src/crypto/cipher/gcm.go - go - Git at Google

 // Copyright 2013 The Go Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.

 package cipher

 import (
 	"crypto/subtle"
 	"errors"
 )

 // AEAD is a cipher mode providing authenticated encryption with associated
 // data. For a description of the methodology, see
 //	https://en.wikipedia.org/wiki/Authenticated_encryption
 type AEAD interface {
 	// NonceSize returns the size of the nonce that must be passed to Seal
 	// and Open.
 	NonceSize() int

 	// Overhead returns the maximum difference between the lengths of a
 	// plaintext and its ciphertext.
 	Overhead() int

 	// Seal encrypts and authenticates plaintext, authenticates the
 	// additional data and appends the result to dst, returning the updated
 	// slice. The nonce must be NonceSize() bytes long and unique for all
 	// time, for a given key.
 	//
 	// The plaintext and dst may alias exactly or not at all. To reuse
 	// plaintext's storage for the encrypted output, use plaintext[:0] as dst.
 	Seal(dst, nonce, plaintext, additionalData []byte) []byte

 	// Open decrypts and authenticates ciphertext, authenticates the
 	// additional data and, if successful, appends the resulting plaintext
 	// to dst, returning the updated slice. The nonce must be NonceSize()
 	// bytes long and both it and the additional data must match the
 	// value passed to Seal.
 	//
 	// The ciphertext and dst may alias exactly or not at all. To reuse
 	// ciphertext's storage for the decrypted output, use ciphertext[:0] as dst.
 	//
 	// Even if the function fails, the contents of dst, up to its capacity,
 	// may be overwritten.
 	Open(dst, nonce, ciphertext, additionalData []byte) ([]byte, error)
 }

 // gcmAble is an interface implemented by ciphers that have a specific optimized
 // implementation of GCM, like crypto/aes. NewGCM will check for this interface
 // and return the specific AEAD if found.
 type gcmAble interface {
 	NewGCM(int) (AEAD, error)
 }

 // gcmFieldElement represents a value in GF(2¹²⁸). In order to reflect the GCM
 // standard and make getUint64 suitable for marshaling these values, the bits
 // are stored backwards. For example:
 //   the coefficient of x⁰ can be obtained by v.low >> 63.
 //   the coefficient of x⁶³ can be obtained by v.low & 1.
 //   the coefficient of x⁶⁴ can be obtained by v.high >> 63.
 //   the coefficient of x¹²⁷ can be obtained by v.high & 1.
 type gcmFieldElement struct {
 	low, high uint64
 }

 // gcm represents a Galois Counter Mode with a specific key. See
 // http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf
 type gcm struct {
 	cipher    Block
 	nonceSize int
 	// productTable contains the first sixteen powers of the key, H.
 	// However, they are in bit reversed order. See NewGCMWithNonceSize.
 	productTable [16]gcmFieldElement
 }

 // NewGCM returns the given 128-bit, block cipher wrapped in Galois Counter Mode
 // with the standard nonce length.
 //
 // In general, the GHASH operation performed by this implementation of GCM is not constant-time.
 // An exception is when the underlying Block was created by aes.NewCipher
 // on systems with hardware support for AES. See the crypto/aes package documentation for details.
 func NewGCM(cipher Block) (AEAD, error) {
 	return NewGCMWithNonceSize(cipher, gcmStandardNonceSize)
 }

 // NewGCMWithNonceSize returns the given 128-bit, block cipher wrapped in Galois
 // Counter Mode, which accepts nonces of the given length.
 //
 // Only use this function if you require compatibility with an existing
 // cryptosystem that uses non-standard nonce lengths. All other users should use
 // NewGCM, which is faster and more resistant to misuse.
 func NewGCMWithNonceSize(cipher Block, size int) (AEAD, error) {
 	if cipher, ok := cipher.(gcmAble); ok {
 		return cipher.NewGCM(size)
 	}

 	if cipher.BlockSize() != gcmBlockSize {
 		return nil, errors.New("cipher: NewGCM requires 128-bit block cipher")
 	}

 	var key [gcmBlockSize]byte
 	cipher.Encrypt(key[:], key[:])

