Network Working Group F. Günther
Internet-Draft IBM Research Europe - Zurich
Intended status: Informational M. Thomson
Expires: 13 April 2025 Mozilla
C. A. Wood
Cloudflare
10 October 2024
Usage Limits on AEAD Algorithms
draft-irtf-cfrg-aead-limits-09
Abstract
An Authenticated Encryption with Associated Data (AEAD) algorithm
provides confidentiality and integrity. Excessive use of the same
key can give an attacker advantages in breaking these properties.
This document provides simple guidance for users of common AEAD
functions about how to limit the use of keys in order to bound the
advantage given to an attacker. It considers limits in both single-
and multi-key settings.
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg.
Source for this draft and an issue tracker can be found at
https://github.com/cfrg/draft-irtf-cfrg-aead-limits.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
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Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
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material or to cite them other than as "work in progress."
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This Internet-Draft will expire on 13 April 2025.
Copyright Notice
Copyright (c) 2024 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights
and restrictions with respect to this document.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 4
3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4. Calculating Limits . . . . . . . . . . . . . . . . . . . . . 6
4.1. Approximations . . . . . . . . . . . . . . . . . . . . . 7
5. Single-Key AEAD Limits . . . . . . . . . . . . . . . . . . . 8
5.1. AEAD_AES_128_GCM and AEAD_AES_256_GCM . . . . . . . . . . 9
5.1.1. Confidentiality Limit . . . . . . . . . . . . . . . . 9
5.1.2. Integrity Limit . . . . . . . . . . . . . . . . . . . 9
5.2. AEAD_CHACHA20_POLY1305 . . . . . . . . . . . . . . . . . 9
5.2.1. Confidentiality Limit . . . . . . . . . . . . . . . . 10
5.2.2. Integrity Limit . . . . . . . . . . . . . . . . . . . 10
5.3. AEAD_AES_128_CCM . . . . . . . . . . . . . . . . . . . . 10
5.3.1. Confidentiality Limit . . . . . . . . . . . . . . . . 10
5.3.2. Integrity Limit . . . . . . . . . . . . . . . . . . . 10
5.4. AEAD_AES_128_CCM_8 . . . . . . . . . . . . . . . . . . . 11
5.5. Single-Key Examples . . . . . . . . . . . . . . . . . . . 11
6. Multi-Key AEAD Limits . . . . . . . . . . . . . . . . . . . . 12
6.1. AEAD_AES_128_GCM and AEAD_AES_256_GCM . . . . . . . . . . 13
6.1.1. Authenticated Encryption Security Limit . . . . . . . 13
6.1.2. Confidentiality Limit . . . . . . . . . . . . . . . . 14
6.1.3. Integrity Limit . . . . . . . . . . . . . . . . . . . 14
6.2. AEAD_CHACHA20_POLY1305 . . . . . . . . . . . . . . . . . 14
6.2.1. Authenticated Encryption Security Limit . . . . . . . 14
6.2.2. Confidentiality Limit . . . . . . . . . . . . . . . . 15
6.2.3. Integrity Limit . . . . . . . . . . . . . . . . . . . 15
6.3. AEAD_AES_128_CCM and AEAD_AES_128_CCM_8 . . . . . . . . . 15
6.4. Multi-Key Examples . . . . . . . . . . . . . . . . . . . 16
7. Security Considerations . . . . . . . . . . . . . . . . . . . 17
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 17
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 17
9.1. Normative References . . . . . . . . . . . . . . . . . . 17
9.2. Informative References . . . . . . . . . . . . . . . . . 19
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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 20
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 20
1. Introduction
An Authenticated Encryption with Associated Data (AEAD) algorithm
provides confidentiality and integrity. [RFC5116] specifies an AEAD
as a function with four inputs -- secret key, nonce, plaintext,
associated data (of which nonce, plaintext, and associated data can
optionally be zero-length) -- that produces ciphertext output and an
error code indicating success or failure. The ciphertext is
typically composed of the encrypted plaintext bytes and an
authentication tag.
The generic AEAD interface does not describe usage limits. Each AEAD
algorithm does describe limits on its inputs, but these are
formulated as strict functional limits, such as the maximum length of
inputs, which are determined by the properties of the underlying AEAD
composition. Degradation of the security of the AEAD as a single key
is used multiple times is not given the same thorough treatment.
