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CFRG Working Group                                            T. Krovetz
INTERNET-DRAFT                                            CSU Sacramento
Expires May 2007                                           November 2006

       VMAC: Message Authentication Code using Universal Hashing
                      <draft-krovetz-vmac-00.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 VMAC message authentication algorithm.  VMAC is designed to
   have exceptional performance in software on 64-bit CPU architectures
   while performing well on 32-bit architectures.  Measured speeds are
   as low as one-half CPU cycle per byte on the 64-bit Intel Core 2
   architecture, and under five cycles per byte on recent 32-bit PowerPC
   and Intel processors.

   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.  VMAC tags can be either 64 or 128 bits
   in length and have proven forgery probabilities on the order of
   1/2^59 and 1/2^94, respectively.





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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 . . . . . . . . . . . . .   5
   2.5  ENDIAN-SWAP: Adjusting endian orientation  . . . . . . . . .   6
3  Key and pad derivation functions  . . . . . . . . . . . . . . . .   6
   3.1  Block cipher choice  . . . . . . . . . . . . . . . . . . . .   7
   3.2  KDF: Key-derivation function . . . . . . . . . . . . . . . .   7
   3.3  PDF: Pad-derivation function . . . . . . . . . . . . . . . .   8
4  VMAC tag generation . . . . . . . . . . . . . . . . . . . . . . .   9
   4.1  VMAC Algorithm . . . . . . . . . . . . . . . . . . . . . . .   9
   4.2  VMAC-64 and VMAC-128 . . . . . . . . . . . . . . . . . . . .   9
5  VHASH: Universal hash function  . . . . . . . . . . . . . . . . .  10
   5.1  VHASH Algorithm  . . . . . . . . . . . . . . . . . . . . . .  10
   5.2  L1-HASH: First-layer hash  . . . . . . . . . . . . . . . . .  10
   5.3  L2-HASH: Second-layer hash . . . . . . . . . . . . . . . . .  13
   5.4  L3-HASH: Third-layer hash  . . . . . . . . . . . . . . . . .  13
6  Security considerations . . . . . . . . . . . . . . . . . . . . .  14
   6.1  Resistance to cryptanalysis  . . . . . . . . . . . . . . . .  14
   6.2  Tag lengths and forging probability  . . . . . . . . . . . .  15
   6.3  Nonce considerations . . . . . . . . . . . . . . . . . . . .  16
   6.4  Replay attacks . . . . . . . . . . . . . . . . . . . . . . .  17
7  IANA Considerations . . . . . . . . . . . . . . . . . . . . . . .  18
Appendix - Test vectors  . . . . . . . . . . . . . . . . . . . . . .  18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18
Author contact information . . . . . . . . . . . . . . . . . . . . .  19
Full Copyright Statement . . . . . . . . . . . . . . . . . . . . . .  19
Intellectual Property  . . . . . . . . . . . . . . . . . . . . . . .  20
Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . . . .  20


















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1  Introduction

   VMAC is a message authentication code (MAC) algorithm designed for
   high performance.  It is backed by a rigorous formal analysis, and
   there are no intellectual property claims made by any of the authors
   to any ideas used in its design.

   VMAC is a MAC in the style of Wegman and Carter [4, 6].  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 pad, resulting in the VMAC tag.  Security depends on the
   sender and receiver sharing a randomly-chosen secret hash function
   and pseudorandom pad.  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 random 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
   agreeing upon the use of some other non-repeating value such as a
   sequence number.  The nonce need not be kept secret, but care needs
   to be taken to ensure that, over the lifetime of a VMAC key, a
   different nonce is used with each message.

