CFRG Working Group T. Krovetz, Editor
INTERNET-DRAFT CSU Sacramento
Expires March 2006 September 2005
UMAC: Message Authentication Code using Universal Hashing
<draft-krovetz-umac-05.txt>
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Abstract
This specification describes how to generate an authentication tag
using the UMAC message authentication algorithm. UMAC is designed to
be very fast to compute in software on contemporary uniprocessors.
Measured speeds are as low as one cycle per byte. UMAC relies on
addition of 32-bit and 64-bit numbers and multiplication of 32-bit
numbers, operations well-supported by contemporary machines.
To generate the authentication tag on a given message, a "universal"
hash function is applied to the message and key to produce a short,
fixed-length hash value, and this hash value is then xor'ed with a
key-derived pseudorandom pad. UMAC enjoys a rigorous security
analysis and its only internal "cryptographic" use is a block cipher
to generate the pseudorandom pads and internal key material.
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Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Notation and basic operations . . . . . . . . . . . . . . . . . . 4
2.1 Operations on strings . . . . . . . . . . . . . . . . . . . 4
2.2 Operations on integers . . . . . . . . . . . . . . . . . . . 5
2.3 String-Integer conversion operations . . . . . . . . . . . . 5
2.4 Mathematical operations on strings . . . . . . . . . . . . . 6
2.5 ENDIAN-SWAP: Adjusting endian orientation . . . . . . . . . 6
3 Key and pad derivation functions . . . . . . . . . . . . . . . . 7
3.1 Block cipher choice . . . . . . . . . . . . . . . . . . . . 7
3.2 KDF: Key-derivation function . . . . . . . . . . . . . . . . 7
3.3 PDF: Pad-derivation function . . . . . . . . . . . . . . . . 8
4 UMAC tag generation . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 UMAC Algorithm . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 UMAC-32, UMAC-64, UMAC-96 and UMAC-128 . . . . . . . . . . . 10
5 UHASH: Universal hash function . . . . . . . . . . . . . . . . . 10
5.1 UHASH Algorithm . . . . . . . . . . . . . . . . . . . . . . 11
5.2 L1-HASH: First-layer hash . . . . . . . . . . . . . . . . . 12
5.3 L2-HASH: Second-layer hash . . . . . . . . . . . . . . . . . 14
5.4 L3-HASH: Third-layer hash . . . . . . . . . . . . . . . . . 16
6 Security considerations . . . . . . . . . . . . . . . . . . . . . 17
6.1 Resistance to cryptanalysis . . . . . . . . . . . . . . . . 17
6.2 Tag lengths and forging probability . . . . . . . . . . . . 17
6.3 Nonce considerations . . . . . . . . . . . . . . . . . . . . 19
6.4 Replay attacks . . . . . . . . . . . . . . . . . . . . . . . 20
6.5 Tag-prefix verification . . . . . . . . . . . . . . . . . . 20
6.6 Side-channel attacks . . . . . . . . . . . . . . . . . . . . 20
7 IANA Considerations . . . . . . . . . . . . . . . . . . . . . . . 21
8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendix - Test vectors . . . . . . . . . . . . . . . . . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Author contact information . . . . . . . . . . . . . . . . . . . . . 23
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1 Introduction
UMAC is a message authentication algorithm (MAC) designed for high
performance. It has been rigorously proven to be secure and there
are no intellectual property claims made to any ideas used in its
design.
The output of UMAC is a string called a tag. UMAC is designed to
produce 32-, 64-, 96- or 128-bit tags, depending on the user's
preference, with 64 bits recommended for most applications. When
UMAC produces 32-, 64-, 96- or 128-bit tags, the probability that an
attacker can produce a correct tag for any message of its choosing is
no more than 1/2^30, 1/2^60, 1/2^90 or 1/2^120, respectively (the
symbol ^ represents exponentiation). This probability increases
linearly with the number of forgery attempts, namely, during N
forgery attempts the probability of getting at least one tag right is
at most N times the above numbers. See Section 6.2 for further
information. Analysis relevant to UMAC security is in [3, 6].
UMAC performs best in environments where 32-bit quantities are
efficiently multiplied into 64-bit results. In producing 64-bit tags
on an Intel Pentium 4 using SSE2 instructions, which do two of these
multiplications in parallel, UMAC processes messages at a peak rate
of about one CPU cycle per byte, with the peak being achieved on
messages of around four kilobytes and longer. On the Pentium III,
without the use of SSE parallelism, UMAC achieves a peak of two
cycles per byte. On shorter messages UMAC still performs well:
around four cycles per byte on 256 byte messages and under two cycles
per byte on 1500 byte messages. The time to produce a 32-bit tag is
a little more than half that needed to produce a 64-bit tag, while
96- and 128-bit tags take one-and-a-half and twice as long.