Effective limits can be influenced by the number of "users" of a
given key. In the traditional setting, there is one key shared
between two parties. Any limits on the maximum length of inputs or
encryption operations apply to that single key. The attacker's goal
is to break security (confidentiality or integrity) of that specific
key. However, in practice, there are often many parties with
independent keys, multiple sessions between two parties, and even
many keys used within a single session due to rekeying. This multi-
key security setting, often referred to as the multi-user setting in
the academic literature, considers an attacker's advantage in
breaking security of any of these many keys, further assuming the
attacker may have done some offline work (measuring time, but not
memory) to help break any key. As a result, AEAD algorithm limits
may depend on offline work and the number of keys. However, given
that a multi-key attacker does not target any specific key,
acceptable advantages may differ from that of the single-key setting.
The number of times a single pair of key and nonce can be used might
also be relevant to security. For some algorithms, such as
AEAD_AES_128_GCM or AEAD_AES_256_GCM, this limit is 1 and using the
same pair of key and nonce has serious consequences for both
confidentiality and integrity; see [NonceDisrespecting]. Nonce-reuse
resistant algorithms like AEAD_AES_128_GCM_SIV can tolerate a limited
amount of nonce reuse. This document focuses on AEAD schemes
requiring non-repeating nonces.
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It is good practice to have limits on how many times the same key (or
pair of key and nonce) are used. Setting a limit based on some
measurable property of the usage, such as number of protected
messages, amount of data transferred, or time passed ensures that it
is easy to apply limits. This might require the application of
simplifying assumptions. For example, TLS 1.3 and QUIC both specify
limits on the number of records that can be protected, using the
simplifying assumption that records are the same size; see
Section 5.5 of [TLS] and Section 6.6 of [RFC9001].
Exceeding the determined usage limit can be avoided using rekeying.
Rekeying uses a lightweight transform to produce new keys. Rekeying
effectively resets progress toward single-key limits, allowing a
session to be extended without degrading security. Rekeying can also
provide a measure of forward and backward (post-compromise) security.
[RFC8645] contains a thorough survey of rekeying and the consequences
of different design choices. When considering rekeying, the multi-
user limits should be applied.
Currently, AEAD limits and usage requirements are scattered among
peer-reviewed papers, standards documents, and other RFCs.
Determining the correct limits for a given setting is challenging as
papers do not use consistent labels or conventions, and rarely apply
any simplifications that might aid in reaching a simple limit.
The intent of this document is to collate all relevant information
about the proper usage and limits of AEAD algorithms in one place.
This may serve as a standard reference when considering which AEAD
algorithm to use, and how to use it.
2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
3. Notation
This document defines limitations in part using the quantities in
Table 1 below.
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+========+===========================================+
| Symbol | Description |
+========+===========================================+
| n | AEAD block length (in bits), of the |
| | underlying block cipher |
+--------+-------------------------------------------+
| k | AEAD key length (in bits) |
+--------+-------------------------------------------+
| r | AEAD nonce length (in bits) |
+--------+-------------------------------------------+
| t | Size of the authentication tag (in bits) |
+--------+-------------------------------------------+
| L | Maximum length of each message, including |
| | both plaintext and AAD (in blocks) |
+--------+-------------------------------------------+
| s | Total plaintext length in all messages |
| | (in blocks) |
+--------+-------------------------------------------+
| q | Number of protected messages (AEAD |
| | encryption invocations) |
+--------+-------------------------------------------+
| v | Number of attacker forgery attempts |
| | (failed AEAD decryption invocations + 1) |
+--------+-------------------------------------------+
| p | Upper bound on adversary attack |
| | probability |
+--------+-------------------------------------------+
| o | Offline adversary work (time, measured in |
| | number of encryption and decryption |
| | queries; multi-key setting only) |
+--------+-------------------------------------------+
| u | Number of keys (multi-key setting only) |
+--------+-------------------------------------------+
| B | Maximum number of blocks encrypted by any |
| | key (multi-key setting only) |
+--------+-------------------------------------------+
Table 1: Notation
For each AEAD algorithm, we define the chosen-plaintext
confidentiality (IND-CPA) and ciphertext integrity (INT-CTXT)
advantage roughly as the advantage an attacker has in breaking the
corresponding classical security property for the algorithm. An IND-
CPA attacker can query ciphertexts for arbitrary plaintexts. An INT-
CTXT attacker can additionally query plaintexts for arbitrary
ciphertexts. Moreover, we define the combined authenticated
encryption advantage guaranteeing both confidentiality and integrity
against an active attacker. Specifically:
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* Confidentiality advantage (CA): The probability of an attacker
succeeding in breaking the IND-CPA (confidentiality) properties of
the AEAD scheme. In this document, the definition of
confidentiality advantage roughly is the probability that an
attacker successfully distinguishes the ciphertext outputs of the
AEAD scheme from the outputs of a random function.
