Internet Draft D. MÆRaihi
Category: Informational Verisign
Document: draft-mraihi-oath-hmac-otp-00.txt M. Bellare
Expires: April 2005 UCSD
F. Hoornaert
Vasco
D. Naccache
Gemplus
O. Ranen
Aladdin
October 2004
HOTP: An HMAC-based One Time Password Algorithm
Status of this Memo
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Abstract
This document describes an algorithm to generate one-time password
values, based on HMAC [BCK1]. A security analysis of the algorithm
is presented, and important parameters related to the secure
deployment of the algorithm are discussed. The proposed algorithm
can be used across a wide range of network applications ranging
from remote VPN access, Wi-Fi network logon to transaction-oriented
Web applications.
This work is a joint effort by the OATH (Open AuTHentication)
membership to specify an algorithm that can be freely distributed
to the technical community. The authors believe that a common and
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shared algorithm will facilitate adoption of two-factor
authentication on the Internet by enabling interoperability across
commercial and open-source implementations.
Table of Contents
1. Overview...................................................2
2. Introduction...............................................3
3. Requirements Terminology...................................4
4. Algorithm Requirements.....................................4
5. HOTP Algorithm.............................................5
5.1 Notation and Symbols.....................................5
5.2 Description..............................................6
5.3 Generating an HOTP value.................................6
5.4 Example of HOTP computation for Digit = 6................7
6. Security and Deployment Considerations.....................8
6.1 Authentication Protocol Requirements.....................8
6.2 Validation of HOTP values................................8
6.3 Throttling at the server.................................9
6.4 Resynchronization of the counter.........................9
7. HOTP Algorithm Security: Overview.........................10
8. Protocol Extensions and Improvements......................10
8.1 Number of Digits........................................10
8.2 Alpha-numeric Values....................................11
8.3 Sequence of HOTP values.................................11
8.4 A Counter-based Re-Synchronization Method...............12
9. Conclusion................................................12
10. Acknowledgements..........................................13
11. Contributors..............................................13
12. References................................................13
12.1 Normative...............................................13
12.2 Informative.............................................13
13. AuthorsÆ Addresses........................................13
Appendix - HOTP Algorithm Security: Detailed Analysis..........14
A.1 Definitions and Notations.................................14
A.2 The idealized algorithm: HOTP-IDEAL.......................15
A.3 Model of Security.........................................15
A.4 Security of the ideal authentication algorithm............17
A.4.1 From bits to digits.....................................17
A.4.2 Brute force attacks.....................................18
A.4.3 Brute force attacks are the best possible attacks.......19
A.5 Security Analysis of HOTP.................................20
1. Overview
The document introduces first the context around the HOTP
algorithm. In section 4, the algorithm requirements are listed and
in section 5, the HOTP algorithm is described. Sections 6 and 7
focus on the algorithm security. Section 8 proposes some extensions
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and improvements, and Section 9 concludes this document. The
interested reader will find in the Appendix a detailed, full-fledge
analysis of the algorithm security: an idealized version of the
algorithm is evaluated, and then the HOTP algorithm security is
analyzed.
2. Introduction
Today, deployment of two-factor authentication remains extremely
limited in scope and scale. Despite increasingly higher levels of
threats and attacks, most Internet applications still rely on weak
authentication schemes for policing user access. The lack of
interoperability among hardware and software technology vendors has
been a limiting factor in the adoption of two-factor authentication
technology. In particular, the absence of open specifications has
led to solutions where hardware and software components are tightly
coupled through proprietary technology, resulting in high cost
solutions, poor adoption and limited innovation.
In the last two years, the rapid rise of network threats has
exposed the inadequacies of static passwords as the primary mean of
authentication on the Internet. At the same time, the current
approach that requires an end-user to carry an expensive, single-
function device that is only used to authenticate to the network is
clearly not the right answer. For two factor authentication to
propagate on the Internet, it will have to be embedded in more
flexible devices that can work across a wide range of applications.
The ability to embed this base technology while ensuring broad
interoperability require that it be made freely available to the
broad technical community of hardware and software developers. Only
an open system approach will ensure that basic two-factor
authentication primitives can be built into the next-generation of
consumer devices such USB mass storage devices, IP phones, and
personal digital assistants).
