		priv.tr = new(big.Int).ModInverse(priv.pq, priv.R)
+	r := new(big.Int).Mul(priv.Primes[0], priv.Primes[1])
+	priv.Precomputed.CRTValues = make([]CRTValue, len(priv.Primes)-2)
+	for i := 2; i < len(priv.Primes); i++ {
+		prime := priv.Primes[i]
+		values := &priv.Precomputed.CRTValues[i-2]
+
+		values.Exp = new(big.Int).Sub(prime, bigOne)
+		values.Exp.Mod(priv.D, values.Exp)
+
+		values.R = new(big.Int).Set(r)
+		values.Coeff = new(big.Int).ModInverse(r, prime)
+
+		r.Mul(r, prime)
 	}
 }
 
@@ -463,53 +437,41 @@ func decrypt(rand io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err os.E
 		}
 		bigE := big.NewInt(int64(priv.E))
 		rpowe := new(big.Int).Exp(r, bigE, priv.N)
-		c.Mul(c, rpowe)
-		c.Mod(c, priv.N)
-	}
-
-	priv.rwMutex.RLock()
-
-	if priv.dP == nil && priv.P != nil {
-		priv.rwMutex.RUnlock()
-		priv.rwMutex.Lock()
-		if priv.dP == nil && priv.P != nil {
-			priv.precompute()
-		}
-		priv.rwMutex.Unlock()
-		priv.rwMutex.RLock()
+		cCopy := new(big.Int).Set(c)
+		cCopy.Mul(cCopy, rpowe)
+		cCopy.Mod(cCopy, priv.N)
+		c = cCopy
 	}
 
-	if priv.dP == nil {
+	if priv.Precomputed.Dp == nil {
 		m = new(big.Int).Exp(c, priv.D, priv.N)
 	} else {
 		// We have the precalculated values needed for the CRT.
-		m = new(big.Int).Exp(c, priv.dP, priv.P)
-		m2 := new(big.Int).Exp(c, priv.dQ, priv.Q)
+		m = new(big.Int).Exp(c, priv.Precomputed.Dp, priv.Primes[0])
+		m2 := new(big.Int).Exp(c, priv.Precomputed.Dq, priv.Primes[1])
 		m.Sub(m, m2)
 		if m.Sign() < 0 {
-			m.Add(m, priv.P)
+			m.Add(m, priv.Primes[0])
 		}
-		m.Mul(m, priv.qInv)
-		m.Mod(m, priv.P)
-		m.Mul(m, priv.Q)
+		m.Mul(m, priv.Precomputed.Qinv)
+		m.Mod(m, priv.Primes[0])
+		m.Mul(m, priv.Primes[1])
 		m.Add(m, m2)
 
-		if priv.dR != nil {
-			// 3-prime CRT.
-			m2.Exp(c, priv.dR, priv.R)
+		for i, values := range priv.Precomputed.CRTValues {
+			prime := priv.Primes[2+i]
+			m2.Exp(c, values.Exp, prime)
 			m2.Sub(m2, m)
-			m2.Mul(m2, priv.tr)
-			m2.Mod(m2, priv.R)
+			m2.Mul(m2, values.Coeff)
+			m2.Mod(m2, prime)
 			if m2.Sign() < 0 {
-				m2.Add(m2, priv.R)
+				m2.Add(m2, prime)
 			}
-			m2.Mul(m2, priv.pq)
+			m2.Mul(m2, values.R)
 			m.Add(m, m2)
 		}
 	}
 
-	priv.rwMutex.RUnlock()
-
 	if ir != nil {
 		// Unblind.
 		m.Mul(m, ir)