 	g := &gcm{cipher: cipher, nonceSize: size}

 	// We precompute 16 multiples of |key|. However, when we do lookups
 	// into this table we'll be using bits from a field element and
 	// therefore the bits will be in the reverse order. So normally one
 	// would expect, say, 4*key to be in index 4 of the table but due to
 	// this bit ordering it will actually be in index 0010 (base 2) = 2.
 	x := gcmFieldElement{
 		getUint64(key[:8]),
 		getUint64(key[8:]),
 	}
 	g.productTable[reverseBits(1)] = x

 	for i := 2; i < 16; i += 2 {
 		g.productTable[reverseBits(i)] = gcmDouble(&g.productTable[reverseBits(i/2)])
 		g.productTable[reverseBits(i+1)] = gcmAdd(&g.productTable[reverseBits(i)], &x)
 	}

 	return g, nil
 }

 const (
 	gcmBlockSize         = 16
 	gcmTagSize           = 16
 	gcmStandardNonceSize = 12
 )

 func (g *gcm) NonceSize() int {
 	return g.nonceSize
 }

 func (*gcm) Overhead() int {
 	return gcmTagSize
 }

 func (g *gcm) Seal(dst, nonce, plaintext, data []byte) []byte {
 	if len(nonce) != g.nonceSize {
 		panic("cipher: incorrect nonce length given to GCM")
 	}
 	if uint64(len(plaintext)) > ((1<<32)-2)*uint64(g.cipher.BlockSize()) {
 		panic("cipher: message too large for GCM")
 	}

 	ret, out := sliceForAppend(dst, len(plaintext)+gcmTagSize)

 	var counter, tagMask [gcmBlockSize]byte
 	g.deriveCounter(&counter, nonce)

 	g.cipher.Encrypt(tagMask[:], counter[:])
 	gcmInc32(&counter)

 	g.counterCrypt(out, plaintext, &counter)
 	g.auth(out[len(plaintext):], out[:len(plaintext)], data, &tagMask)

 	return ret
 }

 var errOpen = errors.New("cipher: message authentication failed")

 func (g *gcm) Open(dst, nonce, ciphertext, data []byte) ([]byte, error) {
 	if len(nonce) != g.nonceSize {
 		panic("cipher: incorrect nonce length given to GCM")
 	}

 	if len(ciphertext) < gcmTagSize {
 		return nil, errOpen
 	}
 	if uint64(len(ciphertext)) > ((1<<32)-2)*uint64(g.cipher.BlockSize())+gcmTagSize {
 		return nil, errOpen
 	}

 	tag := ciphertext[len(ciphertext)-gcmTagSize:]
 	ciphertext = ciphertext[:len(ciphertext)-gcmTagSize]

 	var counter, tagMask [gcmBlockSize]byte
 	g.deriveCounter(&counter, nonce)

 	g.cipher.Encrypt(tagMask[:], counter[:])
 	gcmInc32(&counter)

 	var expectedTag [gcmTagSize]byte
 	g.auth(expectedTag[:], ciphertext, data, &tagMask)

 	ret, out := sliceForAppend(dst, len(ciphertext))

 	if subtle.ConstantTimeCompare(expectedTag[:], tag) != 1 {
 		// The AESNI code decrypts and authenticates concurrently, and
 		// so overwrites dst in the event of a tag mismatch. That
 		// behavior is mimicked here in order to be consistent across
 		// platforms.
 		for i := range out {
 			out[i] = 0
 		}
 		return nil, errOpen
 	}

 	g.counterCrypt(out, ciphertext, &counter)

 	return ret, nil
 }

 // reverseBits reverses the order of the bits of 4-bit number in i.
 func reverseBits(i int) int {
 	i = ((i << 2) & 0xc) | ((i >> 2) & 0x3)
 	i = ((i << 1) & 0xa) | ((i >> 1) & 0x5)
 	return i
 }

 // gcmAdd adds two elements of GF(2¹²⁸) and returns the sum.
 func gcmAdd(x, y *gcmFieldElement) gcmFieldElement {
 	// Addition in a characteristic 2 field is just XOR.
 	return gcmFieldElement{x.low ^ y.low, x.high ^ y.high}
 }

 // gcmDouble returns the result of doubling an element of GF(2¹²⁸).
 func gcmDouble(x *gcmFieldElement) (double gcmFieldElement) {
 	msbSet := x.high&1 == 1

 	// Because of the bit-ordering, doubling is actually a right shift.
 	double.high = x.high >> 1
 	double.high |= x.low << 63
 	double.low = x.low >> 1