One Time Password is certainly one of the simplest and most popular
forms of two-factor authentication for securing network access. For
example, in large enterprises, Virtual Private Network access often
requires the use of One Time Password tokens for remote user
authentication. One Time Passwords are often preferred to stronger
forms of authentication such as PKI or biometrics because an air-
gap device does not require the installation of any client desktop
software on the user machine, therefore allowing them to roam
across multiple machines including home computers, kiosks and
personal digital assistants.
This draft proposes a simple One Time Password algorithm that can
be implemented by any hardware manufacturer or software developer
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to create interoperable authentication devices and software agents.
The algorithm is event-based so that it can be embedded in high
volume devices such as Java smart cards, USB dongles and GSM SIM
cards. The presented algorithm is made freely available to the
developer community under the terms and conditions of the IETF
Intellectual Property Rights [RFC3668].
The authors of this document are members of the Open AuTHentication
initiative [OATH]. The initiative was created in 2004 to facilitate
collaboration among strong authentication technology providers.
3. Requirements Terminology
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.
4. Algorithm Requirements
This section presents the main requirements that drove this
algorithm design. A lot of emphasis was placed on end-consumer
usability as well as the ability for the algorithm to be
implemented by low cost hardware that may provide minimal user
interface capabilities. In particular, the ability to embed the
algorithm into high volume SIM and Java cards was a fundamental
pre-requisite.
R1 - The algorithm MUST be sequence or counter-based: One of the
goals is to have the HOTP algorithm embedded in high volume devices
such as Java smart cards, USB dongles and GSM SIM cards.
R2 - The algorithm SHOULD be economical to implement in hardware by
minimizing requirements on battery, number of buttons,
computational horsepower, and size of LCD display. The algorithm
MUST work with tokens that do not supports any numeric input, but
MAY also be used with more sophisticated devices such as secure
PIN-pads.
R3 - The value displayed on the token MUST be easily read and
entered by the user: This requires the HOTP value to be of
reasonable length. The HOTP value must be at least a 6-digit value.
It is also desirable that the HOTP value be 'numeric only' so that
it can be easily entered on restricted devices such as phones.
R4 - There MUST be user-friendly mechanisms available to
resynchronize the counter. The sections 6.4 and 8.4 detail the
resynchronization mechanism proposed in this draft.
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R5- The algorithm MUST use a strong shared secret. The length of
the shared secret MUST be at least 128 bits. This draft RECOMMENDs
a shared secret length of 160 bits.
5. HOTP Algorithm
In this section, we introduce the notation and describe the HOTP
algorithm basic blocks û the base function to compute an HMAC-SHA-1
value and the truncation method to extract an HOTP value.
5.1 Notation and Symbols
A string always means a binary string, meaning a sequence of zeros
and ones.
If s is a string then |s| denotes its length.
If n is a number then |n| denotes its absolute value.
If s is a string then s[i] denotes its i-th bit. We start numbering
the bits at 0, so s = s[0]s[1]..s[n-1] where n = |s| is the length
of s.
Let StToNum (String to Number) denote the function which as input a
string s returns the number whose binary representation is s.
(For example StToNum(110) = 6).
Here is a list of symbols used in this document.
Symbol Represents
-------------------------------------------------------------------
C 8-byte (Little Endian) counter value, which is the moving
factor. This counter MUST be synchronized between the HOTP
generator (client) and the HOTP validator (server);
K shared secret between the client and the server; each HOTP
generator has a different and unique secret K;
T throttling parameter: the server will refuse connections
from a user after T unsuccessful authentication attempts;
s resynchronization parameter: the server will attempt to
verify a received authenticator across s consecutive
counter values;
Digit number of digits in an HOTP value; system parameter.
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5.2 Description
The HOTP algorithm is based on an increasing counter value and a
static symmetric key known only to the token and the validation
service. In order to create the HOTP value, we will use the HMAC-
SHA-1 algorithm, as defined in RFC 2104 [BCK2].
As the output of the HMAC-SHA1 calculation is 160 bits, we must
truncate this value to something that can be easily entered by a
user.
HOTP(K,C) = Truncate(HMAC-SHA-1(K,C))
Where:
- Truncate represents the function that converts an HMAC-SHA-1
value into an HOTP value as defined in Section 5.3.
The HOTP values generated by the HOTP generator are treated as big
endian.