UMAC is a MAC in the style of Wegman and Carter [4, 7]. A fast
"universal" hash function is used to hash an input message M into a
short string. This short string is then masked by xor'ing with a
pseudorandom string, resulting in the UMAC tag. Security depends on
the sender and receiver sharing a randomly-chosen secret hash
function and pseudorandom string. This is achieved by using keyed
hash function H and pseudorandom function F. A tag is generated by
performing the computation
Tag = H_K1(M) xor F_K2(Nonce)
where K1 and K2 are secret keys shared by sender and receiver, and
Nonce is a value that changes with each generated tag. The receiver
needs to know which nonce was used by the sender, so some method of
synchronizing nonces needs to be used. This can be done by
explicitly sending the nonce along with the message and tag, or
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agreeing upon the use of some other non-repeating value such as
message number. The nonce need not be kept secret, but care needs to
be taken to ensure that, over the lifetime of the UMAC key, a
different nonce is used with each message.
Optimized source code, performance data and papers concerning UMAC
can be found at http://www.cs.ucdavis.edu/~rogaway/umac/.
2 Notation and basic operations
The specification of UMAC involves the manipulation of both strings
and numbers. String variables are denoted with an initial upper-case
letter, whereas numeric variables are denoted in all lower case. The
algorithms of UMAC are denoted in all upper-case letters. Simple
functions, like those for string-length and string-xor, are written
in all lower case.
Whenever a variable is followed by an underscore ("_"), the
underscore is intended to denote a subscript, with the subscripted
expression evaluated to resolve the meaning of the variable. For
example, if i=2, then M_{2 * i} refers to the variable M_4.
2.1 Operations on strings
Messages to be hashed are viewed as strings of bits which get zero-
padded to an appropriate byte-length. Once the message is padded,
all strings are viewed as strings of bytes. A "byte" is an 8-bit
string. The following notation is used to manipulate these strings.
bytelength(S): The length of string S in bytes.
bitlength(S): The length of string S in bits.
zeroes(n): The string made of n zero-bytes.
S xor T: The string which is the bitwise exclusive-or of S
and T. Strings S and T always have the same length.
S and T: The string which is the bitwise conjunction of S and
T. Strings S and T always have the same length.
S[i]: The i-th byte of the string S (indices begin at 1).
S[i...j]: The substring of S consisting of bytes i through j.
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S || T: The string S concatenated with string T.
zeropad(S,n): The string S, padded with zero-bits to the nearest
positive multiple of n bytes. Formally,
zeropad(S,n) = S || T, where T is the shortest
string of zero-bits (possibly empty) so that S || T
is non-empty and 8n divides bitlength(S || T).
2.2 Operations on integers
Standard notation is used for most mathematical operations, such as
"*" for multiplication, "+" for addition and "mod" for modular
reduction. Some less standard notations are defined here.
a^i: The integer a raised to the i-th power.
ceil(x): The smallest integer greater than or equal to x.
prime(n): The largest prime number less than 2^n.
The prime numbers used in UMAC are:
+-----+--------------------+---------------------------------------+
| n | prime(n) [Decimal] | prime(n) [Hexadecimal] |
+-----+--------------------+---------------------------------------+
| 36 | 2^36 - 5 | 0x0000000F FFFFFFFB |
| 64 | 2^64 - 59 | 0xFFFFFFFF FFFFFFC5 |
| 128 | 2^128 - 159 | 0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFF61 |
+-----+--------------------+---------------------------------------+
2.3 String-Integer conversion operations
Conversion between strings and integers is done using the following
functions. Each function treats initial bits as more significant
than later ones.
bit(S,n): Returns the integer 1 if the n-th bit of the string
S is 1, otherwise returns the integer 0 (indices
begin at 1).
str2uint(S): The non-negative integer whose binary representation
is the string S. More formally, if S is t bits long
then str2uint(S) = 2^{t-1} * bit(S,1) + 2^{t-2} *
bit(S,2) + ... + 2^{1} * bit(S,t-1) + bit(S,t).
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uint2str(n,i): The i-byte string S such that str2uint(S) = n.
2.4 Mathematical operations on strings
One of the primary operations in UMAC is repeated application of
addition and multiplication on strings. The operations "+_32",
"+_64" and "*_64" are defined
"S +_32 T" as uint2str(str2uint(S) + str2uint(T) mod 2^32, 4),
"S +_64 T" as uint2str(str2uint(S) + str2uint(T) mod 2^64, 8) and
"S *_64 T" as uint2str(str2uint(S) * str2uint(T) mod 2^64, 8).
These operations correspond well with the addition and multiplication
operations which are performed efficiently by modern computers.
2.5 ENDIAN-SWAP: Adjusting endian orientation
Message data is read little-endian to speed tag generation on little-
endian computers.
2.5.1 ENDIAN-SWAP Algorithm
Input:
S, string with length divisible by 4 bytes.
Output:
T, string S with each 4-byte word endian-reversed.
Compute T using the following algorithm.