 	// If the most-significant bit was set before shifting then it,
 	// conceptually, becomes a term of x^128. This is greater than the
 	// irreducible polynomial so the result has to be reduced. The
 	// irreducible polynomial is 1+x+x^2+x^7+x^128. We can subtract that to
 	// eliminate the term at x^128 which also means subtracting the other
 	// four terms. In characteristic 2 fields, subtraction == addition ==
 	// XOR.
 	if msbSet {
 		double.low ^= 0xe100000000000000
 	}

 	return
 }

 var gcmReductionTable = []uint16{
 	0x0000, 0x1c20, 0x3840, 0x2460, 0x7080, 0x6ca0, 0x48c0, 0x54e0,
 	0xe100, 0xfd20, 0xd940, 0xc560, 0x9180, 0x8da0, 0xa9c0, 0xb5e0,
 }

 // mul sets y to y*H, where H is the GCM key, fixed during NewGCMWithNonceSize.
 func (g *gcm) mul(y *gcmFieldElement) {
 	var z gcmFieldElement

 	for i := 0; i < 2; i++ {
 		word := y.high
 		if i == 1 {
 			word = y.low
 		}

 		// Multiplication works by multiplying z by 16 and adding in
 		// one of the precomputed multiples of H.
 		for j := 0; j < 64; j += 4 {
 			msw := z.high & 0xf
 			z.high >>= 4
 			z.high |= z.low << 60
 			z.low >>= 4
 			z.low ^= uint64(gcmReductionTable[msw]) << 48

 			// the values in |table| are ordered for
 			// little-endian bit positions. See the comment
 			// in NewGCMWithNonceSize.
 			t := &g.productTable[word&0xf]

 			z.low ^= t.low
 			z.high ^= t.high
 			word >>= 4
 		}
 	}

 	*y = z
 }

 // updateBlocks extends y with more polynomial terms from blocks, based on
 // Horner's rule. There must be a multiple of gcmBlockSize bytes in blocks.
 func (g *gcm) updateBlocks(y *gcmFieldElement, blocks []byte) {
 	for len(blocks) > 0 {
 		y.low ^= getUint64(blocks)
 		y.high ^= getUint64(blocks[8:])
 		g.mul(y)
 		blocks = blocks[gcmBlockSize:]
 	}
 }

 // update extends y with more polynomial terms from data. If data is not a
 // multiple of gcmBlockSize bytes long then the remainder is zero padded.
 func (g *gcm) update(y *gcmFieldElement, data []byte) {
 	fullBlocks := (len(data) >> 4) << 4
 	g.updateBlocks(y, data[:fullBlocks])

 	if len(data) != fullBlocks {
 		var partialBlock [gcmBlockSize]byte
 		copy(partialBlock[:], data[fullBlocks:])
 		g.updateBlocks(y, partialBlock[:])
 	}
 }

 // gcmInc32 treats the final four bytes of counterBlock as a big-endian value
 // and increments it.
 func gcmInc32(counterBlock *[16]byte) {
 	for i := gcmBlockSize - 1; i >= gcmBlockSize-4; i-- {
 		counterBlock[i]++
 		if counterBlock[i] != 0 {
 			break
 		}
 	}
 }

 // sliceForAppend takes a slice and a requested number of bytes. It returns a
 // slice with the contents of the given slice followed by that many bytes and a
 // second slice that aliases into it and contains only the extra bytes. If the
 // original slice has sufficient capacity then no allocation is performed.
 func sliceForAppend(in []byte, n int) (head, tail []byte) {
 	if total := len(in) + n; cap(in) >= total {
 		head = in[:total]
 	} else {
 		head = make([]byte, total)
 		copy(head, in)
 	}
 	tail = head[len(in):]
 	return
 }

 // counterCrypt crypts in to out using g.cipher in counter mode.
 func (g *gcm) counterCrypt(out, in []byte, counter *[gcmBlockSize]byte) {
 	var mask [gcmBlockSize]byte

 	for len(in) >= gcmBlockSize {
 		g.cipher.Encrypt(mask[:], counter[:])
 		gcmInc32(counter)

 		xorWords(out, in, mask[:])
 		out = out[gcmBlockSize:]
 		in = in[gcmBlockSize:]
 	}

 	if len(in) > 0 {
 		g.cipher.Encrypt(mask[:], counter[:])
 		gcmInc32(counter)
 		xorBytes(out, in, mask[:])
 	}
 }