5.3 Generating an HOTP value
We can describe the operations in 3 distinct steps:
Step 1: Generate an HMAC-SHA-1 value
Let HS = HMAC-SHA-1(K,C) // HS is a 20 byte string
Step 2: Generate a 4-byte string (Dynamic Truncation)
Let Sbits = DT(HS) // DT, defined in Section 6.3.1
// returns a 31 bit string
Step 3: Compute an HOTP value
Let Snum = StToNum(S) // Convert S to a number in
0...2^{31}-1
Return D = Snum mod 10^Digit // D is a number in the range
0...10^{Digit}-1
The Truncate function performs Step 2 and Step 3, i.e. the dynamic
truncation and then the reduction modulo 10^Digit. The purpose of
the dynamic offset truncation technique is to extract a 4-byte
dynamic binary code from a 160-bit (20-byte) HMAC-SHA1 result.
DT(String) // String = String[0]...String[19]
Let OffsetBits be the low order four bits of String[19]
Offset = StToNum(OffSetBits) // 0 <= OffSet <= 15
Let P = String[OffSet]...String[OffSet+3]
Return the Last 31 bits of P
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The reason for masking the most significant bit of P is to avoid
confusion about signed vs. unsigned modulo computations. Different
processors perform these operations differently, and masking out
the signed bit removes all ambiguity.
Implementations MUST extract a 6-digit code at a minimum and
possibly 7 and 8-digit code. Depending on security requirements,
Digit = 7 or more SHOULD be considered in order to extract a longer
HOTP value.
The following paragraph is an example of using this technique for
Digit = 6, i.e. that a 6-digit HOTP value is calculated from the
HMAC value.
5.4 Example of HOTP computation for Digit = 6
The following code example describes the extraction of a dynamic
binary code given that hmac_result is a byte array with the HMAC-
SHA1 result:
int offset = hmac_result[19] & 0xf ;
int bin_code = (hmac_result[offset] & 0x7f) << 24
| (hmac_result[offset+1] & 0xff) << 16
| (hmac_result[offset+2] & 0xff) << 8
| (hmac_result[offset+3] & 0xff) ;
SHA-1 HMAC Bytes (Example)
-------------------------------------------------------------
| Byte Number |
-------------------------------------------------------------
|00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15|16|17|18|19|
-------------------------------------------------------------
| Byte Value |
-------------------------------------------------------------
|1f|86|98|69|0e|02|ca|16|61|85|50|ef|7f|19|da|8e|94|5b|55|5a|
-------------------------------***********----------------++|
* The last byte (byte 19) has the hex value 0x5a.
* The value of the lower four bits is 0xa (the offset value).
* The offset value is byte 10 (0xa).
* The value of the 4 bytes starting at byte 10 is 0x50ef7f19,
which is the dynamic binary code DBC1
* The MSB of DBC1 is 0x50 so DBC2 = DBC1 = 0x50ef7f19
* HOTP = DBC2 modulo 10^6 = 872921.
We treat the dynamic binary code as a 31-bit, unsigned, big-endian
integer; the first byte is masked with a 0x7f.
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We then take this number modulo 1,000,000 (10^6) to generate the 6-
digit HOTP value 872921 decimal.
6. Security and Deployment Considerations
Any One-Time Password algorithm is only as secure as the
application and the authentication protocols that implement it.
Therefore, this section discusses the critical security
requirements that our choice of algorithm imposes on the
authentication protocol and validation software. The parameters T
and s discussed in this section have a significant impact on the
security û further details in Section 7 elaborate on the relations
between these parameters and their impact on the system security.
6.1 Authentication Protocol Requirements
We introduce in this section some requirements for a protocol P
implementing HOTP as the authentication method between a prover and
a verifier.
RP1 - P MUST be two-factor, i.e. something you know (secret code
such as a Password, Pass phrase, PIN code, etc.) and something you
have (token). The secret code is known only to the user and usually
entered with the one-time password value for authentication purpose
(two-factor authentication).
RP3 - P MUST NOT be vulnerable to brute force attacks. This implies
that a throttling/lockout scheme is REQUIRED on the validation
server side.
RP4 û P SHOULD be implemented with respect to the state of the art
in terms of security, in order to avoid the usual attacks and risks
associated with the transmission of sensitive data over a public
network (privacy, replay attacks, etc.)
6.2 Validation of HOTP values
The HOTP client (hardware or software token) increments its counter
and then calculates the next HOTP value HOTP-client. If the value
received by the authentication server matches the value calculated
by the client, then the HOTP value is validated. In this case, the
server increments the counter value by one.
If the value received by the server does not match the value
calculated by the client, the server initiate the resynch protocol
(look-ahead window) before it requests another pass.