   VMAC uses a function, called VHASH (also specified in this document),
   as the keyed hash function H and uses a pseudorandom function F whose
   default implementation uses the AES algorithm.  VMAC is designed to
   produce 64- or 128-bit tags, depending on the desired security level.
   The theory of Wegman-Carter MACs and the analysis of VMAC show that
   if one "instantiates" VMAC with truly random keys and pads then the
   probability that an attacker (even a computationally unbounded one)
   produces a correct tag for messages of its choosing upto j bits in
   length is less than 1/2^59 or 1/2^117 when the tags output by VMAC
   are of length 64 or 128 bits, respectively (here the symbol ^
   represents exponentiation).  When an attacker makes N forgery
   attempts the probability of getting one or more tags right increases
   linearly to about N/2^59 or Nj/2^117.  In a real implementation of
   VMAC, using AES to produce keys and pads, the forgery probabilities
   listed above increase by a small amount related to the security of
   AES.  As long as AES is secure this small additive term is
   insignificant for any practical attack.  See Section 6.2 for more
   details.  Analysis relevant to VMAC security is in [5].

   VMAC performs best in environments where 64-bit quantities are



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   efficiently multiplied into 128-bit results.  In producing 64-bit
   tags on an Intel Core 2 CPU, VMAC processes messages at a rate of
   about one-half CPU cycle per byte on messages of two kilobytes.  On a
   32-bit Intel Core CPU, which does not support 64-bit multiplication
   well, VMAC achieves a rate of under five cycles per byte.  On shorter
   messages VMAC still performs well: about two cycles per byte on 64
   byte messages on the Core 2.  Tags of 128 bits require slightly less
   than twice the computation as 64-bit tags.

   Optimized source code, performance data and papers concerning VMAC
   can be found at http://www.fastcrypto.org/vmac.


2  Notation and basic operations

   The specification of VMAC involves the manipulation of 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 VMAC 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.  The following
   notation is used to manipulate these strings.

     bitlength(S): The length of string S in bits.

     zeros(n):     The string made of n zero-bits.

     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[i]:         The i-th bit of the string S (indices begin at 1).

     S[i...j]:     The substring of S consisting of bits i through j.

     S || T:       The string S concatenated with string T.

     zeropad(S,n): The string S, padded with zero-bits to the nearest
                   multiple of n bits in length.  If S is empty or



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                   already a multiple of n in length, nothing is
                   appended.  Formally, zeropad(S,n) = S || T, where T
                   is the shortest string of zero-bits so that
                   bitlength(S || T) is a multiple of n.


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 not less than x.

     prime(n): The largest prime number less than 2^n.

   The prime numbers used in VMAC are:

    +-----+--------------------+---------------------------------------+
    |  n  | prime(n) [Decimal] | prime(n) [Hexadecimal]                |
    +-----+--------------------+---------------------------------------+
    | 61  | 2^61  - 1          | 0x1FFFFFFF FFFFFFFF                   |
    | 127 | 2^127 - 1          | 0x7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF |
    +-----+--------------------+---------------------------------------+


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.

     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} * S[1] + 2^{t-2} * S[2] +
                    ... + 2^{1} * S[t-1] + S[t].

     uint2str(n,i): The i-bit string S so that str2uint(S) = n.


2.4  Mathematical operations on strings

   One of the primary operations in VMAC is addition and multiplication
   of strings.  The operations "+_64", "+_128" and "*_128"  are defined

    "S +_64 T" as uint2str(str2uint(S) + str2uint(T) mod 2^64, 64),



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    "S +_128 T" as uint2str(str2uint(S) + str2uint(T) mod 2^128, 128),
    "S *_128 T" as uint2str(str2uint(S) * str2uint(T) mod 2^128, 128).

   These operations correspond well with 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 bitlength divisible by 64.
   Output:
     T, string S with each 64-bit substring endian-reversed.

   Compute T using the following algorithm.

     //
     // Partition S into 64-bit substrings
     //
     n = bitlength(S) / 64
     Let S_1, S_2, ..., S_n be strings of length 64 bits
        so that S_1 || S_2 || ... || S_n = S.

     //
     // Endian-reverse each, and build-up T
     //
     T = <empty string>
     for i = 1 to n do
       Let W_1, W_2, ..., W_8  be strings of length 8 bits
          so that W_1 || W_2 || ... || W_8 = S_i
       SReversed_i = W_4 || W_3 || ... || W_1
       T = T || SReversed_i
     end for

     Return T


3  Key and pad derivation functions

   Pseudorandom bits are needed internally by VHASH and at the time of
   tag generation.  The functions listed in this section use a block
   cipher to generate these bits.



