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cryptol/examples/SIV-rfc5297.md
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```
module SIV where
import AES
// Uses ../bench/data/AES.cry:
// CRYPTOLPATH=../bench/data cryptol SIV-rfc5297.md
type Key = [AESKeySize]
```
Network Working Group D. Harkins
Request for Comments: 5297 Aruba Networks
Category: Informational October 2008
Synthetic Initialization Vector (SIV) Authenticated Encryption
Using the Advanced Encryption Standard (AES)
Status of This Memo
This memo provides information for the Internet community. It does
not specify an Internet standard of any kind. Distribution of this
memo is unlimited.
Abstract
This memo describes SIV (Synthetic Initialization Vector), a block
cipher mode of operation. SIV takes a key, a plaintext, and multiple
variable-length octet strings that will be authenticated but not
encrypted. It produces a ciphertext having the same length as the
plaintext and a synthetic initialization vector. Depending on how it
is used, SIV achieves either the goal of deterministic authenticated
encryption or the goal of nonce-based, misuse-resistant authenticated
encryption.
Table of Contents
1. Introduction ....................................................3
1.1. Background .................................................3
1.2. Definitions ................................................4
1.3. Motivation .................................................4
1.3.1. Key Wrapping ........................................4
1.3.2. Resistance to Nonce Misuse/Reuse ....................4
1.3.3. Key Derivation ......................................5
1.3.4. Robustness versus Performance .......................6
1.3.5. Conservation of Cryptographic Primitives ............6
2. Specification of SIV ............................................6
2.1. Notation ...................................................6
2.2. Overview ...................................................7
2.3. Doubling ...................................................7
2.4. S2V ........................................................8
2.5. CTR .......................................................10
2.6. SIV Encrypt ...............................................10
2.7. SIV Decrypt ...............................................12
3. Nonce-Based Authenticated Encryption with SIV ..................14
4. Deterministic Authenticated Encryption with SIV ................15
5. Optimizations ..................................................15
6. IANA Considerations ............................................15
6.1. AEAD_AES_SIV_CMAC_256 .....................................17
6.2. AEAD_AES_SIV_CMAC_384 .....................................17
6.3. AEAD_AES_SIV_CMAC_512 .....................................18
7. Security Considerations ........................................18
8. Acknowledgments ................................................19
9. References .....................................................19
9.1. Normative References ......................................19
9.2. Informative References ....................................19
Appendix A. Test Vectors ....................................... 22
A.1. Deterministic Authenticated Encryption Example ........... 22
A.2. Nonce-Based Authenticated Encryption Example ............. 23
1. Introduction
1.1. Background
Various attacks have been described (e.g., [BADESP]) when data is
merely privacy protected and not additionally authenticated or
integrity protected. Therefore, combined modes of encryption and
authentication have been developed ([RFC5116], [RFC3610], [GCM],
[JUTLA], [OCB]). These provide conventional authenticated encryption
when used with a nonce ("a number used once") and typically accept
additional inputs that are authenticated but not encrypted,
hereinafter referred to as "associated data" or AD.
A deterministic, nonce-less, form of authenticated encryption has
been used to protect the transportation of cryptographic keys (e.g.,
[X9F1], [RFC3217], [RFC3394]). This is generally referred to as "Key
Wrapping".
This memo describes a new block cipher mode, SIV, that provides both
nonce-based authenticated encryption as well as deterministic, nonce-
less key wrapping. It contains a Pseudo-Random Function (PRF)
construction called S2V and an encryption/decryption construction,
called CTR. SIV was specified by Phillip Rogaway and Thomas
Shrimpton in [DAE]. The underlying block cipher used herein for both
S2V and CTR is AES with key lengths of 128 bits, 192 bits, or 256
bits. S2V uses AES in Cipher-based Message Authentication Code
([CMAC]) mode, CTR uses AES in counter ([MODES]) mode.
Associated data is data input to an authenticated-encryption mode
that will be authenticated but not encrypted. [RFC5116] says that
associated data can include "addresses, ports, sequence numbers,
protocol version numbers, and other fields that indicate how the
plaintext or ciphertext should be handled, forwarded, or processed".
These are multiple, distinct inputs and may not be contiguous. Other
authenticated-encryption cipher modes allow only a single associated
data input. Such a limitation may require implementation of a
scatter/gather form of data marshalling to combine the multiple
components of the associated data into a single input or may require
a pre-processing step where the associated data inputs are
concatenated together. SIV accepts multiple variable-length octet
strings (hereinafter referred to as a "vector of strings") as
associated data inputs. This obviates the need for data marshalling
or pre-processing of associated data to package it into a single
input.
By allowing associated data to consist of a vector of strings SIV
also obviates the requirement to encode the length of component
fields of the associated data when those fields have variable length.
1.2. Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [RFC2119].
1.3. Motivation
1.3.1. Key Wrapping
A key distribution protocol must protect keys it is distributing.
This has not always been done correctly. For example, RADIUS
[RFC2865] uses Microsoft Point-to-Point Encryption (MPPE) [RFC2548]
to encrypt a key prior to transmission from server to client. It
provides no integrity checking of the encrypted key. [RADKEY]
specifies the use of [RFC3394] to wrap a key in a RADIUS request but
because of the inability to pass associated data, a Hashed Message
Authentication Code (HMAC) [RFC2104] is necessary to provide
authentication of the entire request.
SIV can be used as a drop-in replacement for any specification that
uses [RFC3394] or [RFC3217], including the aforementioned use. It is
a more general purpose solution as it allows for associated data to
be specified.
1.3.2. Resistance to Nonce Misuse/Reuse
The nonce-based authenticated encryption schemes described above are
susceptible to reuse and/or misuse of the nonce. Depending on the
specific scheme there are subtle and critical requirements placed on
the nonce (see [SP800-38D]). [GCM] states that it provides
"excellent security" if its nonce is guaranteed to be distinct but
provides "no security" otherwise. Confidentiality guarantees are
voided if a counter in [RFC3610] is reused. In many cases,
guaranteeing no reuse of a nonce/counter/IV is not a problem, but in
others it will be.