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If the resynch fails, the server asks then for another
authentication pass of the protocol to take place, until the
maximum number of authorized attempts is reached.
If and when the maximum number of authorized attempts is reached,
the server SHOULD lock out the account and initiate a procedure to
inform the user.
6.3 Throttling at the server
Truncating the HMAC-SHA1 value to a shorter value makes a brute
force attack possible. Therefore, the authentication server needs
to detect and stop brute force attacks.
We RECOMMEND setting a throttling parameter T, which defines the
maximum number of possible attempts for One-Time-Password
validation. The validation server manages individual counters per
HOTP device in order to take note of any failed attempt. We
RECOMMEND T not to be too large, particularly if the
resynchronization method used on the server is window-based, and
the window size is large. T SHOULD be set as low as possible, while
still ensuring usability is not significantly impacted.
6.4 Resynchronization of the counter
Although the serverÆs counter value is only incremented after a
successful HOTP authentication, the counter on the token is
incremented every time a new HOTP is requested by the user. Because
of this, the counter values on the server and on the token might be
out of synchronization.
We RECOMMEND setting a look-ahead parameter s on the server, which
defines the size of the look-ahead window. In a nutshell, the
server can recalculate the next s HOTP-server values, and check
them against the received HOTP-client.
Synchronization of counters in this scenario simply requires the
server to calculate the next HOTP values and determine if there is
a match. Optionally, the system MAY require the user to send a
sequence of (say 2, 3) HOTP values for resynchronization purpose,
since forging a sequence of consecutive HOTP values is even more
difficult than guessing a single HOTP value.
The upper bound set by the parameter s ensures the server does not
go on checking HOTP values forever (causing a DoS attack) and also
restricts the space of possible solutions for an attacker trying to
manufacture HOTP values. s SHOULD be set as low as possible, while
still ensuring usability is not impacted.
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7. HOTP Algorithm Security: Overview
The conclusion of the security analysis detailed in the Appendix
section is that, for all practical purposes, the outputs of HOTP on
distinct counter inputs are uniformly and independently distributed
strings.
As a result, the best possible attack against the HOTP function is
the brute force attack.
Assuming an adversary is able to observe numerous protocol
exchanges and collect sequences of successful authentication
values. This adversary, trying to build a function F to generate
HOTP values based on his observations, will not have a significant
advantage over a random guess.
The logical conclusion is simply that is best strategy will once
again be to perform a brute force attack to enumerate and try all
the possible values.
Considering the security analysis in the Appendix section of this
document, without loss of generality, we can approximate closely
the security of the HOTP algorithm by the following formula:
Sec = sv/10^Digit
Where:
- Sec is the probability of success of the adversary
- s stands for the look-ahead synchronization window size;
- v stands for the number of verification attempts;
- Digit stands for the number of digits in HOTP values.
Obviously, we can play with s, T (the Throttling parameter that
would limit the number of attempts by an attacker) and Digit until
achieving a certain level of security, still preserving the system
usability.
8. Protocol Extensions and Improvements
We introduce in this section several enhancements and suggestions
to further improve the security of the algorithm HOTP
8.1 Number of Digits
A simple enhancement in terms of security would be to extract more
digits from the HMAC-SHA1 value.
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For instance, calculating the HOTP value modulo 10^8 to build an 8-
digit HOTP value would reduce the probability of success of the
adversary from sv/10^6 to sv/10^8.
This could give the opportunity to improve usability, e.g. by
increasing T and/or s, while still achieving a better security
overall. For instance, s = 10 and 10v/10^8 = v/10^7 < v/10^6 which
is the theoretical optimum for 6-digit code when s = 1.
8.2 Alpha-numeric Values
Another option is to use A-Z and 0-9 values; or rather a subset of
32 symbols taken from the alphanumerical alphabet in order to avoid
any confusion between characters: 0, O and Q as well as l, 1 and I
are very similar, and can look the same on a small display.
The immediate consequence is that the security is now in the order
of sv/32^6 for a 6-digit HOTP value and sv/32^8 for an 8-digit HOTP
value.
32^6 > 10^9 so the security of a 6-alphanumeric HOTP code is
slightly better than a 9-digit HOTP value, which is the maximum
length of an HOTP code supported by the proposed algorithm.
32^8 > 10^12 so the security of an 8-alphanumeric HOTP code is
significantly better than a 9-digit HOTP value.