 // deriveCounter computes the initial GCM counter state from the given nonce.
 // See NIST SP 800-38D, section 7.1. This assumes that counter is filled with
 // zeros on entry.
 func (g *gcm) deriveCounter(counter *[gcmBlockSize]byte, nonce []byte) {
 	// GCM has two modes of operation with respect to the initial counter
 	// state: a "fast path" for 96-bit (12-byte) nonces, and a "slow path"
 	// for nonces of other lengths. For a 96-bit nonce, the nonce, along
 	// with a four-byte big-endian counter starting at one, is used
 	// directly as the starting counter. For other nonce sizes, the counter
 	// is computed by passing it through the GHASH function.
 	if len(nonce) == gcmStandardNonceSize {
 		copy(counter[:], nonce)
 		counter[gcmBlockSize-1] = 1
 	} else {
 		var y gcmFieldElement
 		g.update(&y, nonce)
 		y.high ^= uint64(len(nonce)) * 8
 		g.mul(&y)
 		putUint64(counter[:8], y.low)
 		putUint64(counter[8:], y.high)
 	}
 }

 // auth calculates GHASH(ciphertext, additionalData), masks the result with
 // tagMask and writes the result to out.
 func (g *gcm) auth(out, ciphertext, additionalData []byte, tagMask *[gcmTagSize]byte) {
 	var y gcmFieldElement
 	g.update(&y, additionalData)
 	g.update(&y, ciphertext)

 	y.low ^= uint64(len(additionalData)) * 8
 	y.high ^= uint64(len(ciphertext)) * 8

 	g.mul(&y)

 	putUint64(out, y.low)
 	putUint64(out[8:], y.high)

 	xorWords(out, out, tagMask[:])
 }

 func getUint64(data []byte) uint64 {
 	r := uint64(data[0])<<56 |
 		uint64(data[1])<<48 |
 		uint64(data[2])<<40 |
 		uint64(data[3])<<32 |
 		uint64(data[4])<<24 |
 		uint64(data[5])<<16 |
 		uint64(data[6])<<8 |
 		uint64(data[7])
 	return r
 }

 func putUint64(out []byte, v uint64) {
 	out[0] = byte(v >> 56)
 	out[1] = byte(v >> 48)
 	out[2] = byte(v >> 40)
 	out[3] = byte(v >> 32)
 	out[4] = byte(v >> 24)
 	out[5] = byte(v >> 16)
 	out[6] = byte(v >> 8)
 	out[7] = byte(v)
 }
	// Copyright 2013 The Go Authors. All rights reserved.
	// Use of this source code is governed by a BSD-style
	// license that can be found in the LICENSE file.

	package cipher

	import (
	"crypto/subtle"
	"errors"
	)

	// AEAD is a cipher mode providing authenticated encryption with associated
	// data. For a description of the methodology, see
	// https://en.wikipedia.org/wiki/Authenticated_encryption
	type AEAD interface {
	// NonceSize returns the size of the nonce that must be passed to Seal
	// and Open.
	NonceSize() int

	// Overhead returns the maximum difference between the lengths of a
	// plaintext and its ciphertext.
	Overhead() int

	// Seal encrypts and authenticates plaintext, authenticates the
	// additional data and appends the result to dst, returning the updated
	// slice. The nonce must be NonceSize() bytes long and unique for all
	// time, for a given key.
	//
	// The plaintext and dst may alias exactly or not at all. To reuse
	// plaintext's storage for the encrypted output, use plaintext[:0] as dst.
	Seal(dst, nonce, plaintext, additionalData []byte) []byte

	// Open decrypts and authenticates ciphertext, authenticates the
	// additional data and, if successful, appends the resulting plaintext
	// to dst, returning the updated slice. The nonce must be NonceSize()
	// bytes long and both it and the additional data must match the
	// value passed to Seal.
	//
	// The ciphertext and dst may alias exactly or not at all. To reuse
	// ciphertext's storage for the decrypted output, use ciphertext[:0] as dst.
	//
	// Even if the function fails, the contents of dst, up to its capacity,
	// may be overwritten.
	Open(dst, nonce, ciphertext, additionalData []byte) ([]byte, error)
	}

	// gcmAble is an interface implemented by ciphers that have a specific optimized
	// implementation of GCM, like crypto/aes. NewGCM will check for this interface
	// and return the specific AEAD if found.
	type gcmAble interface {
	NewGCM(int) (AEAD, error)
	}