* Integrity advantage (IA): The probability of an attacker
succeeding in breaking the INT-CTXT (integrity) properties of the
AEAD scheme. In this document, the definition of integrity
advantage roughly is the probability that an attacker is able to
forge a ciphertext that will be accepted as valid.
* Authenticated Encryption advantage (AEA): The probability of an
active attacker succeeding in breaking the authenticated-
encryption properties of the AEAD scheme. In this document, the
definition of authenticated encryption advantage roughly is the
probability that an attacker successfully distinguishes the
ciphertext outputs of the AEAD scheme from the outputs of a random
function or is able to forge a ciphertext that will be accepted as
valid.
Here, we consider advantages beyond distinguishing underyling
primitives from their ideal instances (for example, a pseudorandom
from a truly random function).
See [AEComposition], [AEAD] for the formal definitions of and
relations between IND-CPA (confidentiality), INT-CTXT (integrity),
and authenticated encryption security (AE). The authenticated
encryption advantage subsumes, and can be derived as the combination
of, both CA and IA:
CA <= AEA
IA <= AEA
AEA <= CA + IA
Each application requires an individual determination of limits in
order to keep CA and IA sufficiently small. For instance, TLS aims
to keep CA below 2^-60 and IA below 2^-57 in the single-key setting;
see Section 5.5 of [TLS].
4. Calculating Limits
Once upper bounds on CA, IA, or AEA are determined, this document
defines a process for determining three overall operational limits:
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* Confidentiality limit (CL): The number of messages an application
can encrypt before giving the adversary a confidentiality
advantage higher than CA.
* Integrity limit (IL): The number of ciphertexts an application can
decrypt unsuccessfully before giving the adversary an integrity
advantage higher than IA.
* Authenticated encryption limit (AEL): The combined number of
messages and number of ciphertexts an application can encrypt or
decrypt before giving the adversary an authenticated encryption
advantage higher than AEA.
When limits are expressed as a number of messages an application can
encrypt or decrypt, this requires assumptions about the size of
messages and any authenticated additional data (AAD). Limits can
instead be expressed in terms of the number of bytes, or blocks, of
plaintext and maybe AAD in total.
To aid in translating between message-based and byte/block-based
limits, a formulation of limits that includes a maximum message size
(L) and the AEAD schemes' block length in bits (n) is provided.
All limits are based on the total number of messages, either the
number of protected messages (q) or the number of forgery attempts
(v); which correspond to CL and IL respectively.
Limits are then derived from those bounds using a target attacker
probability. For example, given an integrity advantage of IA = v *
(8L / 2^106) and a targeted maximum attacker success probability of
IA = p, the algorithm remains secure, i.e., the adversary's advantage
does not exceed the targeted probability of success, provided that v
<= (p * 2^106) / 8L. In turn, this implies that v <= (p * 2^103) / L
is the corresponding limit.
To apply these limits, implementations can count the number of
messages that are protected or rejected against the determined limits
(q and v respectively). This requires that messages cannot exceed
the maximum message size (L) that is chosen.
4.1. Approximations
This analysis assumes a message-based approach to setting limits.
Implementations that use byte counting rather than message counting
could use a maximum message size (L) of one to determine a limit for
the number of protected messages (q) that can be applied with byte
counting. This results in attributing per-message overheads to every
byte, so the resulting limit could be significantly lower than
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necessary. Actions, like rekeying, that are taken to avoid the limit
might occur more often as a result.
To simplify formulae, estimates in this document elide terms that
contribute negligible advantage to an attacker relative to other
terms.
In other respects, this document seeks to make conservative choices
that err on the side of overestimating attacker advantage. Some of
these assumptions are present in the papers that this work is based
on. For instance, analyses are simplified by using a single message
size that covers both AAD and plaintext. AAD can contribute less
toward attacker advantage for confidentiality limits, so applications
where AAD comprises a significant proportion of messages might find
the estimates provided to be slightly more conservative than
necessary to meet a given goal.
