CFRG Working Group Ted Krovetz INTERNET-DRAFT CSU Sacramento Expires October 2007 Wei Dai Bitvise Limited April 2007 VMAC: Message Authentication Code using Universal Hashing <draft-krovetz-vmac-01.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 Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet- Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/1id-abstracts.html The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. 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 still performing well on 32-bit architectures. Measured speeds are as fast as one-half CPU cycle per byte (cpb) on 64-bit architectures, under five cpb on desktop 32-bit processors, and around ten cpb on embedded 32-bit architectures. Krovetz & Dai Expires October 2007 [Page 1]

INTERNET-DRAFT VMAC April 2007 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 VMAC tag generation . . . . . . . . . . . . . . . . . . . . . . . 9 4.1 VMAC Algorithm . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 VMAC-64 and VMAC-128 . . . . . . . . . . . . . . . . . . . . 10 5 VHASH: Universal hash function . . . . . . . . . . . . . . . . . 10 5.1 VHASH Constants . . . . . . . . . . . . . . . . . . . . . . 10 5.2 VHASH Algorithm . . . . . . . . . . . . . . . . . . . . . . 11 5.3 L1-HASH: First-layer hash . . . . . . . . . . . . . . . . . 11 5.4 L2-HASH: Second-layer hash . . . . . . . . . . . . . . . . . 13 5.5 L3-HASH: Third-layer hash . . . . . . . . . . . . . . . . . 14 6 Security considerations . . . . . . . . . . . . . . . . . . . . . 15 6.1 Resistance to cryptanalysis . . . . . . . . . . . . . . . . 15 6.2 Tag lengths and forging probability . . . . . . . . . . . . 16 6.3 Nonce considerations . . . . . . . . . . . . . . . . . . . . 17 6.4 Replay attacks . . . . . . . . . . . . . . . . . . . . . . . 18 7 IANA Considerations . . . . . . . . . . . . . . . . . . . . . . . 18 Appendix - Test vectors . . . . . . . . . . . . . . . . . . . . . . 19 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Author contact information . . . . . . . . . . . . . . . . . . . . . 20 Full Copyright Statement . . . . . . . . . . . . . . . . . . . . . . 20 Intellectual Property . . . . . . . . . . . . . . . . . . . . . . . 21 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Krovetz & Dai Expires October 2007 [Page 2]

INTERNET-DRAFT VMAC April 2007 1 Introduction VMAC is a message authentication code (MAC) algorithm designed for high performance. It is backed by a 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, 8]. A fast "universal" hash function is used to hash an input message M into a short string. This short string is then combined by addition 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) + 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 block cipher. VMAC allows for tag lengths of any 64-bit multiple up to the block size of the block cipher in use. When using AES, this means VMAC can produce 64- or 128-bit tags. 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 is less than 1/2^60 or 1/2^120 when the tags 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 less than N/2^60 or N/2^120. In a real implementation of VMAC, using AES to produce keys and pads, these forgery probabilities 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, 6]. Krovetz & Dai Expires October 2007 [Page 3]

INTERNET-DRAFT VMAC April 2007 VMAC performs best in environments where 64-bit quantities are efficiently read from memory "little-endian" and multiplied into 128-bit results. Performance on 32-bit architechtures suppporting Intel's SSE2 instruction-set is also very good. On other 32-bit architectures, each 64-bit multiplication is accomplished via four 32-bit multiplications, resulting in a corresponding slowdown. The data in the following table were generated using the reference implementation available at the VMAC website [7]. The table shows sample performance on several architectures over message lengths of 64, 512 and 4096 bytes. 64-bit Tags 128-bit Tags Bits/Endian/Architecture 64 512 4K 64 512 4K ---------------------------------+-----+----+-----+-----+----+---- 64/LE/AMD Athlon 64 "Manchester" | 6.0 1.1 0.5 | 7.0 1.6 0.9 64/LE/Intel Core 2 "Merom" | 5.9 1.2 0.6 | 6.9 1.7 1.1 64/BE/IBM PowerPC 970FX | 10.1 2.5 1.6 | 11.4 3.8 3.0 32/LE/Intel Core 2 "Merom" | 8.3 2.2 1.4 | 11.1 3.6 2.8 32/LE/Intel NetBurst "Nocona" | 15.0 4.4 3.1 | 18.9 7.1 5.8 32/BE/Freescale PowerPC 7457 | 15.3 6.4 5.3 | 22.1 11.2 10.0 32/LE/Embedded ARM v5te core | 39.9 13.1 10.1 | 53.6 22.9 19.8 ---------------------------------+-----+----+-----+-----+----+---- Table: Tag generation speed measured in CPU cycles per message byte, for cache-resident messages of length 64, 512 and 4K bytes. Architechtures are listed as register-size/endianness/model. 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, such as for string-length, 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. Krovetz & Dai Expires October 2007 [Page 4]