	// gcmFieldElement represents a value in GF(2¹²⁸). In order to reflect the GCM
	// standard and make getUint64 suitable for marshaling these values, the bits
	// are stored backwards. For example:
	// the coefficient of x⁰ can be obtained by v.low >> 63.
	// the coefficient of x⁶³ can be obtained by v.low & 1.
	// the coefficient of x⁶⁴ can be obtained by v.high >> 63.
	// the coefficient of x¹²⁷ can be obtained by v.high & 1.
	type gcmFieldElement struct {
	low, high uint64
	}

	// gcm represents a Galois Counter Mode with a specific key. See
	// http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf
	type gcm struct {
	cipher Block
	nonceSize int
	// productTable contains the first sixteen powers of the key, H.
	// However, they are in bit reversed order. See NewGCMWithNonceSize.
	productTable [16]gcmFieldElement
	}

	// NewGCM returns the given 128-bit, block cipher wrapped in Galois Counter Mode
	// with the standard nonce length.
	//
	// In general, the GHASH operation performed by this implementation of GCM is not constant-time.
	// An exception is when the underlying Block was created by aes.NewCipher
	// on systems with hardware support for AES. See the crypto/aes package documentation for details.
	func NewGCM(cipher Block) (AEAD, error) {
	return NewGCMWithNonceSize(cipher, gcmStandardNonceSize)
	}

	// NewGCMWithNonceSize returns the given 128-bit, block cipher wrapped in Galois
	// Counter Mode, which accepts nonces of the given length.
	//
	// Only use this function if you require compatibility with an existing
	// cryptosystem that uses non-standard nonce lengths. All other users should use
	// NewGCM, which is faster and more resistant to misuse.
	func NewGCMWithNonceSize(cipher Block, size int) (AEAD, error) {
	if cipher, ok := cipher.(gcmAble); ok {
	return cipher.NewGCM(size)
	}

	if cipher.BlockSize() != gcmBlockSize {
	return nil, errors.New("cipher: NewGCM requires 128-bit block cipher")
	}

	var key [gcmBlockSize]byte
	cipher.Encrypt(key[:], key[:])

	g := &gcm{cipher: cipher, nonceSize: size}

	// We precompute 16 multiples of \|key\|. However, when we do lookups
	// into this table we'll be using bits from a field element and
	// therefore the bits will be in the reverse order. So normally one
	// would expect, say, 4*key to be in index 4 of the table but due to
	// this bit ordering it will actually be in index 0010 (base 2) = 2.
	x := gcmFieldElement{
	getUint64(key[:8]),
	getUint64(key[8:]),
	}
	g.productTable[reverseBits(1)] = x

	for i := 2; i < 16; i += 2 {
	g.productTable[reverseBits(i)] = gcmDouble(&g.productTable[reverseBits(i/2)])
	g.productTable[reverseBits(i+1)] = gcmAdd(&g.productTable[reverseBits(i)], &x)
	}

	return g, nil
	}

	const (
	gcmBlockSize = 16
	gcmTagSize = 16
	gcmStandardNonceSize = 12
	)

	func (g *gcm) NonceSize() int {
	return g.nonceSize
	}

	func (*gcm) Overhead() int {
	return gcmTagSize
	}

	func (g *gcm) Seal(dst, nonce, plaintext, data []byte) []byte {
	if len(nonce) != g.nonceSize {
	panic("cipher: incorrect nonce length given to GCM")
	}
	if uint64(len(plaintext)) > ((1<<32)-2)*uint64(g.cipher.BlockSize()) {
	panic("cipher: message too large for GCM")
	}

	ret, out := sliceForAppend(dst, len(plaintext)+gcmTagSize)

	var counter, tagMask [gcmBlockSize]byte
	g.deriveCounter(&counter, nonce)

	g.cipher.Encrypt(tagMask[:], counter[:])
	gcmInc32(&counter)

	g.counterCrypt(out, plaintext, &counter)
	g.auth(out[len(plaintext):], out[:len(plaintext)], data, &tagMask)

	return ret
	}

	var errOpen = errors.New("cipher: message authentication failed")

	func (g *gcm) Open(dst, nonce, ciphertext, data []byte) ([]byte, error) {
	if len(nonce) != g.nonceSize {
	panic("cipher: incorrect nonce length given to GCM")
	}