This document assumes the use of non-repeating nonces (in particular,
non-zero-length nonces). The modes covered here are not robust if
the same nonce and key are used to protect different messages, so
deterministic generation of nonces from a counter or similar
techniques is strongly encouraged. If an application cannot
guarantee that nonces will not repeat, a nonce-misuse resistant AEAD
like AES-GCM-SIV [SIV] is likely to be a better choice.
5. Single-Key AEAD Limits
This section summarizes the confidentiality and integrity bounds and
limits for modern AEAD algorithms used in IETF protocols, including:
AEAD_AES_128_GCM [RFC5116], AEAD_AES_256_GCM [RFC5116],
AEAD_AES_128_CCM [RFC5116], AEAD_CHACHA20_POLY1305 [RFC8439],
AEAD_AES_128_CCM_8 [RFC6655]. The limits in this section apply to
using these schemes with a single key; for settings where multiple
keys are deployed (for example, when rekeying within a connection),
see Section 6.
These algorithms, as cited, all define a nonce length (r) of 96 bits.
Some definitions of these AEAD algorithms allow for other nonce
lengths, but the analyses in this document all fix the nonce length
to r = 96. Using other nonce lengths might result in different
bounds; for example, [GCMProofs] shows that using a variable-length
nonce for AES-GCM results in worse security bounds.
The CL and IL values bound the total number of encryption and forgery
queries (q and v). Alongside each advantage value, we also specify
these bounds.
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5.1. AEAD_AES_128_GCM and AEAD_AES_256_GCM
The CL and IL values for AES-GCM are derived in [AEBounds] and
summarized below. For this AEAD, n = 128 (the AES block length) and
t = 128 [GCM], [RFC5116]. In this example, the length s is the sum
of AAD and plaintext (in blocks of 128 bits), as described in
[GCMProofs].
5.1.1. Confidentiality Limit
CA <= ((s + q + 1)^2) / 2^129
This implies the following usage limit:
q + s <= p^(1/2) * 2^(129/2) - 1
Which, for a message-based protocol with s <= q * L, if we assume
that every packet is size L (in blocks of 128 bits), produces the
limit:
q <= (p^(1/2) * 2^(129/2) - 1) / (L + 1)
5.1.2. Integrity Limit
Applying Equation (22) from [GCMProofs], in which the assumption of s
+ q + v < 2^64 ensures that the delta function cannot produce a value
greater than 2, the following bound applies:
IA <= 2 * (v * (L + 1)) / 2^128
For the assumption of s + q + v < 2^64, observe that this applies
when p > L / 2^63. s + q <= q * (L + 1) is always small relative to
2^64 if the same advantage is applied to the confidentiality limit on
q.
This produces the following limit:
v <= min(2^64, (p * 2^127) / (L + 1))
5.2. AEAD_CHACHA20_POLY1305
The known single-user analyses for AEAD_CHACHA20_POLY1305
[ChaCha20Poly1305-SU], [ChaCha20Poly1305-MU] give a combined AE
limit, which we separate into confidentiality and integrity limits
below. For this AEAD, n = 512 (the ChaCha20 block length), k = 256,
and t = 128; the length L' is the sum of AAD and plaintext (in
Poly1305 blocks of 128 bits), see [ChaCha20Poly1305-MU].
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5.2.1. Confidentiality Limit
CA <= 0
This implies there is no limit beyond the PRF security of the
underlying ChaCha20 block function.
5.2.2. Integrity Limit
IA <= (v * (L' + 1)) / 2^103
This implies the following limit:
v <= (p * 2^103) / (L' + 1)
5.3. AEAD_AES_128_CCM
The CL and IL values for AEAD_AES_128_CCM are derived from
[CCM-ANALYSIS] and specified in the QUIC-TLS mapping specification
[RFC9001]. This analysis uses the total number of underlying block
cipher operations to derive its bound. For CCM, this number is the
sum of: the length of the associated data in blocks, the length of
the ciphertext in blocks, the length of the plaintext in blocks, plus
1.
In the following limits, this is simplified to a value of twice the
length of the packet in blocks, i.e., 2L represents the effective
length, in number of block cipher operations, of a message with L
blocks. This simplification is based on the observation that common
applications of this AEAD carry only a small amount of associated
data compared to ciphertext. For example, QUIC has 1 to 3 blocks of
AAD.