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3.1  Block cipher choice

   VMAC uses the services of a block cipher.  The selection of a block
   cipher defines the following constants and functions.

     BLOCKLEN      The length, in bits, of the plaintext block on which
                   the block cipher operates.

     KEYLEN        The block cipher's key length, in bits.

     ENCIPHER(K,P) The application of the block cipher on P (a string of
                   BLOCKLEN bits) using key K (a string of KEYLEN bits).

   As an example, if AES is used with 192-bit keys, then BLOCKLEN would
   equal 128 (because AES employs 128-bit blocks), KEYLEN would equal
   192, and ENCIPHER would refer to the AES function for 192-bit AES
   keys.

   Unless specified otherwise, AES with 128-bit keys shall be assumed to
   be the chosen block cipher for VMAC.  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 128.
   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

   Input:
     K, string with bitlength KEYLEN.
     index, an integer in the range 0...255.
     numbits, a non-negative integer.
   Output:
     Y, string with bitlength numbits.

   Compute Y using the following algorithm.

     //
     // Calculate number of block cipher iterations
     //
     n = ceil(numbits / BLOCKLEN)
     Y = <empty string>




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     //
     // Build Y using block cipher in a counter mode
     //
     for i = 0 to (n-1) do
       T = uint2str(index, 8) || uint2str(i, BLOCKLEN-8)
       Y = Y || ENCIPHER(K, T)
     end for
     Y = Y[1...numbits]

     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 64 or 128 bits can be
   generated.  Notice that when the block-cipher block-length is twice
   as long as the pad, nonces that differ only in their last bit 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 with bitlength KEYLEN.
     Nonce, string of length in the range 1...BLOCKLEN bits.
     taglen, the integer 64 or 128.
   Output:
     Y, string of length taglen bits.

   Compute Y using the following algorithm.

      //
      // Extract and zero low bits of Nonce if needed.
      // If BLOCKLEN/taglen < 2, this step does nothing but set index=0
      //
      Let i be the greatest integer for which BLOCKLEN/taglen <= 2^i
      index = str2uint(Nonce) mod 2^i
      Nonce = Nonce xor uint2str(index, bitlength(Nonce))

      //
      // Make Nonce BLOCKLEN bits by appending zeros if needed
      //
      Nonce = Nonce || zeros(BLOCKLEN - bitlength(Nonce))

      //
      // Generate subkey, encipher and extract indexed substring



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      //
      T = ENCIPHER(K, Nonce)
      Y = T[index * taglen + 1 ... index * taglen + taglen ]

      Return Y


4  VMAC tag generation

   Tag generation for VMAC proceeds by using VHASH (defined in the next
   section) to hash the message, applying the PDF to the nonce and
   computing the xor of the resulting strings.  The first bit of the
   nonce must be zero to ensure the KDF and PDF functions do not pass
   the same values to the block cipher.  The length of the pad and hash
   can be either 64 or 128 bits.


4.1  VMAC Algorithm

   Input:
     K, string of length KEYLEN bits.
     M, string of any length.
     Nonce, string of length 1 to BLOCKLEN bits  // first bit MUST be 0.
     taglen, the integer 64 or 128.
   Output:
     Tag, string of length taglen bits.

   Compute Tag using the following algorithm.

     HashedMessage = VHASH(K, M, taglen)
     Pad           = PDF(K, Nonce, taglen)
     if taglen = 64 then
       Tag         = Pad +_64  HashedMessage
     else
       Tag         = Pad +_128 HashedMessage
     end if

     Return Tag


4.2  VMAC-64 and VMAC-128

   The preceding VMAC 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.

     VMAC-64(K, M, Nonce) = VMAC(K, M, Nonce, 64)



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     VMAC-128(K, M, Nonce) = VMAC(K, M, Nonce, 128)


5  VHASH: Universal hash function

   VHASH is a keyed hash function, which takes as input a string, and
   produces an 64- or 128-bit output.  VHASH does its work in two or
   three stages, or layers, depending on whether an 64- or 128-bit
   output is requested.  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 taglen is eight.