For example, many applications obtain access to cryptographic
functions via an application program interface to a cryptographic
library. These libraries are typically not stateful and any nonce,
initialization vector, or counter required by the cipher mode is
passed to the cryptographic library by the application. Putting the
construction of a security-critical datum outside the control of the
encryption engine places an onerous burden on the application writer
who may not provide the necessary cryptographic hygiene. Perhaps his
random number generator is not very good or maybe an application
fault causes a counter to be reset. The fragility of the cipher mode
may result in its inadvertent misuse. Also, if one's environment is
(knowingly or unknowingly) a virtual machine, it may be possible to
roll back a virtual state machine and cause nonce reuse thereby
gutting the security of the authenticated encryption scheme (see
[VIRT]).
If the nonce is random, a requirement that it never repeat will limit
the amount of data that can be safely protected with a single key to
one block. More sensibly, a random nonce is required to "almost
always" be non-repeating, but that will drastically limit the amount
of data that can be safely protected.
SIV provides a level of resistance to nonce reuse and misuse. If the
nonce is never reused, then the usual notion of nonce-based security
of an authenticated encryption mode is achieved. If, however, the
nonce is reused, authenticity is retained and confidentiality is only
compromised to the extent that an attacker can determine that the
same plaintext (and same associated data) was protected with the same
nonce and key. See Security Considerations (Section 7).
1.3.3. Key Derivation
A PRF is frequently used as a key derivation function (e.g., [WLAN])
by passing it a key and a single string. Typically, this single
string is the concatenation of a series of smaller strings -- for
example, a label and some context to bind into the derived string.
These are usually multiple strings but are mapped to a single string
because of the way PRFs are typically defined -- two inputs: a key
and data. Such a crude mapping is inefficient because additional
data must be included -- the length of variable-length inputs must be
encoded separately -- and, depending on the PRF, memory allocation
and copying may be needed. Also, if only one or two of the inputs
changed when deriving a new key, it may still be necessary to process
all of the other constants that preceded it every time the PRF is
invoked.
When a PRF is used in this manner its input is a vector of strings
and not a single string and the PRF should handle the data as such.
The S2V ("string to vector") PRF construction accepts a vector of
inputs and provides a more natural mapping of input that does not
require additional lengths encodings and obviates the memory and
processing overhead to marshal inputs and their encoded lengths into
a single string. Constant inputs to the PRF need only be computed
once.
1.3.4. Robustness versus Performance
SIV cannot perform at the same high throughput rates that other
authenticated encryption schemes can (e.g., [GCM] or [OCB]) due to
the requirement for two passes of the data, but for situations where
performance is not a limiting factor -- e.g., control plane
applications -- it can provide a robust alternative, especially when
considering its resistance to nonce reuse.
1.3.5. Conservation of Cryptographic Primitives
The cipher mode described herein can do authenticated encryption, key
wrapping, key derivation, and serve as a generic message
authentication algorithm. It is therefore possible to implement all
these functions with a single tool, instead of one tool for each
function. This is extremely attractive for devices that are memory
and/or processor constrained and that cannot afford to implement
multiple cryptographic primitives to accomplish these functions.
2. Specification of SIV
2.1. Notation
SIV and S2V use the following notation:
len(A)
returns the number of bits in A.
pad(X)
indicates padding of string X, len(X) < 128, out to 128 bits by
the concatenation of a single bit of 1 followed by as many 0 bits
as are necessary.
```
pad : {x} (fin x) => [x] -> [128]
pad x = take `{128} (x # [True] # (zero : [127]))
```
leftmost(A,n)
the n most significant bits of A.
rightmost(A,n)
the n least significant bits of A.
A || B
means concatenation of string A with string B.
A xor B
is the exclusive OR operation on two equal length strings, A and
B.
A xorend B
where len(A) >= len(B), means xoring a string B onto the end of
string A -- i.e., leftmost(A, len(A)-len(B)) || (rightmost(A,
len(B)) xor B).
```
xorend : {a,b} (fin a, fin b, a >= b) => [a] -> [b] -> [a]
xorend a b = a ^ ((zero : [a-b]) # b)
```
A bitand B
is the logical AND operation on two equal length strings, A and B.
```
bitand a b = a && b
```
dbl(S)
is the multiplication of S and 0...010 in the finite field
represented using the primitive polynomial
x^128 + x^7 + x^2 + x + 1. See Doubling (Section 2.3).
a^b
indicates a string that is "b" bits, each having the value "a".
```
// In cryptol tye symbole '^' is taken, we will use the native name 'repeat'
// instead. This should not lead to errors due to the different types, which
// the compiler would detect if one operation were used in place of the other.
```
<zero>
indicates a string that is 128 zero bits.
<one>
indicates a string that is 127 zero bits concatenated with a
single one bit, that is 0^127 || 1^1.
A/B
indicates the greatest integer less than or equal to the real-
valued quotient of A and B.
E(K,X)
indicates AES encryption of string X using key K.
```
E : (Key,[128]) -> [128]
E(K,X) = aesEncrypt (X,K)
// 1. Apply the subkey generation process in Sec. 6.1 to K to produce K1 and K2.
// 2. If Mlen = 0, let n = 1; else, let n = ⎡Mlen/b⎤.
// 3. Let M1, M2, ... , Mn-1, Mn* denote the unique sequence of bit strings such that M = M1 || M2 || ... || Mn-1 || Mn*, where M1, M2,..., Mn-1 are complete blocks.
// 4. If Mn* is a complete block, let Mn = K1 ⊕ Mn* ; else, let Mn = K2 ⊕ (Mn* ||10j), where j = nb-Mlen-1.
