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Versions: 00 01 02 03 04 05 06 07 rfc4418                               
CFRG Working Group                                    T. Krovetz, Editor
INTERNET-DRAFT                                            CSU Sacramento
Expires April 2005                                          October 2004

       UMAC: Message Authentication Code using Universal Hashing
                      <draft-krovetz-umac-02.txt>

   By submitting this Internet-Draft, we certify that any applicable
   patent or other IPR claims of which we are aware have been disclosed,
   or will be disclosed, and any of which we become aware will be
   disclosed, in accordance with RFC 3668

Status of this Memo

   This document is an Internet-Draft and is in full conformance with
   all provisions of Section 10 of RFC2026.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF), its areas, and its working groups.  Note that
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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,
   AES, 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  KDF: Key-derivation function . . . . . . . . . . . . . . . .   7
   3.2  PDF: Pad-derivation function . . . . . . . . . . . . . . . .   8
4  UMAC tag generation . . . . . . . . . . . . . . . . . . . . . . .   9
   4.1  UMAC Algorithm . . . . . . . . . . . . . . . . . . . . . . .   9
   4.2  UMAC-32, UMAC-64 and UMAC-96 . . . . . . . . . . . . . . . .   9
5  UHASH: Universal hash function  . . . . . . . . . . . . . . . . .  10
   5.1  L1-HASH: First-layer hash  . . . . . . . . . . . . . . . . .  11
   5.2  L2-HASH: Second-layer hash . . . . . . . . . . . . . . . . .  13
   5.3  L3-HASH: Third-layer hash  . . . . . . . . . . . . . . . . .  15
6  Security considerations . . . . . . . . . . . . . . . . . . . . .  16
   6.1  Resistance to cryptanalysis  . . . . . . . . . . . . . . . .  16
   6.2  Tag lengths and forging probability  . . . . . . . . . . . .  17
   6.3  Nonce considerations . . . . . . . . . . . . . . . . . . . .  18
   6.4  Guarding against replay attacks  . . . . . . . . . . . . . .  19
7  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .  20
Appendix - Test vectors  . . . . . . . . . . . . . . . . . . . . . .  20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21
Author contact information . . . . . . . . . . . . . . . . . . . . .  21























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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- or 96-bit tags, depending on the user's preference,
   with 64 bits recommended for most applications.  When UMAC produces
   32-, 64- or 96-bit tags, the probability that an attacker could
   produce a correct tag for any message of its choosing is about
   1/2^30, 1/2^60 or 1/2^90, respectively.  These probabilities remains
   the same for each new forgery attempt by the attacker.  Our analysis
   has shown that doing any better would imply that an effective attack
   exists for the Advanced Encryption Standard (AES).  Hence, assuming
   AES is strong, so is UMAC.  Security analysis of UMAC is in [2, 5].

   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-bit tags take about one-and-a-half times as long.

   UMAC is a MAC in the style of Wegman and Carter [3, 6].  A fast
   "universal" hash function is used to hash an input message into a
   short string.  This short string is then encrypted 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 a keyed
   hash function h and pseudorandom function f.  A tag is thus generated
   by performing the computation

     tag = f_k(nonce) xor h_k(message)

   where k is a secret key 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
   agreeing upon the use of some other non-repeating value such as



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   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 needing to be 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 on registers by modern
   computers.


2.5  ENDIAN-SWAP: Adjusting endian orientation

   Message data is read little-endian to speed tag generation on little-
   endian computers.  On little-endian processors, this is a free
   operation.


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



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          so that W_1 || W_2 || W_3 || W_4 = S_i
       SReversed_i = W_4 || W_3 || W_2 || W_1
       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 following two functions use a block cipher to
   generate these bits.  All references to AES refer to the 128-bit key
   encryption function of the Advanced Encryption Standard (AES) [1].


3.1  KDF: Key-derivation function

   The key-derivation function generates pseudorandom bits used to key
   the hash functions.