   Please note that VHASH has certain combinatoric properties making it
   suitable for Wegman-Carter message authentication.  VHASH is not a
   cryptographic hash function and is not a suitable general replacement
   for functions like SHA-1.

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


5.1  VHASH Algorithm

   Input:
     K, string of length KEYLEN bits.
     M, string of any length.
     taglen, the integer 64 or 128.
   Output:
     Y, string of length taglen bits.

   Compute Y using the following algorithm.

     A = L1-HASH(K, M, taglen)
     B = L2-HASH(K, A, taglen)
     if taglen = 64 then
        Y = L3-HASH(K, B)
     else
        Y = B
     end if

     Return Y


5.2  L1-HASH: First-layer hash

   The first-layer hash breaks the message into blocks, each of length
   L1KEYLEN (defined as 128 bytes), and hashes each with a function
   called NH.  Concatenating the results forms a string which is shorter



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   than the original.  One could customize VHASH by changing L1KEYLEN to
   any multiple of 128, achieving different performance characteristics,
   but the resulting algorithm would not be interoperable with the
   standard algorithm defined in this document.


5.2.1  L1-HASH Algorithm

   Input:
     K, string of length KEYLEN bits.
     M, string of any length.
     taglen, the integer 64 or 128.
   Output:
     Y, string of length (2 * taglen * ceil(bitlength(M)/L1KEYLEN))
        bits.

   Compute Y using the following algorithm.

     //
     // Set subkey for L1-HASH
     //
     L1KEYLEN = 1024
     T = KDF(K, 128, L1KEYLEN+128)
     K_1 = T[1 ... L1KEYLEN]
     K_2 = T[129 ... L1KEYLEN + 128]   // Only used when taglen = 128

     //
     // Partition M into L1KEYLEN-bit sgements (last one may be shorter)
     //
     t = max(ceil(bitlength(M) / L1KEYLEN), 1)
     Let M_1, M_2, ..., M_t be strings so that M = M_1 || M_2 || ... ||
        M_t, and bitlength(M_i) = L1KEYLEN for all 0 < i < t.

     //
     // For each segment, except the last: endian-adjust, NH hash,
     // and use the results to build output Y.
     //
     Y = <empty string>
     for i = 1 to t-1 do
        ENDIAN-SWAP(M_i)
        Y = Y || NH(K_1, M_i)
        if taglen = 128 then
           Y = Y || NH(K_2, M_i)  // Hash twice for 128-bit outputs
        end if
     end for

     //
     // For the last block: pad to 128-bit multiple, endian-adjust,



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     // NH hash and add bit-length.  Concatenate the result to Y.
     //
     Len = uint2str(bitlength(M_t), 64) || zeros(64)
     M_t = zeropad(M_t, 128)
     ENDIAN-SWAP(M_t)
     Y = Y || (NH(K_1, M_t) +_128 Len)
     if taglen = 128 then
        Y = Y || NH(K_2, M_t)
     end if

     Return Y


5.2.2  NH Algorithm

        Because this routine is applied directly to every bit of input
        data, an optimized implementation of it yields great benefit.

   Input:
     K, string with length a multiple of 128 bits.
     M, string with length a multiple of 128 bits, but no longer than K.
   Output:
     Y, string of length 128 bits.

   Compute Y using the following algorithm.

     //
     // Partition M and K into 64-bit substrings
     //
     t = bitlength(M) / 64
     Let M_1, M_2, ..., M_t be 64-bit strings
       so that M = M_1 || M_2 || ... || M_t.
     Let K_1, K_2, ..., K_t be 64-bit strings
       so that K_1 || K_2 || ... || K_t  is a prefix of K.

     //
     // Perform NH hash on each.
     //
     Y = zeros(128)
     i = 1
     while (i < t) do
       Y = Y +_128 ((M_i +_64 K_i) *_128 (M_{i+1} +_64 K_{i+1}))
       i = i + 2
     end while
     Y = zeros(2) || Y[3...128]  // Zero first two bits

     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.


5.3.1  L2-HASH Algorithm

   Input:
     K, string of length KEYLEN bits.
     M, string with length a multiple of 128 bits.
   Output:
     Y, string of length 128 bits.

   Compute y using the following algorithm.