Depending on the application and token/interface used for
displaying and entering the HOTP value, the choice of alphanumeric
values could be a simple and efficient way to improve security at a
reduced cost and impact on users.
8.3 Sequence of HOTP values
As we suggested for the resynchronization to enter a short sequence
(say 2 or 3) of HOTP values, we could generalize the concept to the
protocol, and add a parameter L that would define the length of the
HOTP sequence to enter.
Per default, the value L SHOULD be set to 1, but if security needs
to be increased, users might be asked (possibly for a short period
of time, or a specific operation) to enter L HOTP values.
This is another way, without increasing the HOTP length or using
alphanumeric values to tighten security.
Note: The system MAY also be programmed to request synchronization
on a regular basis (e.g. every night, or twice a week, etc.) and to
achieve this purpose, ask for a sequence of L HOTP values.
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8.4 A Counter-based Re-Synchronization Method
In this case, we assume that the client can access and send not
only the HOTP value but also other information, more specifically
the counter value.
A more efficient and secure method for resynchronization is
possible in this case. The client application will not send the
HOTP-client value only, but the HOTP-client and the related C-
client counter value, the HOTP value acting as a message
authentication code of the counter.
Resynchronization Counter-based Protocol (RCP)
----------------------------------------------
The server accepts if the following are all true, where C-server is
its own current counter value:
1) C-client >= C-server
2) C-client û C-server <= s
3) Check that HOTP-client is valid HOTP(K,C-Client)
4) If true, the server sets C to C-client + 1 and client
is authenticated
In this case, there is no need for managing a look-ahead window
anymore. The probability of success of the adversary is only v/10^6
or roughly v in one million. A side benefit is obviously to be able
to increase s ôinfinitelyö and therefore improve the system
usability without impacting the security.
This resynchronization protocol SHOULD be use whenever the related
impact on the client and server applications is deemed acceptable.
9. Conclusion
This draft describes HOTP, a HMAC-based One-Time Password
algorithm. It also recommends the preferred implementation and
related modes of operations for deploying the algorithm.
The draft also exhibits elements of security and demonstrates that
the HOTP algorithm is practical and sound, the best possible attack
being a brute force attack that can be prevented by careful
implementation of countermeasures in the validation server.
Eventually, several enhancements have been proposed, in order to
improve security if needed for specific applications.
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10. Acknowledgements
The authors would like to thank Siddharth Bajaj, Alex Deacon and
Nico Popp for their help during the conception and redaction of
this document.
11. Contributors
The authors of this draft would like to emphasize the role of two
persons who have made a key contribution to this document:
- Laszlo Elteto is system architect with SafeNet, Inc.
- Ernesto Frutos is director of Engineering with Authenex, Inc.
Without their advice and valuable inputs, this draft would not be
the same.
12. References
12.1 Normative
[BCK1] M. Bellare, R. Canetti, and H. Krawczyk, Keyed Hash
Functions and Message Authentication, Proceedings of
Crypto'96, LNCS Vol. 1109, pp. 1-15.
[BCK2] M. Bellare, R. Canetti, and H. Krawczyk, HMAC:
Keyed-Hashing for Message Authentication, IETF Network
Working Group, RFC 2104, February 1997.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC3668] Bradner, S., "Intellectual Propery Rights in IETF
Technologyö, BCP 79, RFC 3668, February 2004.
12.2 Informative
[OATH] www.openauthentication.org, Initiative for Open
AuTHentication
13. AuthorsÆ Addresses
Primary point of contact (for sending comments and question):
David M'Raihi
VeriSign, Inc.
685 E. Middlefield Road Phone: 1-650-426-3832
Mountain View, CA 94043 USA Email: dmraihi@verisign.com
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HOTP: An HMAC-based One Time Password Algorithm October 2004
Other AuthorsÆ contact information:
Mihir Bellare
Dept of Computer Science and Engineering, Mail Code 0114
University of California at San Diego
9500 Gilman Drive
La Jolla, CA 92093, USA Email: mihir@cs.ucsd.edu
Frank Hoornaert
VASCO Data Security, Inc.
Koningin Astridlaan 164
1780 Wemmel, Belgium Email: frh@vasco.com
David Naccache
Gemplus Innovation
34 rue Guynemer, 92447,
Issy les Moulineaux, France Email: david.naccache@gemplus.com
and
Information Security Group,
Royal Holloway,
University of London, Egham,
Surrey TW20 0EX, UK Email: david.naccache@rhul.ac.uk
Ohad Ranen
Aladdin Knowledge Systems Ltd.