// 5. Let C0 = 0b.
// 6. For i = 1 to n, let Ci = CIPHK(Ci-1 ⊕ Mi).
// 7. Let T = MSBTlen(Cn).
// 8. Return T.
aesCMAC : {m} (fin m) => Key -> [m] -> [128]
aesCMAC K m =
cmacBlocks K ((`m%128) == 0 && `m > 0) (split `{each=128,parts=blocks} full)
where
pd = [True] # zero : [128]
full = take `{front=128 * blocks, back = (m + 128) - 128*blocks} (m # pd)
type blocks = max ((m + 127) / 128) 1
cmacBlocks : {blocks} (fin blocks, blocks >= 1) => Key -> Bit -> [blocks][128] -> [128]
cmacBlocks K completeMn ms = ci ! 0
where
(K1,K2) = subkeyGen K
xs = take `{back=1} ms
mn = KN ^ (ms ! 0)
KN = if completeMn then K1 else K2
ci = [zero] # [ E(K,c ^ mi) | c <- ci | mi <- (xs # [mn]) ]
property cmacKAT1 =
~zero ==
[ aesCMAC 0x2b7e151628aed2a6abf7158809cf4f3c [] == 0xbb1d6929e95937287fa37d129b756746
, aesCMAC 0x2b7e151628aed2a6abf7158809cf4f3c 0x6bc1bee22e409f96e93d7e117393172a == 0x070a16b46b4d4144f79bdd9dd04a287c
, aesCMAC 0x2b7e151628aed2a6abf7158809cf4f3c 0x6bc1bee22e409f96e93d7e117393172aae2d8a571e03ac9c9eb76fac45af8e5130c81c46a35ce411 == 0xdfa66747de9ae63030ca32611497c827
, aesCMAC 0x2b7e151628aed2a6abf7158809cf4f3c 0x6bc1bee22e409f96e93d7e117393172aae2d8a571e03ac9c9eb76fac45af8e5130c81c46a35ce411e5fbc1191a0a52eff69f2445df4f9b17ad2b417be66c3710 == 0x51f0bebf7e3b9d92fc49741779363cfe
]
repeat : {n,a} (n >= 1, fin n) => a -> [n]a
repeat a = [a | _ <- [1..n]]
```
```
// 1. Let L = CIPHK(0b).
// 2. If MSB1(L) = 0
// then K1 = L << 1;
// else K1 = (L << 1) ⊕ Rb; see Sec. 5.3 for the definition of Rb.
// 3. If MSB1(K1) = 0
// then K2 = K1 << 1;
// else K2 = (K1 << 1) ⊕ Rb.
// 4. Return K1, K2.
private
subkeyGen : Key -> (Key, Key)
subkeyGen K = (K1,K2)
where
L = E(K,zero)
K1 = ite (L@0 == False) (L << 1) ((L << 1) ^ Rb)
K2 = ite ((K1@0) == False) (K1 << 1) ((K1 << 1) ^ Rb)
Rb = `0b10000111
ite : {n} (fin n, n >= 1) => Bit -> [n] -> [n] -> [n]
ite pred t e = t && m1 || e && m2
where m1 = [pred | _ <- [1..n]]
m2 = ~m1
```
2.2. Overview
SIV-AES uses AES in CMAC mode (S2V) and in counter mode (CTR). SIV-
AES takes either a 256-, 384-, or 512-bit key (which is broken up
into two equal-sized keys, one for S2V and the other for CTR), a
variable length plaintext, and multiple variable-length strings
representing associated data. Its output is a ciphertext that
comprises a synthetic initialization vector concatenated with the
encrypted plaintext.
2.3. Doubling
The doubling operation on a 128-bit input string is performed using a
left-shift of the input followed by a conditional xor operation on
the result with the constant:
> 00000000 00000000 00000000 00000087
The condition under which the xor operation is performed is when the
bit being shifted off is one.
Note that this is the same operation used to generate sub-keys for
CMAC-AES.
```
// dblRef x = if x@0 == False then (x << 1)
// else (x << 1) ^ `0x87
// :prove \x -> dbl x == (dblRef x : [128])
// Q.E.D.
dbl x = ite ((x@0) == False) (x << 1) ((x << 1) ^ `0x87)
```
2.4. S2V
The S2V operation consists of the doubling and xoring of the outputs
of a pseudo-random function, CMAC, operating over individual strings
in the input vector: S1, S2, ... , Sn. It is bootstrapped by
performing CMAC on a 128-bit string of zeros. If the length of the
final string in the vector is greater than or equal to 128 bits, the
output of the double/xor chain is xored onto the end of the final
input string. That result is input to a final CMAC operation to
produce the output V. If the length of the final string is less than
128 bits, the output of the double/xor chain is doubled once more and
it is xored with the final string padded using the padding function
pad(X). That result is input to a final CMAC operation to produce
the output V.