     //
     //  Create subkey
     //
     T = KDF(K, 192, 128)
     k = str2uint(zeros(2) || T[ 3...32] || zeros(2) || T[35... 64] ||
                  zeros(2) || T[67...96] || zeros(2) || T[99...128])

     //
     // Partition M into 128-bit substrings
     //
     n = bitlength(M) / 128
     Let M_1, M_2, ..., M_n be strings of length 128 bits
       so that M = M_1 || M_2 || ... || M_n

     //
     // Polynomial hash M
     //
     y = 1
     for i = 1 to n do
       m_i = str2uint(M_i)
       y = (k * y + m_i) mod prime(127)
     end for
     y = (k * y) mod prime(127)
     Y = uint2str(y, 128)

     Return Y


5.4  L3-HASH: Third-layer hash

   The output from L2-HASH is 128 bits long.  This final hash function
   hashes the 128-bit string to a fixed length of 64 bits.  Note that
   the "do" loop during subkey generation has less than 1/2^58



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   probability of requiring more than one iteration.


5.4.1  L3-HASH Algorithm

   Input:
     K, string of length KEYLEN bits.
     M, string of length 128 bits.
   Output:
     Y, string of length 64 bits.

   Compute Y using the following algorithm.

     i = 0
     repeat
       T    = KDF(K, 224+i, 128)
       k_1  = str2uint(T[ 4... 64])
       k_2  = str2uint(T[68...128])
       i    = i + 1
     until (k_1 < prime(61)) and (k_2 < prime(61))

     m_1  = str2uint(M[ 5... 64])
     m_2  = str2uint(M[69...128])
     y = (m_1 * k_1 + m_2 * k_2) mod prime(61)

     Y = uint2str(y, 64)

     Return Y


6  Security considerations

   Here we describe some security considerations important for the
   proper understanding and use of VMAC.


6.1  Resistance to cryptanalysis

   The strength of VMAC 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, by default the Advanced
   Encryption Standard (AES).  However, the core of the VMAC design, the
   VHASH 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
   unconditionally [5].  This means the strength of VHASH is guaranteed
   regardless of advances in cryptanalysis and that an adversarial



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   attack on VMAC that forges with probability significantly exceeding
   the established collision probability of VHASH 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 VMAC: cryptanalytic efforts might as well focus on
   the block cipher.


6.2  Tag lengths and forging probability

   A MAC algorithm is used to authenticate messages between two parties
   that share a secret MAC key K.  An authentication tag is computed for
   a message using K and, in some MAC algorithms such as VMAC, a nonce.
   Messages transmitted between parties are accompanied by their tag
   and, possibly, 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 (ie, 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.

   In the case of VMAC-64, for example, the above guessing-attack
   strategy is close to optimal.  An adversary can correctly guess a
   64-bit VMAC tag with probability 1/2^64 by simply guessing a random
   value.  The theory of Wegman-Carter MACs and results of [5] show that
   no attack strategy can produce a correct tag with probability better
   than about 1/2^59 if VMAC were to use a random function in its work
   rather than AES.  Another result shows that so long as AES is secure
   as a pseudorandom permutation, it can be used instead of a random
   function without significantly increasing the 1/2^59 forging
   probability, assuming that no more than 2^64 messages are
   authenticated with the same key [2].  Similarly for VMAC-128, the
   per-message forgery probability, when using a random function rather
   than AES to instantiate VMAC and limiting messages to j bits, is no
   more than j/2^117.

   AES has undergone extensive study and is assumed to be very secure as
   a pseudorandom permutation.  If we assume that no attacker with
   feasible computational power can distinguish randomly keyed AES from
   a randomly chosen permutation with probability delta (more precisely,
   delta is a function of the computational resources of the attacker
   and of its ability to sample the function), then we obtain that no
   such attacker can forge j-bit messages in VMAC with probability



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   greater than about 1/2^59 or j/2^117, plus delta.  Over N forgery
   attempts, forgery occurs with probability no more than N/^59 or
   N/2^117, plus delta.  The value delta could possibly be greater than
   1/2^59 or 1/2^88, in which case the probability of VMAC forging is
   dominated by a term representing the security of AES.