15 Beit Oved Street
Tel Aviv, Israel 61110 Email: Ohad.Ranen@ealaddin.com
Appendix - HOTP Algorithm Security: Detailed Analysis
The security analysis of the HOTP algorithm is summarized in this
section. We first detail the best attack strategies, and then
elaborate on the security under various assumptions, the impact of
the truncation and some recommendations regarding the number of
digits.
We focus this analysis on the case where Digit = 6, i.e. an HOTP
function that produces 6-digit values, which is the bare minimum
recommended in this draft.
A.1 Definitions and Notations
We denote by {0,1}^l the set of all strings of length l.
Let Z_{n} = {0,.., n - 1}.
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Let IntDiv(a,b) denote the integer division algorithm that takes
input integers a, b where a >= b >= 1 and returns integers (q,r)
the quotient and remainder, respectively, of the division of a by
b. (Thus a = bq + r and 0 <= r < b.)
Let H: {0,1}^k x {0,1}^c --> {0,1}^n be the base function that
takes a k-bit key K and c-bit counter C and returns an n-bit output
H(K,C). (In the case of HOTP, H is HMAC-SHA-1; we use this formal
definition for generalizing our proof of security)
A.2 The idealized algorithm: HOTP-IDEAL
We now define an idealized counterpart of the HOTP algorithm. In
this algorithm, the role of H is played by a random function that
forms the key.
To be more precise, let Maps(c,n) denote the set of all functions
mapping from {0,1}^c to {0,1}^n. The idealized algorithm has key
space Maps(c,n), so that a ôkeyö for such an algorithm is a
function h from {0,1}^c to {0,1}^n. We imagine this key (function)
to be drawn at random. It is not feasible to implement this
idealized algorithm, since the key, being a function from is way
too large to even store. So why consider it?
Our security analysis will show that as long as H satisfies a
certain well-accepted assumption, the security of the actual and
idealized algorithms is for all practical purposes the same. The
task that really faces us, then, is to assess the security of the
idealized algorithm.
In analyzing the idealized algorithm, we are concentrating on
assessing the quality of the design of the algorithm itself,
independently of HMAC-SHA-1. This is in fact the important issue.
A.3 Model of Security
The model exhibits the type of threats or attacks that are being
considered and enables to asses the security of HOTP and HOTP-
IDEAL. We denote ALG as either HOTP or HOTP-IDEAL for the purpose
of this security analysis.
The scenario we are considering is that a user and server share a
key K for ALG. Both maintain a counter C, initially zero, and the
user authenticates itself by sending ALG(K,C) to the server. The
latter accepts if this value is correct.
In order to protect against accidental increment of the user
counter, the server, upon receiving a value z, will accept as long
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as z equals ALG(K,i) for some i in the range C,...,C + s-1, where s
is the resynchronization parameter and C is the server counter. If
it accepts with some value of i, it then increments its counter to
i+ 1. If it does not accept, it does not change its counter value.
The model we specify captures what an adversary can do and what it
needs to achieve in order to ôwin.ö First, the adversary is assumed
to be able to eavesdrop, meaning see the authenticator transmitted
by the user. Second, the adversary wins if it can get the server to
accept an authenticator relative to a counter value for which the
user has never transmitted an authenticator.
The formal adversary, which we denote by B, starts out knowing
which algorithm ALG is being used, knowing the system design and
knowing all system parameters. The one and only thing it is not
given a priori is the key K shared between the user and the server.
The model gives B full control of the scheduling of events. It has
access to an authenticator oracle representing the user. By calling
this oracle, the adversary can ask the user to authenticate itself
and get back the authenticator in return. It can call this oracle
as often as it wants and when it wants, using the authenticators it
accumulates to perhaps ôlearnö how to make authenticators itself.
At any time, it may also call a verification oracle, supplying the
latter with a candidate authenticator of its choice. It wins if the
server accepts this accumulator.
Consider the following game involving an adversary B that is
attempting to compromise the security of an authentication
algorithm ALG: K x {0,1}^c --> R.
Initializations - A key K is selected at random from K, a counter C
is initialized to 0, and the Boolean value win is set to false.
Game execution - Adversary B is provided with the two following
oracles:
Oracle AuthO()
O = ALG(K,C)
C = C + 1
Return O to B
Oracle VerO()
i = C
While (i <= C + s - 1 and Win = FALSE) do
If O = ALG(K,i) then Win = TRUE; C = i + 1
Else i = i + 1
Return Win to B
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AuthO() is the authenticator oracle and VerO() is the verification
oracle.