S2V with key K on a vector of n inputs S1, S2, ..., Sn-1, Sn, and
len(Sn) >= 128:
> +----+ +----+ +------+ +----+
> | S1 | | S2 | . . . | Sn-1 | | Sn |
> +----+ +----+ +------+ +----+
> <zero> K | | | |
> | | | | | V
> V | V V V /----> xorend
> +-----+ | +-----+ +-----+ +-----+ | |
> | AES-|<----->| AES-| K-->| AES-| K--->| AES-| | |
> | CMAC| | CMAC| | CMAC| | CMAC| | |
> +-----+ +-----+ +-----+ +-----+ | V
> | | | | | +-----+
> | | | | | K-->| AES-|
> | | | | | | CMAC|
> | | | | | +-----+
> \-> dbl -> xor -> dbl -> xor -> dbl -> xor---/ |
> V
> +---+
> | V |
> +---+
>
> Figure 2
S2V with key K on a vector of n inputs S1, S2, ..., Sn-1, Sn, and
len(Sn) < 128:
> +----+ +----+ +------+ +---------+
> | S1 | | S2 | . . . | Sn-1 | | pad(Sn) |
> +----+ +----+ +------+ +---------+
> <zero> K | | | |
> | | | | | V
> V | V V V /------> xor
> +-----+ | +-----+ +-----+ +-----+ | |
> | AES-|<--->| AES-| K-->| AES-| K-->| AES-| | |
> | CMAC| | CMAC| | CMAC| | CMAC| | |
> +-----+ +-----+ +-----+ +-----+ | V
> | | | | | +-----+
> | | | | | K-->| AES-|
> | | | | | | CMAC|
> | | | | | +-----+
> \-> dbl -> xor -> dbl -> xor -> dbl -> xor-> dbl |
> V
> +---+
> | V |
> +---+
>
> Figure 3
Algorithmically S2V can be described as:
> S2V(K, S1, ..., Sn) {
> if n = 0 then
> return V = AES-CMAC(K, <one>)
> fi
> D = AES-CMAC(K, <zero>)
> for i = 1 to n-1 do
> D = dbl(D) xor AES-CMAC(K, Si)
> done
> if len(Sn) >= 128 then
> T = Sn xorend D
> else
> T = dbl(D) xor pad(Sn)
> fi
> return V = AES-CMAC(K, T)
> }
```
// S2V for n=2
S2V : {k, ad, p} (fin ad, fin p) => Key -> [ad] -> [p] -> [128]
S2V K S1 S2 = res
where
D0 = aesCMAC K (zero : [128])
D1 = dbl D0 ^ aesCMAC K S1
res = if `p >= 128
then aesCMAC K (xorend (fixSize S2) D1)
else aesCMAC K (dbl D1 ^ pad S2)
private
// The length of 'p' is >= 128, but Cryptol lacks
// dependent types and can not infer this fact. We
// Provide a no-op computation that results in a
// new type for 'p' that is at least 128 bits
fixSize : {p} (fin p) => [p] -> [max p 128]
fixSize p = take `{front=max p 128, back = p + 128 - max p 128} (p # (zero : [128]))
```
2.5. CTR
CTR is a counter mode of AES. It takes as input a plaintext P of
arbitrary length, a key K of length 128, 192, or 256 bits, and a
counter X that is 128 bits in length, and outputs Z, which represents
a concatenation of a synthetic initialization vector V and the
ciphertext C, which is the same length as the plaintext.
The ciphertext is produced by xoring the plaintext with the first
len(P) bits of the following string:
> E(K, X) || E(K, X+1) || E(K, X+2) || ...
Before beginning counter mode, the 31st and 63rd bits (where the
rightmost bit is the 0th bit) of the counter are cleared. This
enables implementations that support native 32-bit (64-bit) addition
to increment the counter modulo 2^32 (2^64) in a manner that cannot
be distinguished from 128-bit increments, as long as the number of
increment operations is limited by an upper bound that safely avoids
carry to occur out of the respective pre-cleared bit. More formally,
for 32-bit addition, the counter is incremented as:
> SALT=leftmost(X,96)
>
> n=rightmost(X,32)
>
> X+i = SALT || (n + i mod 2^32).
For 64-bit addition, the counter is incremented as:
> SALT=leftmost(X,64)
>
> n=rightmost(X,64)
>
> X+i = SALT || (n + i mod 2^64).
Performing 32-bit or 64-bit addition on the counter will limit the
amount of plaintext that can be safely protected by SIV-AES to 2^39 -
128 bits or 2^71 - 128 bits, respectively.
```
ctr32 : {n} (2^^39 - 128 >= n) => Key -> [128] -> [n] -> [n]
ctr32 k iv pt = pt ^ take stream
where
stream = join [E(k,v) | v <- ivs]
ivs = [take `{96} iv # cnt + i | i <- [0,1..]]
cnt = drop `{back=32} iv
ctr64 : {n} (2^^71 - 128 >= n) => Key -> [128] -> [n] -> [n]
ctr64 k iv pt = pt ^ take stream
where
stream = join [E(k,v) | v <- ivs]
ivs = [take `{64} iv # cnt + i | i <- [0,1..]]
cnt = drop `{back=64} iv
```
2.6. SIV Encrypt
SIV-encrypt takes as input a key K of length 256, 384, or 512 bits,
plaintext of arbitrary length, and a vector of associated data AD[ ]
where the number of components in the vector is not greater than 126
(see Section 7). It produces output, Z, which is the concatenation
of a 128-bit synthetic initialization vector and ciphertext whose
length is equal to the length of the plaintext.
The key is split into equal halves, K1 = leftmost(K, len(K)/2) and K2
= rightmost(K, len(K)/2). K1 is used for S2V and K2 is used for CTR.
In the encryption mode, the associated data and plaintext represent
the vector of inputs to S2V, with the plaintext being the last string
in the vector. The output of S2V is a synthetic IV that represents
the initial counter to CTR.
The encryption construction of SIV is as follows:
> +------+ +------+ +------+ +---+
> | AD 1 | | AD 2 |...| AD n | | P |
> +------+ +------+ +------+ +---+
> | | | |
> | | ... | ------------------|
> \ | / / |
> \ | / / +------------+ |
> \ | / / | K = K1||K2 | |
> \ | / / +------------+ V
> \ | / / | | +-----+
> \ | / / K1 | | K2 | |
> \ | / / ------/ \------>| CTR |
> \ | / / / ------->| |
> | | | | | | +-----+
> V V V V V | |
> +------------+ +--------+ V
> | S2V |------>| V | +----+
> +------------+ +--------+ | C |
> | +----+
> | |
> -----\ |
> \ |
> \ |
> V V
> +-----+
> | Z |
> +-----+
where the plaintext is P, the associated data is AD1 through ADn, V
is the synthetic IV, the ciphertext is C, and Z is the output.
