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Versions: 00 01 02 03 04 05 06 07 RFC 4418

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>

   By submitting this Internet-Draft, each author represents that any
   applicable patent or other IPR claims of which he or she is aware
   have been or will be disclosed, and any of which he or she becomes
   aware will be disclosed, in accordance with Section 6 of BCP 79.

Status of this Memo

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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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Full Copyright Statement

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Acknowledgment

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