3.1.1  KDF Algorithm

   Input:
     K, string of length 16 bytes  // key to AES
     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 AES iterations, set indexed starting point
     //
     n = ceil(numbytes / 16)
     B = uint2str((2 * index + 1)^2 + index, 1) xor uint2str(90, 1)
     T = B repeated 16 times
     Y = <empty string>

     //
     // Build Y using AES in a feedback mode
     //
     for i = 1 to n do
       T = T[1..15] || uint2str(i mod 256, 1)
       T = AES(K, T)



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       Y = Y || T
     end for

     Y = Y[1..numbytes]

     Return Y


3.2  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- or 12-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 AES
   encryption.  This allows caching and sharing a single AES invocation
   for sequential nonces.


3.2.1  PDF Algorithm

   Input:
     K, string of length 16 bytes
     Nonce, string of length 1 to 16 bytes.
     padlen, the integer 4, 8 or 12.
   Output:
     Y, string of length padlen bytes.


   Compute Y using the following algorithm.

      //
      // Extract and zero low bit(s) of Nonce if needed
      //
      if (padlen = 4)
        index = str2uint(Nonce) mod 4
        Nonce = Nonce xor uint2str(index, bytelength(Nonce))
      else if (padlen = 8)
        index = str2uint(Nonce) mod 2
        Nonce = Nonce xor uint2str(index, bytelength(Nonce))
      end if

      //
      // Make Nonce 16 bytes by appending zeroes if needed
      //
      Nonce = Nonce || zeroes(16 - bytelength(Nonce))

      //



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      // Generate subkey, AES and extract indexed substring
      //
      K' = KDF(K, 0, 16)
      T = AES(K', Nonce)
      if (padlength = 4)
        Y = T[1 + (index*4) .. 4 + (index*4)]
      else if (padlength = 8)
        Y = T[1 + (index*8) .. 8 + (index*8)]
      else
        Y = T[1..padlen]
      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 or 12 bytes.


4.1  UMAC Algorithm

   Input:
     K, string of length 16 bytes.
     M, string of length less than 2^67 bits.
     Nonce, string of length 1 to 16 bytes.
     taglen, the integer 4, 8 or 12.
   Output:
     AuthTag, string of length taglen bytes.


   Compute AuthTag using the following algorithm.

     HashedMessage = UHASH(K, M, taglen)
     Pad           = PDF(K, Nonce, taglen)
     AuthTag       = Pad xor HashedMessage

     Return AuthTag


4.2  UMAC-32, UMAC-64 and UMAC-96

   The preceding definition for UMAC has an input parameter "taglen"
   which specifies the length of tag generated by the algorithm.  The
   following aliases define names that make tag length explicit in the



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   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)


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- or 12-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.

   UHASH is presented here in a top-down manner.  First UHASH is
   described, then each of its component hashes are presented.


5.0.1  UHASH Algorithm

   Input:
     K, string of length 16 bytes.
     M, string of length less than 2^67 bits.
     outlen, the integer 4, 8 or 12.
   Output:
     Y, string of length outlen bytes.


   Compute Y using the following algorithm.

     //
     // One internal iteration per 4 bytes of output
     //



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     iters = outlen / 4

     //
     // Define total key needed for all iterations using KDF.
     // L1Key and L3Key1 both reuse most key 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)
       Y = Y || C
     end for

     Return Y


5.1  L1-HASH: First-layer hash

   The first-layer hash breaks the message into 1024 byte chunks and
   hashes each with a function called NH.  The concatenation of these
   hash values results in a string up to 128 times shorter than the
   original.


5.1.1  L1-HASH Algorithm

   Input:
     K, string of length 1024 bytes.
     M, string of length less than 2^67 bits.



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   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)       // See endian discussion in section 3.1.1
       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)
     ENDIAN-SWAP(M_t)
     Y = Y || (NH(K, M_t) +_64 Len)

     return Y


5.1.2  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.




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   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


5.2  L2-HASH: Second-layer hash

   The second-layer rehashes the L1-HASH output using a polynomial hash
   called POLY.  If the output of L1-HASH is long, then POLY is called
   once on a prefix of the L1-HASH output and then called using
   different settings on the remainder.  (This two-step hashing of the
   L1-HASH output is only needed if the initial message length is
   greater than 16 megabytes.)