//
// Break S into 4-byte chunks
//
n = bytelength(S) / 4
Let S_1, S_2, ..., S_n be strings of length 4 bytes
so that S_1 || S_2 || ... || S_n = S.
//
// Byte-reverse each chunk, and build-up T
//
T = <empty string>
for i = 1 to n do
Let W_1, W_2, W_3, W_4 be bytes
so that W_1 || W_2 || W_3 || W_4 = S_i
SReversed_i = W_4 || W_3 || W_2 || W_1
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T = T || SReversed_i
end for
Return T
3 Key and pad derivation functions
Pseudorandom bits are needed internally by UHASH and at the time of
tag generation. The functions listed in this section use a block
cipher to generate these bits.
3.1 Block cipher choice
UMAC uses the services of a block cipher. The selection of a block
cipher defines the following constants and functions.
BLOCKLEN The length, in bytes, of the plaintext block on
which the block cipher operates.
KEYLEN The block cipher's key length, in bytes.
ENCIPHER(K,P) The application of the block cipher on P (a string
of BLOCKLEN bytes) using key K (a string of KEYLEN
bytes).
As an example, if AES is used with 16-byte keys, then BLOCKLEN would
equal 16 (because AES employs 16-byte blocks), KEYLEN would equal
16, and ENCIPHER would refer to the AES function.
Unless specified otherwise, AES with 128-bit keys shall be assumed to
be the chosen block cipher for UMAC. Only if explicitly specified
otherwise, and agreed by communicating parties, shall some other
block cipher be used. In any case, BLOCKLEN must be at least 16 and
a power of two.
AES is defined in another document [1].
3.2 KDF: Key-derivation function
The key-derivation function generates pseudorandom bits used to key
the hash functions.
3.2.1 KDF Algorithm
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Input:
K, string of length KEYLEN bytes // block cipher key
index, an integer in the range 0...7.
numbytes, a positive integer.
Output:
Y, string of length numbytes bytes.
Compute Y using the following algorithm.
//
// Calculate number of block cipher iterations, set starting point
//
n = ceil(numbytes / BLOCKLEN)
B = uint2str((2 * index + 1)^2 + index, 1) xor uint2str(90, 1)
T = B repeated BLOCKLEN times
Y = <empty string>
//
// Build Y using block cipher in a feedback mode
//
for i = 1 to n do
T = T[1...(BLOCKLEN - 1)] || uint2str(i mod 256, 1)
T = ENCIPHER(K, T)
Y = Y || T
end for
Y = Y[1...numbytes]
Return Y
3.3 PDF: Pad-derivation function
This function takes a key and a nonce and returns a pseudorandom pad
for use in tag generation. A pad of length 4, 8, 12 or 16 bytes can
be generated. Notice that pads generated using nonces that differ
only in their last bit (when generating 8-byte pads) or last two bits
(when generating 4-byte pads) are derived from the same block cipher
encryption. This allows caching and sharing a single block cipher
invocation for sequential nonces.
3.3.1 PDF Algorithm
Input:
K, string of length KEYLEN bytes
Nonce, string of length 1 to BLOCKLEN bytes.
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taglen, the integer 4, 8, 12 or 16.
Output:
Y, string of length taglen bytes.
Compute Y using the following algorithm.
//
// Extract and zero low bit(s) of Nonce if needed
//
if (taglen = 4 or taglen = 8)
index = str2uint(Nonce) mod (BLOCKLEN/taglen)
Nonce = Nonce xor uint2str(index, bytelength(Nonce))
end if
//
// Make Nonce BLOCKLEN bytes by appending zeroes if needed
//
Nonce = Nonce || zeroes(BLOCKLEN - bytelength(Nonce))
//
// Generate subkey, encipher and extract indexed substring
//
K' = KDF(K, 0, KEYLEN)
T = ENCIPHER(K', Nonce)
if (taglen = 4 or taglen = 8)
Y = T[1 + (index*taglen) ... taglen + (index*taglen)]
else
Y = T[1...taglen]
end if
Return Y
4 UMAC tag generation
Tag generation for UMAC proceeds by using UHASH (defined in the next
section) to hash the message, applying the PDF to the nonce and
computing the xor of the resulting strings. The length of the pad
and hash can be either 4, 8, 12 or 16 bytes.
4.1 UMAC Algorithm
Input:
K, string of length KEYLEN bytes.
M, string of length less than 2^67 bits.
Nonce, string of length 1 to BLOCKLEN bytes.
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taglen, the integer 4, 8, 12 or 16.
Output:
Tag, string of length taglen bytes.
Compute Tag using the following algorithm.
HashedMessage = UHASH(K, M, taglen)
Pad = PDF(K, Nonce, taglen)
Tag = Pad xor HashedMessage
Return Tag
4.2 UMAC-32, UMAC-64, UMAC-96 and UMAC-128
The preceding UMAC definition has a parameter "taglen" which
specifies the length of tag generated by the algorithm. The
following aliases define names that make tag length explicit in the
name.