src/pkg/crypto/rsa/rsa_test.go

@@ -30,7 +30,20 @@ func Test3PrimeKeyGeneration(t *testing.T) {
 	}
 
 	size := 768
-	priv, err := Generate3PrimeKey(rand.Reader, size)
+	priv, err := GenerateMultiPrimeKey(rand.Reader, 3, size)
+	if err != nil {
+		t.Errorf("failed to generate key")
+	}
+	testKeyBasics(t, priv)
+}
+
+func Test4PrimeKeyGeneration(t *testing.T) {
+	if testing.Short() {
+		return
+	}
+
+	size := 768
+	priv, err := GenerateMultiPrimeKey(rand.Reader, 4, size)
 	if err != nil {
 		t.Errorf("failed to generate key")
 	}
@@ -45,6 +58,7 @@ func testKeyBasics(t *testing.T, priv *PrivateKey) {
 	pub := &priv.PublicKey
 	m := big.NewInt(42)
 	c := encrypt(new(big.Int), pub, m)
+
 	m2, err := decrypt(nil, priv, c)
 	if err != nil {
 		t.Errorf("error while decrypting: %s", err)
@@ -59,7 +73,7 @@ func testKeyBasics(t *testing.T, priv *PrivateKey) {
 		t.Errorf("error while decrypting (blind): %s", err)
 	}
 	if m.Cmp(m3) != 0 {
-		t.Errorf("(blind) got:%v, want:%v", m3, m)
+		t.Errorf("(blind) got:%v, want:%v (%#v)", m3, m, priv)
 	}
 }
 
@@ -77,10 +91,12 @@ func BenchmarkRSA2048Decrypt(b *testing.B) {
 			E: 3,
 		},
 		D: fromBase10("9542755287494004433998723259516013739278699355114572217325597900889416163458809501304132487555642811888150937392013824621448709836142886006653296025093941418628992648429798282127303704957273845127141852309016655778568546006839666463451542076964744073572349705538631742281931858219480985907271975884773482372966847639853897890615456605598071088189838676728836833012254065983259638538107719766738032720239892094196108713378822882383694456030043492571063441943847195939549773271694647657549658603365629458610273821292232646334717612674519997533901052790334279661754176490593041941863932308687197618671528035670452762731"),
-		P: fromBase10("130903255182996722426771613606077755295583329135067340152947172868415809027537376306193179624298874215608270802054347609836776473930072411958753044562214537013874103802006369634761074377213995983876788718033850153719421695468704276694983032644416930879093914927146648402139231293035971427838068945045019075433"),
-		Q: fromBase10("109348945610485453577574767652527472924289229538286649661240938988020367005475727988253438647560958573506159449538793540472829815903949343191091817779240101054552748665267574271163617694640513549693841337820602726596756351006149518830932261246698766355347898158548465400674856021497190430791824869615170301029"),
+		Primes: []*big.Int{
+			fromBase10("130903255182996722426771613606077755295583329135067340152947172868415809027537376306193179624298874215608270802054347609836776473930072411958753044562214537013874103802006369634761074377213995983876788718033850153719421695468704276694983032644416930879093914927146648402139231293035971427838068945045019075433"),
+			fromBase10("109348945610485453577574767652527472924289229538286649661240938988020367005475727988253438647560958573506159449538793540472829815903949343191091817779240101054552748665267574271163617694640513549693841337820602726596756351006149518830932261246698766355347898158548465400674856021497190430791824869615170301029"),
+		},
 	}
-	priv.precompute()
+	priv.Precompute()
 
 	c := fromBase10("1000")
 
@@ -99,11 +115,13 @@ func Benchmark3PrimeRSA2048Decrypt(b *testing.B) {
 			E: 3,
 		},
 		D: fromBase10("10897585948254795600358846499957366070880176878341177571733155050184921896034527397712889205732614568234385175145686545381899460748279607074689061600935843283397424506622998458510302603922766336783617368686090042765718290914099334449154829375179958369993407724946186243249568928237086215759259909861748642124071874879861299389874230489928271621259294894142840428407196932444474088857746123104978617098858619445675532587787023228852383149557470077802718705420275739737958953794088728369933811184572620857678792001136676902250566845618813972833750098806496641114644760255910789397593428910198080271317419213080834885003"),
-		P: fromBase10("1025363189502892836833747188838978207017355117492483312747347695538428729137306368764177201532277413433182799108299960196606011786562992097313508180436744488171474690412562218914213688661311117337381958560443"),
-		Q: fromBase10("3467903426626310123395340254094941045497208049900750380025518552334536945536837294961497712862519984786362199788654739924501424784631315081391467293694361474867825728031147665777546570788493758372218019373"),
-		R: fromBase10("4597024781409332673052708605078359346966325141767460991205742124888960305710298765592730135879076084498363772408626791576005136245060321874472727132746643162385746062759369754202494417496879741537284589047"),
+		Primes: []*big.Int{
+			fromBase10("1025363189502892836833747188838978207017355117492483312747347695538428729137306368764177201532277413433182799108299960196606011786562992097313508180436744488171474690412562218914213688661311117337381958560443"),
+			fromBase10("3467903426626310123395340254094941045497208049900750380025518552334536945536837294961497712862519984786362199788654739924501424784631315081391467293694361474867825728031147665777546570788493758372218019373"),
+			fromBase10("4597024781409332673052708605078359346966325141767460991205742124888960305710298765592730135879076084498363772408626791576005136245060321874472727132746643162385746062759369754202494417496879741537284589047"),
+		},
 	}
-	priv.precompute()
+	priv.Precompute()
 