INTERNET-DRAFT VMAC April 2007 zeros(n): The string made of n zero-bits. 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 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. floor(x): The largest integer not greater than x. a div b: The largest integer i for which b * i <= a. 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. Krovetz & Dai Expires October 2007 [Page 5]

INTERNET-DRAFT VMAC April 2007 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), "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). On many 64-bit architectures, these operations can each be implemented with one or two assembly-language instructions. 2.5 ENDIAN-SWAP: Adjusting endian orientation Message data is normally 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_8 || W_7 || ... || W_1 T = T || SReversed_i end for Return T Krovetz & Dai Expires October 2007 [Page 6]

INTERNET-DRAFT VMAC April 2007 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. 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 block encryption 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. 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 by the hash function. 3.2.1 KDF Algorithm Input: K, string of length KEYLEN bits. index, an integer in the range 0...255. numbits, a non-negative integer. Output: Y, string of length numbits bits. Compute Y using the following algorithm. // Krovetz & Dai Expires October 2007 [Page 7]

INTERNET-DRAFT VMAC April 2007 // Calculate number of block cipher iterations // n = ceil(numbits / BLOCKLEN) // // Build Y using block cipher in a counter mode // Y = <empty string> 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. The length of the pad can be any positive multiple of 64 bits, up to BLOCKLEN bits. 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 of length KEYLEN bits. Nonce, string of length less than BLOCKLEN bits. taglen, positive multiple of 64, no greater than BLOCKLEN. 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 smallest integer for which BLOCKLEN/taglen <= 2^i index = str2uint(Nonce) mod 2^i Nonce = Nonce[1...bitlength(Nonce)-i] || zeros(i) // Krovetz & Dai Expires October 2007 [Page 8]

INTERNET-DRAFT VMAC April 2007 // Make Nonce BLOCKLEN bits by prepending zeros. // At least one zero bit is prepended here. // Nonce = zeros(BLOCKLEN - bitlength(Nonce)) || Nonce // // Encipher and extract indexed substring // 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 then computing the addition of the resulting strings. The length of the pad and hash can be any positive multiple of 64 bits, up to BLOCKLEN bits. 4.1 VMAC Algorithm Input: K, string of length KEYLEN bits. M, string of length up to 2^64 bits. Nonce, string of length less than BLOCKLEN bits. taglen, positive multiple of 64, no greater than BLOCKLEN. Output: Tag, string of length taglen bits. Compute Tag using the following algorithm. HashedMessage = VHASH(K, M, taglen) Pad = PDF(K, Nonce, taglen) Tag = <empty string> for i = 0 to (taglen/64 - 1) do T = Pad [1 + 64 * i ... 64 * (i + 1)] +_64 HashedMessage[1 + 64 * i ... 64 * (i + 1)] Tag = Tag || T end for Return Tag Krovetz & Dai Expires October 2007 [Page 9]

INTERNET-DRAFT VMAC April 2007 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) 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 a string output with length that is a multiple of 64 bits. VHASH is a three-layered hash function. A message is first hashed by L1-HASH, its output is then hashed by L2-HASH, whose output is then hashed by L3-HASH. This process is done once for each 64 bits of output. 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 Constants The following constants are referred to in the definition of VHASH. L1KEYLEN defines how many bits of key material are generated internally for the first layer of hashing. FAVOR-ENDIAN determines which endian orientation is used to read messages. L1KEYLEN = 1024 FAVOR-ENDIAN = LITTLE One could change L1KEYLEN to any positive multiple of 128 or change FAVOR-ENDIAN to BIG. A larger L1KEYLEN improves the speed of the algorithms at the cost of increased memory usage, and changing FAVOR- ENDIAN to BIG improves the speed of the algorithms on big-endian machines at the cost of decreased speed on little-endian machines. The resulting algorithms would be incompatible with the VMAC and VHASH algorithms defined here, but might be useful in custom applications. Krovetz & Dai Expires October 2007 [Page 10]