For this AEAD, n = 128 (the AES block length) and t = 128.
5.3.1. Confidentiality Limit
CA <= (2L * q)^2 / 2^n
<= (2L * q)^2 / 2^128
This implies the following limit:
q <= sqrt(p) * 2^63 / L
5.3.2. Integrity Limit
IA <= v / 2^t + (2L * (v + q))^2 / 2^n
<= v / 2^128 + (2L * (v + q))^2 / 2^128
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This implies the following limit:
v + (2L * (v + q))^2 <= p * 2^128
In a setting where v or q is sufficiently large, v is negligible
compared to (2L * (v + q))^2, so this this can be simplified to:
v + q <= sqrt(p) * 2^63 / L
5.4. AEAD_AES_128_CCM_8
The analysis in [CCM-ANALYSIS] also applies to this AEAD, but the
reduced tag length of 64 bits changes the integrity limit calculation
considerably.
IA <= v / 2^t + (2L * (v + q))^2 / 2^n
<= v / 2^64 + (2L * (v + q))^2 / 2^128
This results in reducing the limit on v by a factor of 2^64.
v * 2^64 + (2L * (v + q))^2 <= p * 2^128
Note that, to apply this result, two inequalities can be produced,
with the first applied to determine v, then applying the second to
find q:
v * 2^64 <= p * 2^127
(2L * (v + q))^2 <= p * 2^127
This approach produces much smaller values for v than for q.
Alternative allocations tend to greatly reduce q without
significantly increasing v.
5.5. Single-Key Examples
An example protocol might choose to aim for a single-key CA and IA
that is at most 2^-50. If the messages exchanged in the protocol are
at most a common Internet MTU of around 1500 bytes, then a value for
L might be set to 2^7. Table 2 shows limits for q and v that might
be chosen under these conditions.
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+========================+===========+===========+
| AEAD | Maximum q | Maximum v |
+========================+===========+===========+
| AEAD_AES_128_GCM | 2^32.5 | 2^64 |
+------------------------+-----------+-----------+
| AEAD_AES_256_GCM | 2^32.5 | 2^64 |
+------------------------+-----------+-----------+
| AEAD_CHACHA20_POLY1305 | n/a | 2^46 |
+------------------------+-----------+-----------+
| AEAD_AES_128_CCM | 2^30 | 2^30 |
+------------------------+-----------+-----------+
| AEAD_AES_128_CCM_8 | 2^30.4 | 2^13 |
+------------------------+-----------+-----------+
Table 2: Example single-key limits
AEAD_CHACHA20_POLY1305 provides no limit to q based on the provided
single-user analyses.
The limit for q on AEAD_AES_128_CCM and AEAD_AES_128_CCM_8 is reduced
due to a need to reduce the value of q to ensure that IA does not
exceed the target. This assumes equal proportions for q and v for
AEAD_AES_128_CCM. AEAD_AES_128_CCM_8 permits a much smaller value of
v due to the shorter tag, which permits a higher limit for q.
Some protocols naturally limit v to 1, such as TCP-based variants of
TLS, which terminate sessions on decryption failure. If v is limited
to 1, q can be increased to 2^31 for both CCM AEADs.
6. Multi-Key AEAD Limits
In the multi-key setting, each user is assumed to have an independent
and uniformly distributed key, though nonces may be re-used across
users with some very small probability. The success probability in
attacking one of these many independent keys can be generically
bounded by the success probability of attacking a single key
multiplied by the number of keys present [MUSecurity], [GCM-MU].
Absent concrete multi-key bounds, this means the attacker advantage
in the multi-key setting is the product of the single-key advantage
and the number of keys.
This section summarizes the confidentiality and integrity bounds and
limits for the same algorithms as in Section 5 for the multi-key
setting. The CL and IL values bound the total number of encryption
and forgery queries (q and v). Alongside each value, we also specify
these bounds.
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6.1. AEAD_AES_128_GCM and AEAD_AES_256_GCM
Concrete multi-key bounds for AEAD_AES_128_GCM and AEAD_AES_256_GCM
exist due to Theorem 4.3 in [GCM-MU2], which covers protocols with
nonce randomization, like TLS 1.3 [TLS] and QUIC [RFC9001]. Here,
the full nonce is XORed with a secret, random offset. The bound for
nonce randomization was further improved in [ChaCha20Poly1305-MU].