UMAC-32(K, M, Nonce) = UMAC(K, M, Nonce, 4)
UMAC-64(K, M, Nonce) = UMAC(K, M, Nonce, 8)
UMAC-96(K, M, Nonce) = UMAC(K, M, Nonce, 12)
UMAC-128(K, M, Nonce) = UMAC(K, M, Nonce, 16)
5 UHASH: Universal hash function
UHASH is a keyed hash function, which takes as input a string of
arbitrary length, and produces a 4-, 8-, 12- or 16-byte output.
UHASH does its work in three stages, or layers. A message is first
hashed by L1-HASH, its output is then hashed by L2-HASH, whose output
is then hashed by L3-HASH. If the message being hashed is no longer
than 1024 bytes, then L2-HASH is skipped as an optimization. Because
L3-HASH outputs a string whose length is only four bytes long,
multiple iterations of this three-layer hash are used if a total
hash-output longer than four bytes is requested. To reduce memory
use, L1-HASH reuses most of its key material between iterations. A
significant amount of internal key is required for UHASH, but it
remains constant so long as UMAC's key is unchanged. It is the
implementor's choice whether to generate the internal keys each time
a message is hashed, or to cache them between messages.
Please note that UHASH has certain combinatoric properties making it
suitable for Wegman-Carter message authentication. UHASH is not a
cryptographic hash function and is not a suitable general replacement
for functions like SHA-1.
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UHASH is presented here in a top-down manner. First UHASH is
described, then each of its component hashes are presented.
5.1 UHASH Algorithm
Input:
K, string of length KEYLEN bytes.
M, string of length less than 2^67 bits.
taglen, the integer 4, 8, 12 or 16.
Output:
Y, string of length taglen bytes.
Compute Y using the following algorithm.
//
// One internal iteration per 4 bytes of output
//
iters = taglen / 4
//
// Define total key needed for all iterations using KDF.
// L1Key and L3Key1 reuse most key material between iterations.
//
L1Key = KDF(K, 1, 1024 + (iters - 1) * 16)
L2Key = KDF(K, 2, iters * 24)
L3Key1 = KDF(K, 3, iters * 64)
L3Key2 = KDF(K, 4, iters * 4)
//
// For each iteration, extract key and three-layer hash.
// If bytelength(M) <= 1024, then skip L2-HASH.
//
Y = <empty string>
for i = 1 to iters do
L1Key_i = L1Key [(i-1) * 16 + 1 ... (i-1) * 16 + 1024]
L2Key_i = L2Key [(i-1) * 24 + 1 ... i * 24]
L3Key1_i = L3Key1[(i-1) * 64 + 1 ... i * 64]
L3Key2_i = L3Key2[(i-1) * 4 + 1 ... i * 4]
A = L1-HASH(L1Key_i, M)
if (bitlength(M) <= bitlength(L1Key_i)) then
B = zeroes(8) || A
else
B = L2-HASH(L2Key_i, A)
end if
C = L3-HASH(L3Key1_i, L3Key2_i, B)
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Y = Y || C
end for
Return Y
5.2 L1-HASH: First-layer hash
The first-layer hash breaks the message into 1024-byte chunks and
hashes each with a function called NH. Concatenating the results
forms a string which is up to 128 times shorter than the original.
5.2.1 L1-HASH Algorithm
Input:
K, string of length 1024 bytes.
M, string of length less than 2^67 bits.
Output:
Y, string of length (8 * ceil(bytelength(M)/1024)) bytes.
Compute Y using the following algorithm.
//
// Break M into 1024 byte chunks (final chunk may be shorter)
//
t = ceil(bytelength(M) / 1024)
Let M_1, M_2, ..., M_t be strings so that M = M_1 || M_2 || ... ||
M_t, and bytelength(M_i) = 1024 for all 0 < i < t.
//
// For each chunk, except the last: endian-adjust, NH hash
// and add bit-length. Use results to build Y.
//
Len = uint2str(1024 * 8, 8)
Y = <empty string>
for i = 1 to t-1 do
ENDIAN-SWAP(M_i)
Y = Y || (NH(K, M_i) +_64 Len)
end for
//
// For the last chunk: pad to 32-byte boundary, endian-adjust,
// NH hash and add bit-length. Concatenate the result to Y.
//
Len = uint2str(bitlength(M_t), 8)
M_t = zeropad(M_t, 32)
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ENDIAN-SWAP(M_t)
Y = Y || (NH(K, M_t) +_64 Len)
return Y
5.2.2 NH Algorithm
Because this routine is applied directly to every bit of input
data, optimized implementation of it yields great benefit.
Input:
K, string of length 1024 bytes.
M, string with length divisible by 32 bytes.
Output:
Y, string of length 8 bytes.
Compute Y using the following algorithm.