INTERNET-DRAFT VMAC April 2007 5.2 VHASH Algorithm Input: K, string of length KEYLEN bits. M, string of length up to 2^64 bits. taglen, positive multiple of 64. Output: Y, string of length taglen bits. Compute Y using the following algorithm. Y = <empty string> for i = 0 to (taglen/64 - 1) do A = L1-HASH(K, M, i) B = L2-HASH(K, A, bitlength(M), i) Y = Y || L3-HASH(K, B, i) end for Return Y 5.3 L1-HASH: First-layer hash The first-layer hash breaks the message into blocks, each of length up to L1KEYLEN (normally defined as 1024 bits), and hashes each with a function called NH. Concatenating the results forms a string which is shorter than the original (unless the original length was no greater than 128 bits). 5.2.1 L1-HASH Algorithm Input: K, string of length KEYLEN bits. M, string of any length. iter, non-negative integer. Output: Y, string of length ceil(bitlength(M)/L1KEYLEN) * 128 bits. Compute Y using the following algorithm. // // Set subkey for L1-HASH // T = KDF(K, 128, L1KEYLEN + 128 * iter) K = T[1 + 128 * iter ... L1KEYLEN + 128 * iter] Y = <empty string> Krovetz & Dai Expires October 2007 [Page 11]

INTERNET-DRAFT VMAC April 2007 if bitlength(M) > 0 then // // Break M into L1KEYLEN-bit segments (last one may be shorter) // t = ceil(bitlength(M) / L1KEYLEN) Let M_1, M_2, ..., M_t be strings so that M_1 || M_2 || ... || M_t = M, and bitlength(M_i) = L1KEYLEN for all 0 < i < t. // // For each segment: pad, endian-adjust, NH hash, and use // results to build output Y. Note that padding only effects // the final segment because all other segment lengths are // already a multiple of 128. // for i = 1 to t do M_i = zeropad(M_i, 128) if FAVOR-ENDIAN = LITTLE then ENDIAN-SWAP(M_i) end if Y = Y || NH(K, M_i) end for 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 Krovetz & Dai Expires October 2007 [Page 12]

INTERNET-DRAFT VMAC April 2007 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 two bits (ie, mod 2^126) Return Y 5.4 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. len, non-negative integer. iter, non-negative integer. Output: Y, string of length 128 bits. Compute y using the following algorithm. // // Create subkey - the (iter+1)'st 128-bit chunk of the // string generated by KDF(K, 192) // T = KDF(K, 192, 128 * (iter + 1)) T = T[1 + 128 * iter ... 128 * (iter + 1)] k = str2uint(zeros(3) || T[ 4...32] || zeros(3) || T[ 36... 64] || zeros(3) || T[68...96] || zeros(3) || T[100...128]) n = bitlength(M) / 128 if n > 0 then // // Partition M into 128-bit substrings and polynomial hash // Krovetz & Dai Expires October 2007 [Page 13]

INTERNET-DRAFT VMAC April 2007 Let M_1, M_2, ..., M_n be strings of length 128 bits so that M = M_1 || M_2 || ... || M_n p127 = 2^127 - 1 // 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF in hex y = 1 for i = 1 to n do m_i = str2uint(M_i) y = (y * k + m_i) mod p127 end for else // // M is an empty string; handled as a special case // y = k end if y = (y + (len mod L1KEYLEN) * 2^64) mod p127 Y = uint2str(y, 128) Return Y 5.5 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. 5.4.1 L3-HASH Algorithm Input: K, string of length KEYLEN bits. M, string of length 128 bits. iter, non-negative integer. Output: Y, string of length 64 bits. Compute Y using the following algorithm. // // Create subkey - the (iter+1)'st 128-bit chunk of the // string generated by KDF(K, 224) that passes a test // p64 = 2^64 - 257 // 0xFFFFFFFFFFFFFEFF in hex i = 0 Krovetz & Dai Expires October 2007 [Page 14]

INTERNET-DRAFT VMAC April 2007 need = iter + 1 repeat T = KDF(K, 224, 128 * (i + 1)) T = T[1 + 128 * i ... 128 * (i + 1)] k_1 = str2uint(T[ 0... 64]) k_2 = str2uint(T[65...128]) i = i + 1 if (k_1 < p64) and (k_2 < p64) then need = need - 1 end if until (need = 0) // // Transform M into two integers less than p64 and hash // m_1 = str2uint(M) div (2^64 - 2^32) m_2 = str2uint(M) mod (2^64 - 2^32) y = ((m_1 + k_1) * (m_2 + k_2)) mod p64 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, 6]. This means the strength of VHASH is guaranteed regardless of advances in cryptanalysis and that an adversarial 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. Krovetz & Dai Expires October 2007 [Page 15]

INTERNET-DRAFT VMAC April 2007 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, 6] show that no attack strategy can produce a correct tag with probability better than 1/2^60 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 1/2^60 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 is no more than 1/2^120. 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 messages in VMAC with probability greater than about 1/2^60 or 1/2^120, plus delta. Over N forgery attempts, forgery occurs with probability no more than N/^60 or N/2^120, plus delta. The value delta could possibly be greater than 1/2^60 or 1/2^120, 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 Krovetz & Dai Expires October 2007 [Page 16]