 	c := fromBase10("1000")
 

src/pkg/crypto/tls/handshake_server_test.go

@@ -188,8 +188,10 @@ var testPrivateKey = &rsa.PrivateKey{
 		E: 65537,
 	},
 	D: bigFromString("29354450337804273969007277378287027274721892607543397931919078829901848876371746653677097639302788129485893852488285045793268732234230875671682624082413996177431586734171663258657462237320300610850244186316880055243099640544518318093544057213190320837094958164973959123058337475052510833916491060913053867729"),
-	P: bigFromString("11969277782311800166562047708379380720136961987713178380670422671426759650127150688426177829077494755200794297055316163155755835813760102405344560929062149"),
-	Q: bigFromString("10998999429884441391899182616418192492905073053684657075974935218461686523870125521822756579792315215543092255516093840728890783887287417039645833477273829"),
+	Primes: []*big.Int{
+		bigFromString("11969277782311800166562047708379380720136961987713178380670422671426759650127150688426177829077494755200794297055316163155755835813760102405344560929062149"),
+		bigFromString("10998999429884441391899182616418192492905073053684657075974935218461686523870125521822756579792315215543092255516093840728890783887287417039645833477273829"),
+	},
 }
 
 // Script of interaction with gnutls implementation.

src/pkg/crypto/x509/x509.go

@@ -8,6 +8,7 @@ package x509
 import (
 	"asn1"
 	"big"
+	"bytes"
 	"container/vector"
 	"crypto"
 	"crypto/rsa"
@@ -26,6 +27,20 @@ type pkcs1PrivateKey struct {
 	D       asn1.RawValue
 	P       asn1.RawValue
 	Q       asn1.RawValue
+	// We ignore these values, if present, because rsa will calculate them.
+	Dp   asn1.RawValue "optional"
+	Dq   asn1.RawValue "optional"
+	Qinv asn1.RawValue "optional"
+
+	AdditionalPrimes []pkcs1AddtionalRSAPrime "optional"
+}
+
+type pkcs1AddtionalRSAPrime struct {
+	Prime asn1.RawValue
+
+	// We ignore these values because rsa will calculate them.
+	Exp   asn1.RawValue
+	Coeff asn1.RawValue
 }
 
 // rawValueIsInteger returns true iff the given ASN.1 RawValue is an INTEGER type.
@@ -45,6 +60,10 @@ func ParsePKCS1PrivateKey(der []byte) (key *rsa.PrivateKey, err os.Error) {
 		return
 	}
 
+	if priv.Version > 1 {
+		return nil, os.ErrorString("x509: unsupported private key version")
+	}
+
 	if !rawValueIsInteger(&priv.N) ||
 		!rawValueIsInteger(&priv.D) ||
 		!rawValueIsInteger(&priv.P) ||
@@ -60,26 +79,66 @@ func ParsePKCS1PrivateKey(der []byte) (key *rsa.PrivateKey, err os.Error) {
 	}
 
 	key.D = new(big.Int).SetBytes(priv.D.Bytes)
-	key.P = new(big.Int).SetBytes(priv.P.Bytes)
-	key.Q = new(big.Int).SetBytes(priv.Q.Bytes)
+	key.Primes = make([]*big.Int, 2+len(priv.AdditionalPrimes))
+	key.Primes[0] = new(big.Int).SetBytes(priv.P.Bytes)
+	key.Primes[1] = new(big.Int).SetBytes(priv.Q.Bytes)
+	for i, a := range priv.AdditionalPrimes {
+		if !rawValueIsInteger(&a.Prime) {
+			return nil, asn1.StructuralError{"tags don't match"}
+		}
+		key.Primes[i+2] = new(big.Int).SetBytes(a.Prime.Bytes)
+		// We ignore the other two values because rsa will calculate
+		// them as needed.
+	}
 