//
// Break M and K into 4-byte chunks
//
t = bytelength(M) / 4
Let M_1, M_2, ..., M_t be 4-byte strings
so that M = M_1 || M_2 || ... || M_t.
Let K_1, K_2, ..., K_t be 4-byte strings
so that K_1 || K_2 || ... || K_t is a prefix of K.
//
// Perform NH hash on the chunks, pairing words for multiplication
// which are 4 apart to accommodate vector-parallelism.
//
Y = zeroes(8)
i = 1
while (i < t) do
Y = Y +_64 ((M_{i+0} +_32 K_{i+0}) *_64 (M_{i+4} +_32 K_{i+4}))
Y = Y +_64 ((M_{i+1} +_32 K_{i+1}) *_64 (M_{i+5} +_32 K_{i+5}))
Y = Y +_64 ((M_{i+2} +_32 K_{i+2}) *_64 (M_{i+6} +_32 K_{i+6}))
Y = Y +_64 ((M_{i+3} +_32 K_{i+3}) *_64 (M_{i+7} +_32 K_{i+7}))
i = i + 8
end while
Return Y
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5.3 L2-HASH: Second-layer hash
The second-layer rehashes the L1-HASH output using a polynomial hash
called POLY. If the L1-HASH output is long, then POLY is called once
on a prefix of the L1-HASH output and called using different settings
on the remainder. (This two-step hashing of the L1-HASH output is
only needed if the message length is greater than 16 megabytes.)
Careful implementation of POLY is necessary to avoid a possible
timing attack (see Section 6.6 for more information).
5.3.1 L2-HASH Algorithm
Input:
K, string of length 24 bytes.
M, string of length less than 2^64 bytes.
Output:
Y, string of length 16 bytes.
Compute y using the following algorithm.
//
// Extract keys and restrict to special key-sets
//
Mask64 = uint2str(0x01ffffff01ffffff, 8)
Mask128 = uint2str(0x01ffffff01ffffff01ffffff01ffffff, 16)
k64 = str2uint(K[1...8] and Mask64)
k128 = str2uint(K[9...24] and Mask128)
//
// If M is no more than 2^17 bytes, hash under 64-bit prime,
// otherwise, hash first 2^17 bytes under 64-bit prime and
// remainder under 128-bit prime.
//
if (bytelength(M) <= 2^17) then // 2^14 64-bit words
//
// View M as an array of 64-bit words, and use POLY modulo
// prime(64) (and with bound 2^64 - 2^32) to hash it.
//
y = POLY(64, 2^64 - 2^32, k64, M)
else
M_1 = M[1...2^17]
M_2 = M[2^17 + 1 ... bytelength(M)]
M_2 = zeropad(M_2 || uint2str(0x80,1), 16)
y = POLY(64, 2^64 - 2^32, k64, M_1)
y = POLY(128, 2^128 - 2^96, k128, uint2str(y, 16) || M_2)
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end if
Y = uint2str(y, 16)
Return Y
5.3.2 POLY Algorithm
Input:
wordbits, The integer 64 or 128.
maxwordrange, positive integer less than 2^wordbits.
k, integer in the range 0 ... prime(wordbits) - 1.
M, string with length divisible by (wordbits / 8) bytes.
Output:
y, integer in the range 0 ... prime(wordbits) - 1.
Compute y using the following algorithm.
//
// Define constants used for fixing out-of-range words
//
wordbytes = wordbits / 8
p = prime(wordbits)
offset = 2^wordbits - p
marker = p - 1
//
// Break M into chunks of length wordbytes bytes
//
n = bytelength(M) / wordbytes
Let M_1, M_2, ..., M_n be strings of length wordbytes bytes
so that M = M_1 || M_2 || ... || M_n
//
// Each input word m is compared with maxwordrange. If not smaller
// then 'marker' and (m - offset), both in range, are hashed.
//
y = 1
for i = 1 to n do
m = str2uint(M_i)
if (m >= maxwordrange) then
y = (k * y + marker) mod p
y = (k * y + (m - offset)) mod p
else
y = (k * y + m) mod p
end if
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end for
Return y
5.4 L3-HASH: Third-layer hash
The output from L2-HASH is 16 bytes long. This final hash function
hashes the 16-byte string to a fixed length of 4 bytes.
5.4.1 L3-HASH Algorithm
Input:
K1, string of length 64 bytes.
K2, string of length 4 bytes.
M, string of length 16 bytes.
Output:
Y, string of length 4 bytes.
Compute Y using the following algorithm.
y = 0
//
// Break M and K1 into 8 chunks and convert to integers
//
for i = 1 to 8 do
M_i = M [(i - 1) * 2 + 1 ... i * 2]
K_i = K1[(i - 1) * 8 + 1 ... i * 8]
m_i = str2uint(M_i)
k_i = str2uint(K_i) mod prime(36)
end for
//
// Inner-product hash, extract last 32 bits and affine-translate
//
y = (m_1 * k_1 + ... + m_8 * k_8) mod prime(36)
y = y mod 2^32
Y = uint2str(y, 4)
Y = Y xor K2
Return Y
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6 Security considerations
As a message authentication code specification, this entire document
is about security. Here we describe some security considerations
important for the proper understanding and use of UMAC.