   With VMAC, off-line computation aimed at exceeding the forging
   probability is hopeless as long as the underlying cipher is not
   broken.  An attacker attempting to forge VMAC tags will need to
   interact with the entity that verifies message tags and try a large
   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 reported.

   Let us reemphasize: a forging probability of 1/2^59 does not mean
   that there is an attack that runs in 2^59 time; to the contrary, as
   long as the block cipher in use is not broken there is no such attack
   for VMAC.  Instead, a 1/2^59 forging probability means that if an
   attacker could have N forgery attempts, then the attacker would have
   no more than N/2^59 probability of getting one or more of them 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 more easily forged.  This is important to
   understand in gauging the severity of a successful forgery, even
   though no such attack on VMAC is known to date.  Due to the short-
   lived nature of most authentication sessions, 64-bit tags seem
   appropriate for many security architectures and commercial
   applications.  If, however, one wants a more conservative option, at
   a cost of about double the computation, VMAC's 128-bit tags may be
   more appropriate.


6.3  Nonce considerations

   VMAC requires a nonce with length upto BLOCKLEN bits.  (For technical
   reasons, the first bit of every nonce must be zero.)  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 VMAC session-key.  There is no guarantee of message
   authenticity when a nonce is repeated, and so messages accompanied by
   a repeated nonce should be considered not authenticated.




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   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 a 64-bit 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 2^63 messages within a session.

   2.  The nonce is a BLOCKLEN-bit 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-bit random value with first bit zero,
       communicated along with the messgae and tag.  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 random nonces should not be allowed to approach 2^(n/2).

   We emphasize that the value of the nonce need not be kept secret.
   When VMAC 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 VMAC.


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 VMAC, 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



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   allows will result in rejecting otherwise valid authentication tags.
   We emphasize that it is up to the receiver to determine 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 VMAC,
   but the message may still be deemed inauthentic because the nonce is
   detected to be a replay.


7  IANA Considerations

   This document has no actions for IANA.


Appendix - Test vectors

   Following are some sample VMAC outputs over a collection of input
   values, using AES with 128-bit keys.  Let key K and nonce N be
   defined by the following ASCII strings.

     K  = "abcdefghijklmnop"              // A 128-bit VMAC key
     N  = "bcdefghi"                      // A 64-bit nonce

   The tags generated by VMAC using key K and nonce N are:

     Message          64-bit Tag               128-bit Tag
     -------          ----------               -----------
     <empty>       4EDE4AE94EDD87E1  E87569084EFF3E1CCA1500C5A6A89CE6
     'abc' * 1     4157A6D46E3EC1A1  E5B10669E5B61668A11E3351CC1A7211
     'abc' * 16    4D3C8A9C2A09E2DE  12A64330F81D8B6407CE90667303FEE2
     'abc' * 100   4FD5EC2FCFE31FBE  10A63F27D4B292723739B4BB6F17A4C1
     'abc' * 10^6  4E13F57841D33D58  22C65CC2CFE9BED72E485CA6EB8A48BE

   The first column lists a small sample of messages which are strings
   of repeated ASCII 'abc' strings.  The remaining columns give in
   hexadecimal the tags generated when VMAC is called with the
   corresponding message, nonce N and key K.


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",



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         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, A. Hevia, H. Krawczyk, T. Krovetz, and P.
         Rogaway, "UMAC: Message authentication code using universal
         hashing", RFC 4418, IETF, 2006.

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

   [5]   T. Krovetz, "Message auhentication on 64-bit architectures",
         Selected Areas in Cryptography - SAC 2006, Springer-Verlag,
         2006.

   [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

   Author's Address

     Ted Krovetz
     Department of Computer Science
     California State University
     Sacramento CA 95819
     USA

     EMail: tdk@acm.org


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Acknowledgments

   This document borrows much text from RFC 4418 [3].  That document
   describes another message authentication scheme, UMAC, and was co-
   written by John Black, Shai Halevi, Alejandro Hevia, Hugo Krawczyk,
   Ted Krovetz and Phillip Rogaway. Funding for the RFC Editor function
   is currently provided by the Internet Society.















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