 	err = key.Validate()
 	if err != nil {
 		return nil, err
 	}
+	key.Precompute()
 
 	return
 }
 
+// rawValueForBig returns an asn1.RawValue which represents the given integer.
+func rawValueForBig(n *big.Int) asn1.RawValue {
+	b := n.Bytes()
+	if n.Sign() >= 0 && len(b) > 0 && b[0]&0x80 != 0 {
+		// This positive number would be interpreted as a negative
+		// number in ASN.1 because the MSB is set.
+		padded := make([]byte, len(b)+1)
+		copy(padded[1:], b)
+		b = padded
+	}
+	return asn1.RawValue{Tag: 2, Bytes: b}
+}
+
 // MarshalPKCS1PrivateKey converts a private key to ASN.1 DER encoded form.
 func MarshalPKCS1PrivateKey(key *rsa.PrivateKey) []byte {
+	key.Precompute()
+
+	version := 0
+	if len(key.Primes) > 2 {
+		version = 1
+	}
+
 	priv := pkcs1PrivateKey{
-		Version: 1,
-		N:       asn1.RawValue{Tag: 2, Bytes: key.PublicKey.N.Bytes()},
+		Version: version,
+		N:       rawValueForBig(key.N),
 		E:       key.PublicKey.E,
-		D:       asn1.RawValue{Tag: 2, Bytes: key.D.Bytes()},
-		P:       asn1.RawValue{Tag: 2, Bytes: key.P.Bytes()},
-		Q:       asn1.RawValue{Tag: 2, Bytes: key.Q.Bytes()},
+		D:       rawValueForBig(key.D),
+		P:       rawValueForBig(key.Primes[0]),
+		Q:       rawValueForBig(key.Primes[1]),
+		Dp:      rawValueForBig(key.Precomputed.Dp),
+		Dq:      rawValueForBig(key.Precomputed.Dq),
+		Qinv:    rawValueForBig(key.Precomputed.Qinv),
+	}
+
+	priv.AdditionalPrimes = make([]pkcs1AddtionalRSAPrime, len(key.Precomputed.CRTValues))
+	for i, values := range key.Precomputed.CRTValues {
+		priv.AdditionalPrimes[i].Prime = rawValueForBig(key.Primes[2+i])
+		priv.AdditionalPrimes[i].Exp = rawValueForBig(values.Exp)
+		priv.AdditionalPrimes[i].Coeff = rawValueForBig(values.Coeff)
 	}
 
 	b, _ := asn1.Marshal(priv)
@@ -396,6 +455,10 @@ func (ConstraintViolationError) String() string {
 	return "invalid signature: parent certificate cannot sign this kind of certificate"
 }
 
+func (c *Certificate) Equal(other *Certificate) bool {
+	return bytes.Equal(c.Raw, other.Raw)
+}
+
 // CheckSignatureFrom verifies that the signature on c is a valid signature
 // from parent.
 func (c *Certificate) CheckSignatureFrom(parent *Certificate) (err os.Error) {

src/pkg/crypto/x509/x509_test.go

@@ -20,12 +20,13 @@ func TestParsePKCS1PrivateKey(t *testing.T) {
 	priv, err := ParsePKCS1PrivateKey(block.Bytes)
 	if err != nil {
 		t.Errorf("Failed to parse private key: %s", err)
+		return
 	}
 	if priv.PublicKey.N.Cmp(rsaPrivateKey.PublicKey.N) != 0 ||
 		priv.PublicKey.E != rsaPrivateKey.PublicKey.E ||
 		priv.D.Cmp(rsaPrivateKey.D) != 0 ||
-		priv.P.Cmp(rsaPrivateKey.P) != 0 ||
-		priv.Q.Cmp(rsaPrivateKey.Q) != 0 {
+		priv.Primes[0].Cmp(rsaPrivateKey.Primes[0]) != 0 ||
+		priv.Primes[1].Cmp(rsaPrivateKey.Primes[1]) != 0 {
 		t.Errorf("got:%+v want:%+v", priv, rsaPrivateKey)
 	}
 }
@@ -47,14 +48,54 @@ func bigFromString(s string) *big.Int {
 	return ret
 }
 