Results for AES-GCM with random, partially implicit nonces [RFC5288]
are captured by Theorem 5.3 in [GCM-MU2], which apply to protocols
such as TLS 1.2 [RFC5246]. Here, the implicit part of the nonce is a
random value, of length at least 32 bits and fixed per key, while we
assume that the explicit part of the nonce is chosen using a non-
repeating process. The full nonce is the concatenation of the two
parts. This produces similar limits under most conditions. Note
that implementations that choose the explicit part at random have a
higher chance of nonce collisions and are not considered for the
limits in this section.
For this AEAD, n = 128 (the AES block length), t = 128, and r = 96;
the key length is k = 128 or k = 256 for AEAD_AES_128_GCM and
AEAD_AES_256_GCM respectively.
6.1.1. Authenticated Encryption Security Limit
Protocols with nonce randomization have a limit of:
AEA <= (q+v)*L*B / 2^127
This implies the following limit:
q + v <= p * 2^127 / (L * B)
This assumes that B is much larger than 100; that is, each user
enciphers significantly more than 1600 bytes of data. Otherwise, B
should be increased by 161 for AEAD_AES_128_GCM and by 97 for
AEAD_AES_256_GCM.
Protocols with random, partially implicit nonces have the following
limit, which is similar to that for nonce randomization:
AEA <= (((q+v)*o + (q+v)^2) / 2^(k+26)) + ((q+v)*L*B / 2^127)
The first term is negligible if k = 256; this implies the following
simplified limits:
AEA <= (q+v)*L*B / 2^127
q + v <= p * 2^127 / (L * B)
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For k = 128, assuming o <= q + v (i.e., that the attacker does not
spend more work than all legitimate protocol users together), the
limits are:
AEA <= (((q+v)*o + (q+v)^2) / 2^154) + ((q+v)*L*B / 2^127)
q + v <= min( sqrt(p) * 2^76, p * 2^126 / (L * B) )
6.1.2. Confidentiality Limit
The confidentiality advantage is essentially dominated by the same
term as the AE advantage for protocols with nonce randomization:
CA <= q*L*B / 2^127
This implies the following limit:
q <= p * 2^127 / (L * B)
Similarly, the limits for protocols with random, partially implicit
nonces are:
CA <= ((q*o + q^2) / 2^(k+26)) + (q*L*B / 2^127)
q <= min( sqrt(p) * 2^76, p * 2^126 / (L * B) )
6.1.3. Integrity Limit
There is currently no dedicated integrity multi-key bound available
for AEAD_AES_128_GCM and AEAD_AES_256_GCM. The AE limit can be used
to derive an integrity limit as:
IA <= AEA
Section 6.1.1 therefore contains the integrity limits.
6.2. AEAD_CHACHA20_POLY1305
Concrete multi-key bounds for AEAD_CHACHA20_POLY1305 are given in
Theorem 7.2 in [ChaCha20Poly1305-MU], covering protocols with nonce
randomization like TLS 1.3 [TLS] and QUIC [RFC9001].
For this AEAD, n = 512 (the ChaCha20 block length), k = 256, t = 128,
and r = 96; the length (L') is the sum of AAD and plaintext (in
Poly1305 blocks of 128 bits).
6.2.1. Authenticated Encryption Security Limit
Protocols with nonce randomization have a limit of:
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AEA <= (v * (L' + 1)) / 2^103
It implies the following limit:
v <= (p * 2^103) / (L' + 1)
Note that this is the same limit as in the single-user case except
that the total number of forgery attempts (v) and maximum message
length in Poly1305 blocks (L') is calculated across all used keys.
6.2.2. Confidentiality Limit
While the AE advantage is dominated by the number of forgery attempts
v, those are irrelevant for the confidentiality advantage. The
relevant limit for protocols with nonce randomization becomes
dominated, at a very low level, by the adversary's offline work o and
the number of protected messages q across all used keys:
CA <= (o + q) / 2^247
This implies the following simplified limit, which for most
reasonable values of p is dominated by a technical limitation of
approximately q = 2^100:
q <= min( p * 2^247 - o, 2^100 )
6.2.3. Integrity Limit
The AE limit for AEAD_CHACHA20_POLY1305 essentially is the integrity
(multi-key) bound. The former hence also applies to the latter:
IA <= AEA
Section 6.2.1 therefore contains the integrity limits.