6.1 Resistance to cryptanalysis
The strength of UMAC depends on the strength of its underlying
cryptographic functions: the key-derivation function (KDF) and the
pad-derivation function (PDF). In this specification both operations
are implemented using a block cipher, presumably the Advanced
Encryption Standard (AES). However, the design of UMAC allows for
the replacement of these components. Indeed, it is straightforward
to use other block ciphers or other cryptographic objects, such as
(properly keyed) SHA-1 or HMAC for the realization of the KDF or PDF.
The core of the UMAC design, the UHASH function, does not depend on
cryptographic assumptions: its strength is specified by a purely
mathematical property stated in terms of collision probability, and
this property is proven in an absolute sense [3, 6]. This means the
strength of UHASH is guaranteed regardless of advances in
cryptanalysis.
The analysis of UMAC [3, 6] shows this scheme to have provable
security, in the sense of modern cryptography, by way of tight
reductions. What this means is that an adversarial attack on UMAC
that forges with probability significantly exceeding the established
collision probability will give rise to an attack of comparable
complexity which breaks the block cipher, in the sense of
distinguishing the block cipher from a family of random permutations.
This design approach essentially obviates the need for cryptanalysis
on UMAC: cryptanalytic efforts might as well focus on the block
cipher, the results imply.
6.2 Tag lengths and forging probability
A MAC algorithm is used between two parties that share a secret MAC
key, K. Messages transmitted between these parties are accompanied
by authentication tags computed using K and a given nonce. Breaking
the MAC means that the attacker is able to generate, on its own, with
no knowledge of the key K, a new message M (i.e. one not previously
transmitted between the legitimate parties) and to compute on M a
correct authentication tag under the key K. This is called a
forgery. Note that if the authentication tag is specified to be of
length t then the attacker can trivially break the MAC with
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probability 1/2^t. For this the attacker can just generate any
message of its choice and try a random tag; obviously, the tag is
correct with probability 1/2^t. By repeated guesses the attacker can
increase linearly its probability of success.
In the case of UMAC the above guessing-attack strategy is close to
optimal. For example, in the case of UMAC-64 an adversary can
correctly guess an 8-byte UMAC tag with probability 1/2^64 by simply
guessing a random value. The results of [3, 6] show that no feasible
attack strategy can produce a correct tag with probability better
than 1/2^60 if UMAC were to use a random function in its work rather
than AES. Another result [2], when combined with [3, 6], shows that
so long as AES is secure as a pseudorandom permutation, it can be
used instead of a random function without losing the 1/2^60 forging
probability, assuming that no more than 2^64 messages are
authenticated. Likewise 32-, 96- and 128-bit tags cannot be forged
with more than 1/2^30, 1/2^90 and 1/2^120 probability.
With UMAC, off-line computation aimed at exceeding the forging
probability is hopeless as long as the underlying cipher is not
broken. The only way to forge is to interact with the entity that
verifies the MAC and to try a huge number of forgeries before one is
likely to succeed. The system architecture will determine the extent
to which this is possible. In a well-architected system there should
not be any high-bandwidth capability for presenting forged MACs and
determining if they are valid. In particular, the number of
authentication failures at the verifying party should be limited. If
a large number of such attempts are detected the session key in use
should be dropped and the event be recorded in an audit log.
Let us reemphasize: a forging probability of 1/2^60 does not mean
that there is an attack that runs in 2^60 time; to the contrary, as
long as the block cipher in use maintains its believed security there
is no such attack for UMAC. Instead, a 1/2^60 forging probability
means that if an attacker could have N forgery attempts, then the
attacker would have no more than N/2^60 probability of getting one or
more of them right. In conclusion, 64-bit tags seem appropriate for
most security architectures and applications.
If one wants more security, at a cost of about 50% or 100% more
computation, UMAC can produce 96- or 128-bit tags that have collision
probabilities of at most 1/2^90 and 1/2^120. If one needs less
security, with the benefit of about 50% less computation, UMAC can
produce 32-bit tags. In this case, under the same assumptions as
before, one can not forge a message with probability better than
1/2^30. Special care must be taken when using 32-bit tags because
1/2^30 forgery probability is considered fairly high. Still, high-
speed low-security authentication can be applied usefully on low-
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value data or rapidly-changing key environments.
It should be pointed out that once an attempted forgery is
successful, it is possible, in principle, that subsequent messages
under this key may be forged, too. This is important to understand
in gauging the severity of a successful forgery, even though no such
attack on UMAC is known to date.