Algorithmically, SIV Encrypt can be described as:
> SIV-ENCRYPT(K, P, AD1, ..., ADn) {
> K1 = leftmost(K, len(K)/2)
> K2 = rightmost(K, len(K)/2)
> V = S2V(K1, AD1, ..., ADn, P)
> Q = V bitand (1^64 || 0^1 || 1^31 || 0^1 || 1^31)
> m = (len(P) + 127)/128
>
> for i = 0 to m-1 do
> Xi = AES(K2, Q+i)
> done
> X = leftmost(X0 || ... || Xm-1, len(P))
> C = P xor X
>
> return V || C
> }
```
sivEncrypt : {p, ad} (fin ad, 2^^71-128 >= p) => [256] -> [ad] -> [p] -> [128 + p]
sivEncrypt K ad P = V # C
where
[K1,K2] = split K
V = S2V K1 ad P
Q = bitand V (repeat `{64} True # [False] # repeat `{31} True # [False] # repeat `{31} True)
C = ctr64 K2 Q P
```
where the key length used by AES in CTR and S2V is len(K)/2 and will
each be either 128 bits, 192 bits, or 256 bits.
The 31st and 63rd bit (where the rightmost bit is the 0th) of the
counter are zeroed out just prior to being used by CTR for
optimization purposes, see Section 5.
2.7. SIV Decrypt
SIV-decrypt takes as input a key K of length 256, 384, or 512 bits,
Z, which represents a synthetic initialization vector V concatenated
with a ciphertext C, and a vector of associated data AD[ ] where the
number of components in the vector is not greater than 126 (see
Section 7). It produces either the original plaintext or the special
symbol FAIL.
The key is split as specified in Section 2.6
The synthetic initialization vector acts as the initial counter to
CTR to decrypt the ciphertext. The associated data and the output of
CTR represent a vector of strings that is passed to S2V, with the CTR
output being the last string in the vector. The output of S2V is
then compared against the synthetic IV that accompanied the original
ciphertext. If they match, the output from CTR is returned as the
decrypted and authenticated plaintext; otherwise, the special symbol
FAIL is returned.
The decryption construction of SIV is as follows:
> +------+ +------+ +------+ +---+
> | AD 1 | | AD 2 |...| AD n | | P |
> +------+ +------+ +------+ +---+
> | | | ^
> | | ... / |
> | | / /----------------|
> | | / / |
> \ | / / +------------+ |
> \ | / / | K = K1||k2 | |
> \ | / / +------------+ |
> \ | / / | | +-----+
> \ | / / K1 | | K2 | |
> \ | | | /-----/ \----->| CTR |
> \ | | | | ------->| |
> | | | | | | +-----+
> V V V V V | ^
> +-------------+ +--------+ |
> | S2V | | V | +---+
> +-------------+ +--------+ | C |
> | | ^ +---+
> | | | ^
> | | \ |
> | | \___ |
> V V \ |
> +-------+ +---------+ +---+
> | T |----->| if != | | Z |
> +-------+ +---------+ +---+
> |
> |
> V
> FAIL
>
> Figure 10
Algorithmically, SIV-Decrypt can be described as:
> SIV-DECRYPT(K, Z, AD1, ..., ADn) {
> V = leftmost(Z, 128)
> C = rightmost(Z, len(Z)-128)
> K1 = leftmost(K, len(K)/2)
> K2 = rightmost(K, len(K)/2)
> Q = V bitand (1^64 || 0^1 || 1^31 || 0^1 || 1^31)
>
> m = (len(C) + 127)/128
> for i = 0 to m-1 do
> Xi = AES(K2, Q+i)
> done
> X = leftmost(X0 || ... || Xm-1, len(C))
> P = C xor X
> T = S2V(K1, AD1, ..., ADn, P)
>
> if T = V then
> return P
> else
> return FAIL
> fi
> }
```
sivDecrypt : {p, ad} (2^^71 - 128 >= p, fin ad) => [256] -> [ad] -> [p+128] -> (Bit, [p])
sivDecrypt K ad Z = (T == V, P)
where
V = take `{128} Z
C = drop `{back=p} Z
[K1,K2] = split `{parts=2} K
Q = bitand V (repeat `{64} True # [False] # repeat `{31} True # [False] # repeat `{31} True)
P = ctr64 K2 Q C
T = S2V K1 ad P
```
where the key length used by AES in CTR and S2V is len(K)/2 and will
each be either 128 bits, 192 bits, or 256 bits.
The 31st and 63rd bit (where the rightmost bit is the 0th) of the
counter are zeroed out just prior to being used in CTR mode for
optimization purposes, see Section 5.
3. Nonce-Based Authenticated Encryption with SIV
SIV performs nonce-based authenticated encryption when a component of
the associated data is a nonce. For purposes of interoperability the
final component -- i.e., the string immediately preceding the
plaintext in the vector input to S2V -- is used for the nonce. Other
associated data are optional. It is up to the specific application
of SIV to specify how the rest of the associated data are input.
If the nonce is random, it SHOULD be at least 128 bits in length and
be harvested from a pool having at least 128 bits of entropy. A non-
random source MAY also be used, for instance, a time stamp, or a
counter. The definition of a nonce precludes reuse, but SIV is
resistant to nonce reuse. See Section 1.3.2 for a discussion on the
security implications of nonce reuse.
It MAY be necessary to transport this nonce with the output generated
by S2V.
4. Deterministic Authenticated Encryption with SIV
When the plaintext to encrypt and authenticate contains data that is
unpredictable to an adversary -- for example, a secret key -- SIV can
be used in a deterministic mode to perform "key wrapping". Because
S2V allows for associated data and imposes no unnatural size
restrictions on the data it is protecting, it is a more useful and
general purpose solution than [RFC3394]. Protocols that use SIV for
deterministic authenticated encryption (i.e., for more than just
wrapping of keys) MAY define associated data inputs to SIV. It is
not necessary to add a nonce component to the AD in this case.