6.3. AEAD_AES_128_CCM and AEAD_AES_128_CCM_8
There are currently no concrete multi-key bounds for AEAD_AES_128_CCM
or AEAD_AES_128_CCM_8. Thus, to account for the additional factor u,
i.e., the number of keys, each p term in the confidentiality and
integrity limits is replaced with p / u.
The multi-key integrity limit for AEAD_AES_128_CCM is as follows.
v + q <= sqrt(p / u) * 2^63 / L
Likewise, the multi-key integrity limit for AEAD_AES_128_CCM_8 is as
follows.
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v * 2^64 + (2L * (v + q))^2 <= (p / u) * 2^128
6.4. Multi-Key Examples
An example protocol might choose to aim for a multi-key AEA, CA, and
IA that is at most 2^-50. If the messages exchanged in the protocol
are at most a common Internet MTU of around 1500 bytes, then a value
for L might be set to 2^7. Table 3 shows limits for q and v across
all keys that might be chosen under these conditions.
+========================+================+==============+
| AEAD | Maximum q | Maximum v |
+========================+================+==============+
| AEAD_AES_128_GCM | 2^69/B | 2^69/B |
+------------------------+----------------+--------------+
| AEAD_AES_256_GCM | 2^69/B | 2^69/B |
+------------------------+----------------+--------------+
| AEAD_CHACHA20_POLY1305 | 2^100 | 2^46 |
+------------------------+----------------+--------------+
| AEAD_AES_128_CCM | 2^30/sqrt(u) | 2^30/sqrt(u) |
+------------------------+----------------+--------------+
| AEAD_AES_128_CCM_8 | 2^30.9/sqrt(u) | 2^13/u |
+------------------------+----------------+--------------+
Table 3: Example multi-key limits
The limits for AEAD_AES_128_GCM, AEAD_AES_256_GCM, AEAD_AES_128_CCM,
and AEAD_AES_128_CCM_8 assume equal proportions for q and v. The
limits for AEAD_AES_128_GCM, AEAD_AES_256_GCM and
AEAD_CHACHA20_POLY1305 assume the use of nonce randomization, like in
TLS 1.3 [TLS] and QUIC [RFC9001].
The limits for AEAD_AES_128_GCM and AEAD_AES_256_GCM further depend
on the maximum number (B) of 128-bit blocks encrypted by any single
key. For example, limiting the number of messages (of size <= 2^7
blocks) to at most 2^20 (about a million) per key results in B of
2^27, which limits both q and v to 2^42 messages.
Only the limits for AEAD_AES_128_CCM and AEAD_AES_128_CCM_8 depend on
the number of used keys (u), which further reduces them considerably.
If v is limited to 1, q can be increased to 2^31/sqrt(u) for both CCM
AEADs.
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7. Security Considerations
The different analyses of AEAD functions that this work is based upon
generally assume that the underlying primitives are ideal. For
example, that a pseudorandom function (PRF) used by the AEAD is
indistinguishable from a truly random function or that a pseudorandom
permutation (PRP) is indistinguishable from a truly random
permutation. Thus, the advantage estimates assume that the attacker
is not able to exploit a weakness in an underlying primitive.
Many of the formulae in this document depend on simplifying
assumptions, from differing models, which means that results are not
universally applicable. When using this document to set limits, it
is necessary to validate all these assumptions for the setting in
which the limits might apply. In most cases, the goal is to use
assumptions that result in setting a more conservative limit, but
this is not always the case. As an example of one such
simplification, this document defines v as the total number of
decryption queries leading to a successful forgery (that is, the
number of failed forgery attempts plus one), whereas models usually
include all forgery attempts when determining v.
The CA, IA, and AEA values defined in this document are upper bounds
based on existing cryptographic research. Future analysis may
introduce tighter bounds. Applications SHOULD NOT assume these
bounds are rigid, and SHOULD accommodate changes.
Note that the limits in this document apply to the adversary's
ability to conduct a single successful forgery. For some algorithms
and in some cases, an adversary's success probability in repeating
forgeries may be noticeably larger than that of the first forgery.
As an example, [MF05] describes such multiple forgery attacks in the
context of AES-GCM in more detail.
8. IANA Considerations
This document does not make any request of IANA.
9. References
9.1. Normative References
[AEAD] Rogaway, P., "Authenticated-Encryption with Associated-
Data", September 2002,
.