6.3 Nonce considerations
UMAC requires a nonce with length in the range 1 to BLOCKLEN bytes.
All nonces in an authentication session must be equal in length. For
secure operation, no nonce value should be repeated within the life
of a single UMAC session-key.
To authenticate messages over a duplex channel (where two parties
send messages to each other), a different key could be used for each
direction. If the same key is used in both directions, then it is
crucial that all nonces be distinct. For example, one party can use
even nonces while the other party uses odd ones. The receiving party
must verify that the sender is using a nonce of the correct form.
This specification does not indicate how nonce values are created,
updated, or communicated between the entity producing a tag and the
entity verifying a tag. The following are possibilities:
1. The nonce is an eight-byte unsigned number, Counter, which is
initialized to zero, which is incremented by one following the
generation of each authentication tag, and which is always
communicated along with the message and the authentication tag.
An error occurs at the sender if there is an attempt to
authenticate more than 2^64 messages within a session.
2. The nonce is a BLOCKLEN-byte unsigned number, Counter, which is
initialized to zero and which is incremented by one following the
generation of each authentication tag. The Counter is not
explicitly communicated between the sender and receiver.
Instead, the two are assumed to communicate over a reliable
transport, and each maintains its own counter so as to keep track
of what the current nonce value is.
3. The nonce is a BLOCKLEN-byte random value. (Because repetitions
in a random n-bit value are expected at around 2^(n/2) trials,
the number of messages to be communicated in a session using n-
bit nonces should not be allowed to approach 2^(n/2).)
We emphasize that the value of the nonce need not be kept secret.
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When UMAC is used within a higher-level protocol there may already be
a field, such as a sequence number, which can be co-opted so as to
specify the nonce needed by UMAC [5]. The application will then
specify how to construct the nonce from this already-existing field.
6.4 Replay attacks
A replay attack entails the attacker repeating a message, nonce, and
authentication tag. In many applications, replay attacks may be
quite damaging and must be prevented. In UMAC, this would normally
be done at the receiver by having the receiver check that no nonce
value is used twice. On a reliable connection, when the nonce is a
counter, this is trivial. On an unreliable connection, when the
nonce is a counter, one would normally cache some window of recent
nonces. Out-of-order message delivery in excess of what the window
allows will result in rejecting otherwise valid authentication tags.
We emphasize that it is up to the receiver when a given (message,
nonce, tag) triple will be deemed authentic. Certainly the tag
should be valid for the message and nonce, as determined by UMAC, but
the message may still be deemed inauthentic because the nonce is
detected to be a replay.
6.5 Tag-prefix verification
UMAC's definition makes it possible to implement tag-prefix
verification; for example, a receiver might verify only the 32-bit
prefix of a 64-bit tag if its computational load is high. Or a
receiver might reject out-of-hand a 64-bit tag whose 32-bit prefix is
incorrect. Such practices are potentially dangerous and can lead to
attacks that reduce the security of the session to the length of the
verified prefix. A UMAC key (or session) must have an associated and
immutable tag length and the implementation should not leak
information that would reveal if a given proper prefix of a tag is
valid or invalid.
6.6 Side-channel attacks
Side-channel attacks have the goal of subverting the security of a
cryptographic system by exploiting its implementation
characteristics. One common side-channel attack is to measure system
response time and derive information regarding conditions met by the
data being processed. Such attacks are known as "timing attacks".
Discussion of timing and other side-channel attacks is outside of
this document's scope. However, we warn that there are places in the
UMAC algorithm where timing information could be unintentionally
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leaked. In particular, the POLY algorithm (Section 5.3.2) tests
whether a value m is out of a particular range, and the behavior of
the algorithm differs depending on the result. If timing attacks are
to be avoided, care should be given to equalize the computation time
in both cases. Timing attacks can also occur for more subtle
reasons, including caching effects.
7 IANA Considerations
This document has no actions for IANA.
8 Acknowledgments
David McGrew and Scott Fluhrer, of Cisco Systems, played a
significant role in improving UMAC by encouraging us to pay more
attention to the performance of short messages. Black, Krovetz, and
Rogaway have received support for this work under NSF awards 0208842,
0240000, 9624560, and a gift from Cisco Systems. Funding for the RFC
Editor function is currently provided by the Internet Society.