5. Optimizations
Implementations that cannot or do not wish to support addition modulo
2^128 can take advantage of the fact that the 31st and 63rd bits
(where the rightmost bit is the 0th bit) in the counter are cleared
before being used by CTR. This allows implementations that natively
support 32-bit or 64-bit addition to increment the counter naturally.
Of course, in this case, the amount of plaintext that can be safely
protected by SIV is reduced by a commensurate amount -- addition
modulo 2^32 limits plaintext to (2^39 - 128) bits, addition modulo
2^64 limits plaintext to (2^71 - 128) bits.
It is possible to optimize an implementation of S2V when it is being
used as a key derivation function (KDF), for example in [WLAN]. This
is because S2V operates on a vector of distinct strings and typically
the data passed to a KDF contains constant strings. Depending on the
location of variant components of the input different optimizations
are possible. The CMACed output of intermediate and invariant
components can be computed once and cached. This can then be doubled
and xored with the running sum to produce the output. Or an
intermediate value that represents the doubled and xored output of
multiple components, up to the variant component, can be computed
once and cached.
6. IANA Considerations
[RFC5116] defines a uniform interface to cipher modes that provide
nonce-based Authenticated Encryption with Associated Data (AEAD). It
does this via a registry of AEAD algorithms.
The Internet Assigned Numbers Authority (IANA) assigned three entries
from the AEAD Registry for AES-SIV-CMAC-256 (15), AES-SIV-CMAC-384
(16), and AES-SIV-CMAC-512 (17) based upon the following AEAD
algorithm definitions. [RFC5116] defines operations in octets, not
bits. Limits in this section will therefore be specified in octets.
The security analysis for each of these algorithms is in [DAE].
Unfortunately, [RFC5116] restricts AD input to a single component and
limits the benefit SIV offers for dealing in a natural fashion with
AD consisting of multiple distinct components. Therefore, when it is
required to access SIV through the interface defined in [RFC5116], it
is necessary to marshal multiple AD inputs into a single string (see
Section 1.1) prior to invoking SIV. Note that this requirement is
not unique to SIV. All cipher modes using [RFC5116] MUST similarly
marshal multiple AD inputs into a single string, and any technique
used for any other AEAD mode (e.g., a scatter/gather technique) can
be used with SIV.
[RFC5116] requires AEAD algorithm specifications to include maximal
limits to the amount of plaintext, the amount of associated data, and
the size of a nonce that the AEAD algorithm can accept.
SIV uses AES in counter mode and the security guarantees of SIV would
be lost if the counter was allowed to repeat. Since the counter is
128 bits, a limit to the amount of plaintext that can be safely
protected by a single invocation of SIV is 2^128 blocks.
To prevent the possibility of collisions, [CMAC] recommends that no
more than 2^48 invocations be made to CMAC with the same key. This
is not a limit on the amount of data that can be passed to CMAC,
though. There is no practical limit to the amount of data that can
be made to a single invocation of CMAC, and likewise, there is no
practical limit to the amount of associated data or nonce material
that can be passed to SIV.
A collision in the output of S2V would mean the same counter would be
used with different plaintext in counter mode. This would void the
security guarantees of SIV. The "Birthday Paradox" (see [APPCRY])
would imply that no more than 2^64 distinct invocations to SIV be
made with the same key. It is prudent to follow the example of
[CMAC] though, and further limit the number of distinct invocations
of SIV using the same key to 2^48. Note that [RFC5116] does not
provide a variable to describe this limit.
The counter-space for SIV is 2^128. Each invocation of SIV consumes
a portion of that counter-space and the amount consumed depends on
the amount of plaintext being passed to that single invocation.
Multiple invocations of SIV with the same key can increase the
possibility of distinct invocations overlapping the counter-space.
The total amount of plaintext that can be safely protected with a
single key is, therefore, a function of the number of distinct
invocations and the amount of plaintext protected with each
invocation.
6.1. AEAD_AES_SIV_CMAC_256
The AES-SIV-CMAC-256 AEAD algorithm works as specified in Sections
2.6 and 2.7. The input and output lengths for AES-SIV-CMAC-256 as
defined by [RFC5116] are:
K_LEN is 32 octets.
P_MAX is 2^132 octets.
A_MAX is unlimited.
N_MIN is 1 octet.
N_MAX is unlimited.
C_MAX is 2^132 + 16 octets.
The security implications of nonce reuse and/or misuse are described
in Section 1.3.2.
6.2. AEAD_AES_SIV_CMAC_384
The AES-SIV-CMAC-384 AEAD algorithm works as specified in Sections
2.6 and 2.7. The input and output lengths for AES-SIV-CMAC-384 as
defined by [RFC5116] are:
K_LEN is 48 octets.
P_MAX is 2^132 octets.
A_MAX is unlimited.
N_MIN is 1 octet.
N_MAX is unlimited.
C_MAX is 2^132 + 16 octets.
The security implications of nonce reuse and/or misuse are described
in Section 1.3.2.
6.3. AEAD_AES_SIV_CMAC_512
The AES-SIV-CMAC-512 AEAD algorithm works as specified in Sections
2.6 and 2.7. The input and output lengths for AES-SIV-CMAC-512 as
defined by [RFC5116] are:
K_LEN is 64 octets.
P_MAX is 2^132 octets.
A_MAX is unlimited.
N_MIN is 1 octet.
N_MAX is unlimited.
C_MAX is 2^132 + 16 octets.
The security implications of nonce reuse and/or misuse are described
in Section 1.3.2.