	if len(ciphertext) < gcmTagSize {
	return nil, errOpen
	}
	if uint64(len(ciphertext)) > ((1<<32)-2)*uint64(g.cipher.BlockSize())+gcmTagSize {
	return nil, errOpen
	}

	tag := ciphertext[len(ciphertext)-gcmTagSize:]
	ciphertext = ciphertext[:len(ciphertext)-gcmTagSize]

	var counter, tagMask [gcmBlockSize]byte
	g.deriveCounter(&counter, nonce)

	g.cipher.Encrypt(tagMask[:], counter[:])
	gcmInc32(&counter)

	var expectedTag [gcmTagSize]byte
	g.auth(expectedTag[:], ciphertext, data, &tagMask)

	ret, out := sliceForAppend(dst, len(ciphertext))

	if subtle.ConstantTimeCompare(expectedTag[:], tag) != 1 {
	// The AESNI code decrypts and authenticates concurrently, and
	// so overwrites dst in the event of a tag mismatch. That
	// behavior is mimicked here in order to be consistent across
	// platforms.
	for i := range out {
	out[i] = 0
	}
	return nil, errOpen
	}

	g.counterCrypt(out, ciphertext, &counter)

	return ret, nil
	}

	// reverseBits reverses the order of the bits of 4-bit number in i.
	func reverseBits(i int) int {
	i = ((i << 2) & 0xc) \| ((i >> 2) & 0x3)
	i = ((i << 1) & 0xa) \| ((i >> 1) & 0x5)
	return i
	}

	// gcmAdd adds two elements of GF(2¹²⁸) and returns the sum.
	func gcmAdd(x, y *gcmFieldElement) gcmFieldElement {
	// Addition in a characteristic 2 field is just XOR.
	return gcmFieldElement{x.low ^ y.low, x.high ^ y.high}
	}

	// gcmDouble returns the result of doubling an element of GF(2¹²⁸).
	func gcmDouble(x *gcmFieldElement) (double gcmFieldElement) {
	msbSet := x.high&1 == 1

	// Because of the bit-ordering, doubling is actually a right shift.
	double.high = x.high >> 1
	double.high \|= x.low << 63
	double.low = x.low >> 1

	// If the most-significant bit was set before shifting then it,
	// conceptually, becomes a term of x^128. This is greater than the
	// irreducible polynomial so the result has to be reduced. The
	// irreducible polynomial is 1+x+x^2+x^7+x^128. We can subtract that to
	// eliminate the term at x^128 which also means subtracting the other
	// four terms. In characteristic 2 fields, subtraction == addition ==
	// XOR.
	if msbSet {
	double.low ^= 0xe100000000000000
	}

	return
	}

	var gcmReductionTable = []uint16{
	0x0000, 0x1c20, 0x3840, 0x2460, 0x7080, 0x6ca0, 0x48c0, 0x54e0,
	0xe100, 0xfd20, 0xd940, 0xc560, 0x9180, 0x8da0, 0xa9c0, 0xb5e0,
	}

	// mul sets y to y*H, where H is the GCM key, fixed during NewGCMWithNonceSize.
	func (g gcm) mul(y gcmFieldElement) {
	var z gcmFieldElement

	for i := 0; i < 2; i++ {
	word := y.high
	if i == 1 {
	word = y.low
	}

	// Multiplication works by multiplying z by 16 and adding in
	// one of the precomputed multiples of H.
	for j := 0; j < 64; j += 4 {
	msw := z.high & 0xf
	z.high >>= 4
	z.high \|= z.low << 60
	z.low >>= 4
	z.low ^= uint64(gcmReductionTable[msw]) << 48

	// the values in \|table\| are ordered for
	// little-endian bit positions. See the comment
	// in NewGCMWithNonceSize.
	t := &g.productTable[word&0xf]

	z.low ^= t.low
	z.high ^= t.high
	word >>= 4
	}
	}

	*y = z
	}

	// updateBlocks extends y with more polynomial terms from blocks, based on
	// Horner's rule. There must be a multiple of gcmBlockSize bytes in blocks.
	func (g gcm) updateBlocks(y gcmFieldElement, blocks []byte) {
	for len(blocks) > 0 {
	y.low ^= getUint64(blocks)
	y.high ^= getUint64(blocks[8:])
	g.mul(y)
	blocks = blocks[gcmBlockSize:]
	}
	}