INTERNET-DRAFT VMAC April 2007 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. 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 is not broken there is no such attack for VMAC. 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. It should be pointed out that once an attacker knows that 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 are appropriate for many security architectures and 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 less than BLOCKLEN bits. All nonces in an authentication session must be unique and equal in length. The security of VMAC depends on the assumption that no nonce is ever used to generate tags for more than one message under the same key. If an attacker is able to observe two VMAC tags that were generated using the same key, the same nonce, and different messages, he may be able to easily forge other VMAC tags. While such an attack is not known to date, VMAC was not designed to offer any protection in this scenario, and nonce reuse must be prevented through appropriate system architecture. 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 Krovetz & Dai Expires October 2007 [Page 17]

INTERNET-DRAFT VMAC April 2007 entity verifying a tag, but there are many possibilities. Nonce values could be randomly generated, could come from an incrementing counter, or could be co-opted from some non-repeating part of the messages being authenticated (such as a sequence number). The nonce can then be sent along with the message if necessary, or if the receiver is able to deduce the nonce in use, the nonce need not be sent. We emphasize that the nonce need not be kept secret, but that no nonce should be used more than once in any session by either sender or receiver. Designers of systems and applications that use VMAC should be aware that modern virtual machine software such as VMware may allow the user of a virtual machine to roll back its state (both persistent storage and volatile memory) to earlier snapshots or checkpoints. This rollback may be part of an attack, or simply due to an unrelated decision by an authorized user. In any case, if VMAC is used in such a virtual machine, a rollback may cause a nonce to be reused, intentionally or unintentionally, thus violating an important security assumption. The system or application must either prevent the occurrence of a state rollback, or be designed to ensure that nonces are not reused on different messages even when state rollbacks are possible. For example, one possible design is to generate a fresh random string as the nonce for each message, after the content of that message has been fixed. 6.4 Replay attacks A replay attack occurs when an attacker repeats a message, nonce, and authentication tag. If the replay of a previously authenticated message would have negative consequences, then the receiver should identify repeated message-nonce pairs and ignore them. One way to do this is to look for a nonce that has already been used to authenticate a prior message, and ignore it. 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 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. Krovetz & Dai Expires October 2007 [Page 18]

INTERNET-DRAFT VMAC April 2007 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> 2576BE1C56D8B81B 472766C70F74ED23481D6D7DE4E80DAC 'abc' * 1 2D376CF5B1813CE5 4EE815A06A1D71EDD36FC75D51188A42 'abc' * 16 E8421F61D573D298 09F2C80C8E1007A0C12FAE19FE4504AE 'abc' * 100 4492DF6C5CAC1BBE 66438817154850C61D8A412164803BCB 'abc' * 1000000 09BA597DD7601113 2B6B02288FFC461B75485DE893C629DC 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", 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. Krovetz & Dai Expires October 2007 [Page 19]

INTERNET-DRAFT VMAC April 2007 [5] W. Dai and T. Krovetz, "VMAC high-speed message authentication", in progress. [6] T. Krovetz, "Message auhentication on 64-bit architectures", Selected Areas in Cryptography - SAC 2006, Springer-Verlag, 2006. [7] VMAC Website, http://fastcrypto.com/vmac, as seen April 2007. [8] 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 Addresses Ted Krovetz Department of Computer Science California State University Sacramento CA 95819 USA Email: tdk@acm.org Wei Dai Bitvise Limited Email: rfc@weidai.com Full Copyright Statement Copyright (C) The IETF Trust (2007). This document is subject to the rights, licenses and restrictions contained in BCP 78, and 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, THE IETF TRUST AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Krovetz & Dai Expires October 2007 [Page 20]

INTERNET-DRAFT VMAC April 2007 Intellectual Property The IETF takes no position regarding the validity or scope of any Intellectual Property Rights or other rights that might be claimed to pertain to the implementation or use of the technology described in this document or the extent to which any license under such rights might or might not be available; nor does it represent that it has made any independent effort to identify any such rights. Information on the ISOC's procedures with respect to rights in ISOC Documents can be found in BCP 78 and BCP 79. Copies of IPR disclosures made to the IETF Secretariat and any assurances of licenses to be made available, or the result of an attempt made to obtain a general license or permission for the use of such proprietary rights by implementers or users of this specification can be obtained from the IETF on-line IPR repository at http://www.ietf.org/ipr. The IETF invites any interested party to bring to its attention any copyrights, patents or patent applications, or other proprietary rights that may cover technology that may be required to implement this standard. Please address the information to the IETF at ietf- ipr@ietf.org. 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. Krovetz & Dai Expires October 2007 [Page 21]