7. Security Considerations
SIV provides confidentiality in the sense that the output of SIV-
Encrypt is indistinguishable from a random string of bits. It
provides authenticity in the sense that an attacker is unable to
construct a string of bits that will return other than FAIL when
input to SIV-Decrypt. A proof of the security of SIV with an "all-
in-one" notion of security for an authenticated encryption scheme is
provided in [DAE].
SIV provides deterministic "key wrapping" when the plaintext contains
data that is unpredictable to an adversary (for instance, a
cryptographic key). Even when this key is made available to an
attacker the output of SIV-Encrypt is indistinguishable from random
bits. Similarly, even when this key is made available to an
attacker, she is unable to construct a string of bits that when input
to SIV-Decrypt will return anything other than FAIL.
When the nonce used in the nonce-based authenticated encryption mode
of SIV-AES is treated with the care afforded a nonce or counter in
other conventional nonce-based authenticated encryption schemes --
i.e., guarantee that it will never be used with the same key for two
distinct invocations -- then SIV achieves the level of security
described above. If, however, the nonce is reused SIV continues to
provide the level of authenticity described above but with a slightly
reduced amount of privacy (see Section 1.3.2).
If S2V is used as a key derivation function, the secret input MUST be
generated uniformly at random. S2V is a pseudo-random function and
is not suitable for use as a random oracle as defined in [RANDORCL].
The security bound set by the proof of security of S2V in [DAE]
depends on the number of vector-based queries made by an adversary
and the total number of all components in those queries. The
security is only proven when the number of components in each query
is limited to n-1, where n is the blocksize of the underlying pseudo-
random function. The underlying pseudo-random function used here is
based on AES whose blocksize is 128 bits. Therefore, S2V must not be
passed more than 127 components. Since SIV includes the plaintext as
a component to S2V, that limits the number of components of
associated data that can be safely passed to SIV to 126.
8. Acknowledgments
Thanks to Phil Rogaway for patiently answering numerous questions on
SIV and S2V and for useful critiques of earlier versions of this
paper. Thanks also to David McGrew for numerous helpful comments and
suggestions for improving this paper. Thanks to Jouni Malinen for
reviewing this paper and producing another independent implementation
of SIV, thereby confirming the correctness of the test vectors.
9. References
9.1. Normative References
[CMAC] Dworkin, M., "Recommendation for Block Cipher Modes of
Operation: The CMAC Mode for Authentication", NIST
Special Pulication 800-38B, May 2005.
[MODES] Dworkin, M., "Recommendation for Block Cipher Modes of
Operation: Methods and Techniques", NIST Special
Pulication 800-38A, 2001 edition.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC5116] McGrew, D., "An Interface and Algorithms for
Authenticated Encryption", RFC 5116, January 2008.
9.2. Informative References
[APPCRY] Menezes, A., van Oorshot, P., and S. Vanstone, "Handbook
of Applied Cryptography", CRC Press Series on Discrete
Mathematics and Its Applications, 1996.
[BADESP] Bellovin, S., "Problem Areas for the IP Security
Protocols", Proceedings from the 6th Usenix UNIX Security
Symposium, July 22-25 1996.
[RFC3610] Whiting, D., Housley, R., and N. Ferguson, "Counter with
CBC-MAC (CCM)", RFC 3610, September 2003.
[DAE] Rogaway, P. and T. Shrimpton, "Deterministic
Authenticated Encryption, A Provable-Security Treatment
of the Key-Wrap Problem", Advances in Cryptology --
EUROCRYPT '06 St. Petersburg, Russia, 2006.
[GCM] McGrew, D. and J. Viega, "The Galois/Counter Mode of
Operation (GCM)".
[JUTLA] Jutla, C., "Encryption Modes With Almost Free Message
Integrity", Proceedings of the International Conference
on the Theory and Application of Cryptographic
Techniques: Advances in Cryptography.
[OCB] Krovetz, T. and P. Rogaway, "The OCB Authenticated
Encryption Algorithm", Work in Progress, March 2005.
[RADKEY] Zorn, G., Zhang, T., Walker, J., and J. Salowey, "RADIUS
Attributes for the Delivery of Keying Material", Work in
Progress, April 2007.
[RANDORCL] Bellare, M. and P. Rogaway, "Random Oracles are
Practical: A Paradigm for Designing Efficient
Protocols", Proceeding of the First ACM Conference on
Computer and Communications Security, November 1993.
[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
Hashing for Message Authentication", RFC 2104, February
1997.
[RFC2548] Zorn, G., "Microsoft Vendor-specific RADIUS Attributes",
RFC 2548, March 1999.
[RFC2865] Rigney, C., Willens, S., Rubens, A., and W. Simpson,
"Remote Authentication Dial In User Service (RADIUS)",
RFC 2865, June 2000.
[RFC3217] Housley, R., "Triple-DES and RC2 Key Wrapping", RFC 3217,
December 2001.
[RFC3394] Schaad, J. and R. Housley, "Advanced Encryption Standard
(AES) Key Wrap Algorithm", RFC 3394, September 2002.
[SP800-38D] Dworkin, M., "Recommendations for Block Cipher Modes of
Operation: Galois Counter Mode (GCM) and GMAC", NIST
Special Pulication 800-38D, June 2007.
[VIRT] Garfinkel, T. and M. Rosenblum, "When Virtual is Harder
than Real: Security Challenges in Virtual Machine Based
Computing Environments" In 10th Workshop on Hot Topics in
Operating Systems, May 2005.
[WLAN] "Draft Standard for IEEE802.11: Wireless LAN Medium
Access Control (MAC) and Physical Layer (PHY)
Specification", 2007.
[X9F1] Dworkin, M., "Wrapping of Keys and Associated Data",
Request for review of key wrap algorithms. Cryptology
ePrint report 2004/340, 2004. Contents are excerpts from
a draft standard of the Accredited Standards Committee,
X9, entitled ANS X9.102.
Appendix A. Test Vectors
The following test vectors are for the mode defined in Section 6.1.