+func fromBase10(base10 string) *big.Int {
+	i := new(big.Int)
+	i.SetString(base10, 10)
+	return i
+}
+
 var rsaPrivateKey = &rsa.PrivateKey{
 	PublicKey: rsa.PublicKey{
 		N: bigFromString("9353930466774385905609975137998169297361893554149986716853295022578535724979677252958524466350471210367835187480748268864277464700638583474144061408845077"),
 		E: 65537,
 	},
 	D: bigFromString("7266398431328116344057699379749222532279343923819063639497049039389899328538543087657733766554155839834519529439851673014800261285757759040931985506583861"),
-	P: bigFromString("98920366548084643601728869055592650835572950932266967461790948584315647051443"),
-	Q: bigFromString("94560208308847015747498523884063394671606671904944666360068158221458669711639"),
+	Primes: []*big.Int{
+		bigFromString("98920366548084643601728869055592650835572950932266967461790948584315647051443"),
+		bigFromString("94560208308847015747498523884063394671606671904944666360068158221458669711639"),
+	},
+}
+
+func TestMarshalRSAPrivateKey(t *testing.T) {
+	priv := &rsa.PrivateKey{
+		PublicKey: rsa.PublicKey{
+			N: fromBase10("16346378922382193400538269749936049106320265317511766357599732575277382844051791096569333808598921852351577762718529818072849191122419410612033592401403764925096136759934497687765453905884149505175426053037420486697072448609022753683683718057795566811401938833367954642951433473337066311978821180526439641496973296037000052546108507805269279414789035461158073156772151892452251106173507240488993608650881929629163465099476849643165682709047462010581308719577053905787496296934240246311806555924593059995202856826239801816771116902778517096212527979497399966526283516447337775509777558018145573127308919204297111496233"),
+			E: 3,
+		},
+		D: fromBase10("10897585948254795600358846499957366070880176878341177571733155050184921896034527397712889205732614568234385175145686545381899460748279607074689061600935843283397424506622998458510302603922766336783617368686090042765718290914099334449154829375179958369993407724946186243249568928237086215759259909861748642124071874879861299389874230489928271621259294894142840428407196932444474088857746123104978617098858619445675532587787023228852383149557470077802718705420275739737958953794088728369933811184572620857678792001136676902250566845618813972833750098806496641114644760255910789397593428910198080271317419213080834885003"),
+		Primes: []*big.Int{
+			fromBase10("1025363189502892836833747188838978207017355117492483312747347695538428729137306368764177201532277413433182799108299960196606011786562992097313508180436744488171474690412562218914213688661311117337381958560443"),
+			fromBase10("3467903426626310123395340254094941045497208049900750380025518552334536945536837294961497712862519984786362199788654739924501424784631315081391467293694361474867825728031147665777546570788493758372218019373"),
+			fromBase10("4597024781409332673052708605078359346966325141767460991205742124888960305710298765592730135879076084498363772408626791576005136245060321874472727132746643162385746062759369754202494417496879741537284589047"),
+		},
+	}
+
+	derBytes := MarshalPKCS1PrivateKey(priv)
+
+	priv2, err := ParsePKCS1PrivateKey(derBytes)
+	if err != nil {
+		t.Errorf("error parsing serialized key: %s", err)
+		return
+	}
+	if priv.PublicKey.N.Cmp(priv2.PublicKey.N) != 0 ||
+		priv.PublicKey.E != priv2.PublicKey.E ||
+		priv.D.Cmp(priv2.D) != 0 ||
+		len(priv2.Primes) != 3 ||
+		priv.Primes[0].Cmp(priv2.Primes[0]) != 0 ||
+		priv.Primes[1].Cmp(priv2.Primes[1]) != 0 ||
+		priv.Primes[2].Cmp(priv2.Primes[2]) != 0 {
+		t.Errorf("got:%+v want:%+v", priv, priv2)
+	}
 }
 
 type matchHostnamesTest struct {