Upon execution, B queries the two oracles at will. Let Adv(B) be
the probability that win gets set to true in the above game. This
is the probability that the adversary successfully impersonates the
user.
Our goal is to assess how large this value can be as a function of
the number v of verification queries made by B, the number a of
authenticator oracle queries made by B, and the running time t of
B. This will tell us how to set the throttle, which effectively
upper bounds v.
A.4 Security of the ideal authentication algorithm
This section summarizes the security analysis of HOTP-IDEAL,
starting with the impact of the conversion modulo 10^Digit and
then, focusing on the different possible attacks.
A.4.1 From bits to digits
The dynamic offset truncation of a random n-bit string yields a
random 31-bit string. What happens to the distribution when it is
taken modulo m = 10^Digit, as done in HOTP?
The following lemma estimates the biases in the outputs in this
case.
Lemma 1
Let N >= m >= 1 be integers, and let (q,r) = IntDiv(N,m). For z in
Z_{m} let:
P_{N,m}(z) = Pr [x mod m = z : x randomly pick in Z_{n}]
Then for any z in Z_{m}
P_{N,m}(z) = (q + 1) / N if 0 <= z < r
q / N if r <= z < m
Proof of Lemma 1
Let the random variable X be uniformly distributed over Z_{N}.
Then:
P_{N,m}(z) = Pr [X mod m = z]
= Pr [X < mq] ¸ Pr [X mod m = z| X < mq]
+ Pr [mq <= X < N] ¸ Pr [X mod m = z| mq <= X < N]
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= mq/N ¸ 1/m +
(N - mq)/N ¸ 1 / (N û mq) if 0 <= z < N û mq
0 if N û mq <= z <= m
= q/N +
r/N ¸ 1 / r if 0 <= z < N û mq
0 if r <= z <= m
Simplifying yields the claimed equation.
Let N = 2^31, d = 6 and m = 10^d. If x is chosen at random from
Z_{N} (meaning, is a random 31-bit string), then reducing it to a
6-digit number by taking x mod m does not yield a random 6-digit
number.
Rather, x mod m is distributed as shown in the following table:
Values Probability that each appears as output
----------------------------------------------------------------
0,1,...,483647 2148/2^31 roughly equals to 1.00024045/10^6
483648,...,999999 2147/2^31 roughly equals to 0.99977478/10^6
If X is uniformly distributed over Z_{2^31} (meaning is a random
31-bit string) then the above shows the probabilities for different
outputs of X mod 10^6. The first set of values appear with
probability slightly greater than 10^-6, the rest with probability
slightly less, meaning the distribution is slightly non-uniform.
However, as the Figure indicates, the bias is small and as we will
see later, negligible: the probabilities are very close to 10^-6.
A.4.2 Brute force attacks
If the authenticator consisted of d random digits, then a brute
force attack using v verification attempts would succeed with
probability sv/10^Digit.
However, an adversary can exploit the bias in the outputs of HOTP-
IDEAL, predicted by Lemma 1, to mount a slightly better attack.
Namely, it makes authentication attempts with authenticators which
are the most likely values, meaning the ones in the range 0,...,r -
1, where (q,r) = IntDiv(2^31,10^Digit).
The following specifies an adversary in our model of security that
mounts the attack. It estimates the success probability as a
function of the number of verification queries.
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For simplicity, we assume the number of verification queries is at
most r. With N = 2^31 and m = 10^6 we have r = 483,648, and the
throttle value is certainly less than this, so this assumption is
not much of a restriction.
Proposition 1
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Assume
s <= m. The brute-force attack adversary B-bf attacks HOTP using v
<= r verification oracle queries. This adversary makes no
authenticator oracle queries, and succeeds with probability
Adv(B-bf) = 1 - (1 - v(q+1)/2^31)^s
which is roughly equals to
sv ¸ (q+1)/2^31
With m = 10^6 we get q = 2,147. In that case, the brute force
attack using v verification attempts succeeds with probability
Adv(B-bf) roughly = sv ¸ 2148/2^31 = sv ¸ 1.00024045/10^6
As this equation shows, the resynchronization parameter s has a
significant impact in that the adversaryÆs success probability is
proportional to s. This means that s cannot be made too large
without compromising security.