5.2.1  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.



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     //
     //  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)
      end if

     Y = uint2str(y, 16)

     Return Y


5.2.2  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



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     //
     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
     end for

     Return y


5.3  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.


5.3.1  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.



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     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


6  Security considerations

   As a specification of a message authentication code, 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 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
   any 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 [2, 5].  In this way,
   the strength of UHASH is guaranteed regardless of future advances in
   cryptanalysis.  The UHASH function was not designed to provide



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   cryptographic collision resistance properties, as does SHA-1, and so
   should not be used as a substitute for it.

   The analysis of UMAC [2, 5] 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
   which forges with probability significantly exceeding the established
   collision probability will give rise to an attack of comparable
   complexity which breaks the AES, in the sense of distinguishing AES
   from a family of random permutations.  This design approach
   essentially obviates the need for cryptanalysis on UMAC:
   cryptanalytic efforts might as well focus on AES, 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
   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.

   UMAC is designed to make this guessing strategy the best possible
   attack against UMAC as long as the attacker does not invest the
   computational effort needed to break the underlying cipher, AES, used
   to produce the one time pads used in UMAC computation.  More
   precisely, under the assumed strength of this cipher UMAC provides
   for close-to-optimal security with regards to forgery probability.
   An adversary could guess an 8-byte UMAC tag correctly with
   probability 1/2^64 by simply guessing a random value.  The proofs of
   [2, 5] show that an adversary being more clever in tag guessing can
   do no better than about 1/2^60.

   This means that for 8-byte tags the ideal forging probability is
   2^-64 while UMAC produces an actual forging probability of at most
   2^-60.  This probability of forging a message is well under the
   chance that a randomly guessed DES key is correct.  DES is now widely
   seen as vulnerable, but the problem has never been that some
   extraordinarily lucky attacker might, in a single guess, find the
   right key.  Instead, the problem is that large amounts of computation



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   can be thrown at the problem until, off-line, the attacker finds the
   right key.

   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 amount 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 AES maintains its believed security there is no such attack
   for UMAC.  Instead, a 1 / 2^60 forging probability means that if an
   attacker could try out 2^60 MACs, then the attacker would probably
   get one right.

   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.

   In conclusion, 64-bit tags seem appropriate for most security
   architectures and applications.  If one wants more security, at a
   cost of 50% more computation, UMAC can produce 96-bit tags which
   cannot be forged with probability better than 1/2^90.  Likewise, if
   less security is required, with the benefit of 50% less computation,
   UMAC can produce 32-bit tags which cannot be forged with probability
   better than 1/2^30.  Great 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-value data or rapidly-changing key environments.


6.3  Nonce considerations

   UMAC requires a nonce with length in the range 1 to 16 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



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   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 exemplify some of the
   possibilities:

   1.  The nonce is an eight-byte [resp., four-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 [resp., 2^32] messages
       within a session.

   2.  The nonce is a 16-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 16-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.

   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  Guarding against 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



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   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.


7  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.

   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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References

Normative References

   [1]   FIPS-197, "Advanced Encryption Standard (AES)", National
         Institute of Standards and Technology, 2001.

Informative References

   [2]   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.

   [3]   L. Carter and M. Wegman, "Universal classes of hash functions",
         Journal of Computer and System Sciences, 18 (1979), pp.
         143-154.

   [4] S. Kent and R. Atkinson, "IP Encapsulating Security Payload
         (ESP)", RFC 2406, IETF, 1998.

   [5]  T. Krovetz, "Software-optimized universal hashing and message
         authentication", UMI Dissertation Services, 2000.

   [6]  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

     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



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     Alejandro Hevia
     Department of Computer Science & Engineering
     University of California at San Diego
     La Jolla CA 92093
     USA

     EMail: ahevia@cs.ucsd.edu

     Hugo Krawczyk
     Department of Electrical Engineering
     Technion
     Haifa 32000
     ISRAEL

     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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   except as set forth therein, the authors retain all their rights.

   This document and the information contained herein are provided on an
   "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
   OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET



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   ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
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