	// update extends y with more polynomial terms from data. If data is not a
	// multiple of gcmBlockSize bytes long then the remainder is zero padded.
	func (g gcm) update(y gcmFieldElement, data []byte) {
	fullBlocks := (len(data) >> 4) << 4
	g.updateBlocks(y, data[:fullBlocks])

	if len(data) != fullBlocks {
	var partialBlock [gcmBlockSize]byte
	copy(partialBlock[:], data[fullBlocks:])
	g.updateBlocks(y, partialBlock[:])
	}
	}

	// gcmInc32 treats the final four bytes of counterBlock as a big-endian value
	// and increments it.
	func gcmInc32(counterBlock *[16]byte) {
	for i := gcmBlockSize - 1; i >= gcmBlockSize-4; i-- {
	counterBlock[i]++
	if counterBlock[i] != 0 {
	break
	}
	}
	}

	// sliceForAppend takes a slice and a requested number of bytes. It returns a
	// slice with the contents of the given slice followed by that many bytes and a
	// second slice that aliases into it and contains only the extra bytes. If the
	// original slice has sufficient capacity then no allocation is performed.
	func sliceForAppend(in []byte, n int) (head, tail []byte) {
	if total := len(in) + n; cap(in) >= total {
	head = in[:total]
	} else {
	head = make([]byte, total)
	copy(head, in)
	}
	tail = head[len(in):]
	return
	}

	// counterCrypt crypts in to out using g.cipher in counter mode.
	func (g gcm) counterCrypt(out, in []byte, counter [gcmBlockSize]byte) {
	var mask [gcmBlockSize]byte

	for len(in) >= gcmBlockSize {
	g.cipher.Encrypt(mask[:], counter[:])
	gcmInc32(counter)

	xorWords(out, in, mask[:])
	out = out[gcmBlockSize:]
	in = in[gcmBlockSize:]
	}

	if len(in) > 0 {
	g.cipher.Encrypt(mask[:], counter[:])
	gcmInc32(counter)
	xorBytes(out, in, mask[:])
	}
	}

	// deriveCounter computes the initial GCM counter state from the given nonce.
	// See NIST SP 800-38D, section 7.1. This assumes that counter is filled with
	// zeros on entry.
	func (g gcm) deriveCounter(counter [gcmBlockSize]byte, nonce []byte) {
	// GCM has two modes of operation with respect to the initial counter
	// state: a "fast path" for 96-bit (12-byte) nonces, and a "slow path"
	// for nonces of other lengths. For a 96-bit nonce, the nonce, along
	// with a four-byte big-endian counter starting at one, is used
	// directly as the starting counter. For other nonce sizes, the counter
	// is computed by passing it through the GHASH function.
	if len(nonce) == gcmStandardNonceSize {
	copy(counter[:], nonce)
	counter[gcmBlockSize-1] = 1
	} else {
	var y gcmFieldElement
	g.update(&y, nonce)
	y.high ^= uint64(len(nonce)) * 8
	g.mul(&y)
	putUint64(counter[:8], y.low)
	putUint64(counter[8:], y.high)
	}
	}

	// auth calculates GHASH(ciphertext, additionalData), masks the result with
	// tagMask and writes the result to out.
	func (g gcm) auth(out, ciphertext, additionalData []byte, tagMask [gcmTagSize]byte) {
	var y gcmFieldElement
	g.update(&y, additionalData)
	g.update(&y, ciphertext)

	y.low ^= uint64(len(additionalData)) * 8
	y.high ^= uint64(len(ciphertext)) * 8

	g.mul(&y)

	putUint64(out, y.low)
	putUint64(out[8:], y.high)

	xorWords(out, out, tagMask[:])
	}

	func getUint64(data []byte) uint64 {
	r := uint64(data[0])<<56 \|
	uint64(data[1])<<48 \|
	uint64(data[2])<<40 \|
	uint64(data[3])<<32 \|
	uint64(data[4])<<24 \|
	uint64(data[5])<<16 \|
	uint64(data[6])<<8 \|
	uint64(data[7])
	return r
	}

	func putUint64(out []byte, v uint64) {
	out[0] = byte(v >> 56)
	out[1] = byte(v >> 48)
	out[2] = byte(v >> 40)
	out[3] = byte(v >> 32)
	out[4] = byte(v >> 24)
	out[5] = byte(v >> 16)
	out[6] = byte(v >> 8)
	out[7] = byte(v)
	}