A.1. Deterministic Authenticated Encryption Example
Input:
-----
Key:
fffefdfc fbfaf9f8 f7f6f5f4 f3f2f1f0
f0f1f2f3 f4f5f6f7 f8f9fafb fcfdfeff
AD:
10111213 14151617 18191a1b 1c1d1e1f
20212223 24252627
Plaintext:
11223344 55667788 99aabbcc ddee
```
property KAT1 =
sivEncrypt 0xfffefdfcfbfaf9f8f7f6f5f4f3f2f1f0f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff
0x101112131415161718191a1b1c1d1e1f2021222324252627
0x112233445566778899aabbccddee
== 0x85632d07c6e8f37f950acd320a2ecc9340c02b9690c4dc04daef7f6afe5c
enc_dec : {ad,p} (fin ad, fin p, 2^^71 - 128 >= p) => [256] -> [ad] -> [p] -> Bit
property enc_dec k ad p = (sivDecrypt k ad (sivEncrypt k ad p)).1 == p
// `encrypt . decrypt ~ ident` for inputs larger, equal to, and smaller than the block size
// Expect ~20 minutes for ABC to prove each property
property enc_dec_ll k ad p = enc_dec `{431,500} k ad p
property enc_dec_el k ad p = enc_dec `{128,140} k ad p
property enc_dec_le k ad p = enc_dec `{148,128} k ad p
property enc_dec_se k ad p = enc_dec `{18,128} k ad p
property enc_dec_es k ad p = enc_dec `{128,18} k ad p
property enc_dec_sl k ad p = enc_dec `{12,180} k ad p
property enc_dec_ls k ad p = enc_dec `{140,18} k ad p
property enc_dec_ee k ad p = enc_dec `{128,128} k ad p
property enc_dec_ss k ad p = enc_dec `{50,80} k ad p
property enc_dec_block k ad p = (sivDecrypt k (ad : [50]) (sivEncrypt k ad p)).1 == (p : [500])
```
S2V-CMAC-AES
------------
CMAC(zero):
0e04dfaf c1efbf04 01405828 59bf073a
double():
1c09bf5f 83df7e08 0280b050 b37e0e74
CMAC(ad):
f1f922b7 f5193ce6 4ff80cb4 7d93f23b
xor:
edf09de8 76c642ee 4d78bce4 ceedfc4f
double():
dbe13bd0 ed8c85dc 9af179c9 9ddbf819
pad:
11223344 55667788 99aabbcc ddee8000
xor:
cac30894 b8eaf254 035bc205 40357819
CMAC(final):
85632d07 c6e8f37f 950acd32 0a2ecc93
CTR-AES
-------
CTR:
85632d07 c6e8f37f 150acd32 0a2ecc93
E(K,CTR):
51e218d2 c5a2ab8c 4345c4a6 23b2f08f
ciphertext:
40c02b96 90c4dc04 daef7f6a fe5c
output
------
IV || C:
85632d07 c6e8f37f 950acd32 0a2ecc93
40c02b96 90c4dc04 daef7f6a fe5c
A.2. Nonce-Based Authenticated Encryption Example
Input:
-----
Key:
7f7e7d7c 7b7a7978 77767574 73727170
40414243 44454647 48494a4b 4c4d4e4f
AD1:
00112233 44556677 8899aabb ccddeeff
deaddada deaddada ffeeddcc bbaa9988
77665544 33221100
AD2:
10203040 50607080 90a0
Nonce:
09f91102 9d74e35b d84156c5 635688c0
Plaintext:
74686973 20697320 736f6d65 20706c61
696e7465 78742074 6f20656e 63727970
74207573 696e6720 5349562d 414553
S2V-CMAC-AES
------------
CMAC(zero):
c8b43b59 74960e7c e6a5dd85 231e591a
double():
916876b2 e92c1cf9 cd4bbb0a 463cb2b3
CMAC(ad1)
3c9b689a b41102e4 80954714 1dd0d15a
xor:
adf31e28 5d3d1e1d 4ddefc1e 5bec63e9
double():
5be63c50 ba7a3c3a 9bbdf83c b7d8c755
CMAC(ad2)
d98c9b0b e42cb2d7 aa98478e d11eda1b
xor:
826aa75b 5e568eed 3125bfb2 66c61d4e
double():
04d54eb6 bcad1dda 624b7f64 cd8c3a1b
CMAC(nonce)
128c62a1 ce3747a8 372c1c05 a538b96d
xor:
16592c17 729a5a72 55676361 68b48376
xorend:
74686973 20697320 736f6d65 20706c61
696e7465 78742074 6f20656e 63727966
2d0c6201 f3341575 342a3745 f5c625
CMAC(final)
7bdb6e3b 432667eb 06f4d14b ff2fbd0f
CTR-AES
-------
CTR:
7bdb6e3b 432667eb 06f4d14b 7f2fbd0f
E(K,CTR):
bff8665c fdd73363 550f7400 e8f9d376
CTR+1:
7bdb6e3b 432667eb 06f4d14b 7f2fbd10
E(K,CTR+1):
b2c9088e 713b8617 d8839226 d9f88159
CTR+2
7bdb6e3b 432667eb 06f4d14b 7f2fbd11
E(K,CTR+2):
9e44d827 234949bc 1b12348e bc195ec7
ciphertext:
cb900f2f ddbe4043 26601965 c889bf17
dba77ceb 094fa663 b7a3f748 ba8af829
ea64ad54 4a272e9c 485b62a3 fd5c0d
output
------
IV || C:
7bdb6e3b 432667eb 06f4d14b ff2fbd0f
cb900f2f ddbe4043 26601965 c889bf17
dba77ceb 094fa663 b7a3f748 ba8af829
ea64ad54 4a272e9c 485b62a3 fd5c0d
Author's Address
Dan Harkins
Aruba Networks
EMail: dharkins@arubanetworks.com
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