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[AEBounds] Luykx, A. and K. Paterson, "Limits on Authenticated
Encryption Use in TLS", 8 March 2016,
.
[AEComposition]
Bellare, M. and C. Namprempre, "Authenticated Encryption:
Relations among notions and analysis of the generic
composition paradigm", July 2007,
.
[CCM-ANALYSIS]
Jonsson, J., "On the Security of CTR + CBC-MAC", Springer
Berlin Heidelberg, Lecture Notes in Computer Science pp.
76-93, DOI 10.1007/3-540-36492-7_7, ISBN ["9783540006220",
"9783540364924"], 2003,
.
[ChaCha20Poly1305-MU]
Degabriele, J. P., Govinden, J., Günther, F., and K. G.
Paterson, "The Security of ChaCha20-Poly1305 in the Multi-
user Setting", 24 January 2023,
.
[ChaCha20Poly1305-SU]
Procter, G., "A Security Analysis of the Composition of
ChaCha20 and Poly1305", 11 August 2014,
.
[GCM] Dworkin, M., "Recommendation for Block Cipher Modes of
Operation: Galois/Counter Mode (GCM) and GMAC",
NIST Special Publication 800-38D, November 2007.
[GCM-MU] Bellare, M. and B. Tackmann, "The Multi-User Security of
Authenticated Encryption: AES-GCM in TLS 1.3", 27 November
2017, .
[GCM-MU2] Hoang, V. T., Tessaro, S., and A. Thiruvengadam, "The
Multi-user Security of GCM, Revisited: Tight Bounds for
Nonce Randomization", 15 October 2018,
.
[GCMProofs]
Iwata, T., Ohashi, K., and K. Minematsu, "Breaking and
Repairing GCM Security Proofs", 1 August 2012,
.
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[MUSecurity]
Bellare, M., Boldyreva, A., and S. Micali, "Public-Key
Encryption in a Multi-user Setting: Security Proofs and
Improvements", May 2000,
.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC5116] McGrew, D., "An Interface and Algorithms for Authenticated
Encryption", RFC 5116, DOI 10.17487/RFC5116, January 2008,
.
[RFC6655] McGrew, D. and D. Bailey, "AES-CCM Cipher Suites for
Transport Layer Security (TLS)", RFC 6655,
DOI 10.17487/RFC6655, July 2012,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[RFC8439] Nir, Y. and A. Langley, "ChaCha20 and Poly1305 for IETF
Protocols", RFC 8439, DOI 10.17487/RFC8439, June 2018,
.
9.2. Informative References
[MF05] McGrew, D. A. and S. R. Fluhrer, "Multiple forgery attacks
against Message Authentication Codes", 31 May 2005,
.
[NonceDisrespecting]
Bock, H., Zauner, A., Devlin, S., Somorovsky, J., and P.
Jovanovic, "Nonce-Disrespecting Adversaries -- Practical
Forgery Attacks on GCM in TLS", 17 May 2016,
.
[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.2", RFC 5246,
DOI 10.17487/RFC5246, August 2008,
.
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[RFC5288] Salowey, J., Choudhury, A., and D. McGrew, "AES Galois
Counter Mode (GCM) Cipher Suites for TLS", RFC 5288,
DOI 10.17487/RFC5288, August 2008,
.
[RFC8645] Smyshlyaev, S., Ed., "Re-keying Mechanisms for Symmetric
Keys", RFC 8645, DOI 10.17487/RFC8645, August 2019,
.
[RFC9001] Thomson, M., Ed. and S. Turner, Ed., "Using TLS to Secure
QUIC", RFC 9001, DOI 10.17487/RFC9001, May 2021,
.
[SIV] Gueron, S., Langley, A., and Y. Lindell, "AES-GCM-SIV:
Nonce Misuse-Resistant Authenticated Encryption",
RFC 8452, DOI 10.17487/RFC8452, April 2019,
.
[TLS] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
.
Acknowledgments
In addition to the authors of papers performing analysis of ciphers,
thanks are owed to Scott Fluhrer, Thomas Fossati, John Mattsson,
David McGrew, Yoav Nir, Thomas Pornin, and Alexander Tereschenko for
helping making this document better.
Authors' Addresses
Felix Günther
IBM Research Europe - Zurich
Email: mail@felixguenther.info
Martin Thomson
Mozilla
Email: mt@lowentropy.net
Christopher A. Wood
Cloudflare
Email: caw@heapingbits.net
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