Appendix - Test vectors
Following are some sample UMAC outputs over a collection of input
values, using AES with 16-byte keys. Let
K = "abcdefghijklmnop" // A 16-byte UMAC key
N = "bcdefghi" // An 8-byte nonce
The tags generated by UMAC using key K and nonce N are:
Message 32-bit Tag 64-bit Tag 96-bit Tag
------- ---------- ---------- ----------
<empty> EC085847 B9FE492F357C6DF8 3383059D11C13E532BD1E310
'a' * 3 5DA7EE32 0851FF5A9FFA52A0 822CB3E8BB47010BAEC943F8
'a' * 2^10 C8B389F9 9D459891837A7B7D 1738D423A7C728D603BE1725
'a' * 2^15 7B4291BF 2EB480D7EB0EFACA A4C9CC65CFB3A961C5456D6D
'a' * 2^20 A1AB1E5D F45D0F35F15E64D4 7E204387D5E3377F131EF03D
'a' * 2^25 961CA14D C3EAB025C055F3DB 4997FC97E4E8A0709A5842DD
'abc' * 1 CA507696 9FA667FE61D9E4C8 15DB2B4C4564B763303B8E31
'abc' * 500 87347438 D2C26550692E16F1 58BF29E24D93455AE5A05F07
The first column lists a small sample of messages which are strings
of repeated ASCII 'a' bytes or 'abc' strings. The remaining columns
give in hexadecimal the tags generated when UMAC is called with the
corresponding message, nonce N and key K.
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When using key K and producing a 64-bit tag, the following relevant
keys are generated:
Iteration 1 Iteration 2
----------- -----------
NH (Sec 5.2.2)
K_1 6A4CCAC8 45B6482B
K_2 7D284F5D A1D1EEE0
K_3 574E8F49 588794C4
K_4 6CF44825 D7B6EE25
K_5 45B6482B 6496AA93
...
K_256 E0FB2534 D1D8FCA2
L2-HASH (Sec 5.3.1)
k64 019B156C01BC4DB9 01323C7B0131A9AA
L3-HASH (Sec 5.4.1)
k_5 3851BD978 232452052
k_6 9BE0BCC94 2C6BE19F3
k_7 E9AADA709 03F8215DA
k_8 73E9A9BD5 91779998C
K2 C289A7F1 EF65916D
(Note that k_1 ... k_4 are not used in this example because they are
multiplied by zero.)
When generating a 64-bit tag on input "'abc' * 500", the following
intermediate results are produced:
Iteration 1
-----------
L1-HASH 96658AFE85E951B0C7C22E940AC965FC
L2-HASH 0000000000000000FF0048B71C1C3A14
L3-HASH 7CC9A8E1
Iteration 2
-----------
L1-HASH F75648691D2CD179B2AA9169138CA69F
L2-HASH 00000000000000006171E964AA02F73F
L3-HASH EA25DB2B
Concatenating the two L3-HASH results produces a final UHASH result
of 7CC9A8E1EA25DB2B. The pad generated for nonce N is
AE0BCDB1830BCDDA, which when xor'ed with the L3-HASH result yields a
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Tag of D2C26550692E16F1.
References
Normative References
[1] FIPS-197, "Advanced Encryption Standard (AES)", National
Institute of Standards and Technology, 2001.
Informative References
[2] D. Bernstein, "Stronger security bounds for permutations",
unpublished manuscript, 2005. This work refines "Stronger
security bounds for Wegman-Carter-Shoup authenticators",
Advances in Cryptology - EUROCRYPT 2005, LNCS vol. 3494, pp.
164-180, Springer-Verlag, 2005.
[3] J. Black, S. Halevi, H. Krawczyk, T. Krovetz, and P. Rogaway,
"UMAC: Fast and provably secure message authentication",
Advances in Cryptology - CRYPTO '99, LNCS vol. 1666, pp.
216-233, Springer-Verlag, 1999.
[4] L. Carter and M. Wegman, "Universal classes of hash functions",
Journal of Computer and System Sciences, 18 (1979), pp.
143-154.
[5] S. Kent and R. Atkinson, "IP Encapsulating Security Payload
(ESP)", RFC 2406, IETF, 1998.
[6] T. Krovetz, "Software-optimized universal hashing and message
authentication", UMI Dissertation Services, 2000.
[7] M. Wegman and L. Carter, "New hash functions and their use in
authentication and set equality", Journal of Computer and
System Sciences, 22 (1981), pp. 265-279.
Author contact information
Authors' Addresses
John Black
Department of Computer Science
University of Colorado
Boulder CO 80309
USA
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EMail: jrblack@cs.colorado.edu
Shai Halevi
IBM T.J. Watson Research Center
P.O. Box 704
Yorktown Heights NY 10598
USA
EMail: shaih@alum.mit.edu
Alejandro Hevia
Department of Computer Science
University of Chile
Santiago 837-0459
CHILE
EMail: ahevia@dcc.uchile.cl
Hugo Krawczyk
IBM Research
19 Skyline Dr
Hawthorne, NY 10533
USA
EMail: hugo@ee.technion.ac.il
Ted Krovetz
Department of Computer Science
California State University
Sacramento CA 95819
USA
EMail: tdk@acm.org
Phillip Rogaway
Department of Computer Science
University of California
Davis CA 95616
USA
and
Department of Computer Science
Faculty of Science
Chiang Mai University
Chiang Mai 50200
THAILAND
EMail: rogaway@cs.ucdavis.edu
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Krovetz, et al. Expires March 2006 [Page 25]