A.4.3 Brute force attacks are the best possible attacks
A central question is whether there are attacks any better than the
brute force one. In particular, the brute force attack did not
attempt to collect authenticators sent by the user and try to
cryptanalyze them in an attempt to learn how to better construct
authenticators. Would doing this help? Is there some way to ôlearnö
how to build authenticators that result in a higher success rate
than given by the brute-force attack?
The following says the answer to these questions is no. No matter
what strategy the adversary uses, and even if it sees, and tries to
exploit, the authenticators from authentication attempts of the
user, its success probability will not be above that of the brute
force attack - this is true as long as the number of
authentications it observes is not incredibly large. This is
valuable information regarding the security of the scheme.
Proposition 2
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Let B
be any adversary attacking HOTP-IDEAL using v verification oracle
queries and a <= 2^c û s authenticator oracle queries. Then
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Adv(B) < = sv ¸ (q+1)/ 2^31
Note: This result is conditional on the adversary not seeing more
than 2^c - s authentications performed by the user, which is hardly
restrictive as long as c is large enough.
With m = 10^6 we get q = 2,147. In that case, Proposition 2 says
that any adversary B attacking HOTP-IDEAL and making v verification
attempts succeeds with probability at most
Equation 1
sv ¸ 2148/2^31 roughly = sv ¸ 1.00024045/10^6
Meaning, BÆs success rate is not more than that achieved by the
brute force attack.
A.5 Security Analysis of HOTP
We have analyzed in the previous sections, the security of the
idealized counterparts HOTP-IDEAL of the actual authentication
algorithm HOTP. We now show that, under appropriate and well-
believed assumption on H, the security of the actual algorithms is
essentially the same as that of its idealized counterpart.
The assumption in question is that H is a secure pseudorandom
function, or PRF, meaning that its input-output values are
indistinguishable from those of a random function in practice.
Consider an adversary A that is given an oracle for a function f:
{0,1}^c --> {0, 1}^n and eventually outputs a bit. We denote Adv(A)
as the prf-advantage of A, which represents how well the adversary
does at distinguishing the case where its oracle is H(K,.) from the
case where its oracle is a random function of {0,1}^c to {0,1}^n.
One possible attack is based on exhaustive search for the key K. If
A runs for t steps and T denotes the time to perform one
computation of H, its prf-advantage from this attack turns out to
be (t/T)2^-k . Another possible attack is a birthday one [3],
whereby A can attain advantage p^2/2^n in p oracle queries and
running time about pT.
Our assumption is that these are the best possible attacks. This
translates into the following.
Assumption 1
Let T denotes the time to perform one computation of H. Then if A
is any adversary with running time at most t and making at most p
oracle queries,
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Adv(A) <= (t/T)/2^k + p^2/2^n
In practice this assumption means that H is very secure as PRF. For
example, given that k = n = 160, an attacker with running time 2^60
and making 2^40 oracle queries has advantage at most (about) 2^-80.
Theorem 1
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Let B
be any adversary attacking HOTP using v verification oracle
queries, a <= 2^c - s authenticator oracle queries, and running
time t. Let T denote the time to perform one computation of H. If
Assumption 1 is true then
Adv(B) <= sv ¸ (q + 1)/2^31 + (t/T)/2^k + ((sv + a)^2)/2^n
In practice, the (t/T)2^-k + ((sv + a)^2)2^-n term is much smaller
than the sv(q + 1)/2^n term, so that the above says that for all
practical purposes the success rate of an adversary attacking HOTP
is sv(q + 1)/2^n, just as for HOTP-IDEAL, meaning the HOTP
algorithm is in practice essentially as good as its idealized
counterpart.
In the case m = 10^6 of a 6-digit output this means that an
adversary making v authentication attempts will have a success rate
that is at most that of Equation 1.
For example, consider an adversary with running time at most 2^60
that sees at most 2^40 authentication attempts of the user. Both
these choices are very generous to the adversary, who will
typically not have these resources, but we are saying that even
such a powerful adversary will not have more success than indicated
by Equation 1.
We can safely assume sv <= 2^40 due to the throttling and bounds on
s. So:
(t/T)/2^k + ((sv + a)^2)/2^n <= 2^60/2^160 + (2^41)^2/2^160
roughly <= 2^-78
which is much smaller than the success probability of Equation 1
and negligible compared to it.
Full Copyright Statement
Copyright (C) The Internet Society 2004. 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.
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This document and the information contained herein are provided on
an "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE
REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND
THE INTERNET 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.
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