ISO/IEC 18031:2005

INTERNATIONAL ISO/IEC
STANDARD 18031
First edition
2005-11-01
Information technology — Security
techniques — Random bit generation
Technologies de l'information — Techniques de sécurité — Génération
de bits aléatoires
Reference number
©
ISO/IEC 2005
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ii © ISO/IEC 2005 – All rights reserved

Contents Page
Foreword. vi
Introduction . vii
1 Scope. 1
2 Normative references. 1
3 Terms and definitions. 2
4 Symbols. 5
5 Overarching objectives and requirements of a random bit generator. 5
5.1 Required properties of randomness. 6
5.2 Backward and forward secrecy. 6
5.3 Top-level objectives and requirements for a random bit generator (RBG) output . 7
5.4 Top-level objectives and requirements for RBG operation. 7
5.5 Random bit generator functional requirements .8
6 General functional model for random bit generation. 8
6.1 Basic components. 8
6.1.1 Entropy source. 9
6.1.2 Additional inputs. 10
6.1.3 Internal state. 10
6.1.4 Internal state transition functions. 11
6.1.5 Output generation function . 12
6.1.6 Support functions. 13
7 Types of random bit generators. 14
7.1 Non-deterministic random bit generators (NRBGs). 14
7.2 Deterministic random bit generators (DRBGs). 15
7.3 The RBG spectrum . 15
8 Overview and requirements for a non-deterministic random bit generator. 16
8.1 Overview. 16
8.2 Functional model of a non-deterministic random bit generator. 16
8.2.1 Overview of the model. 16
8.3 Entropy sources. 18
8.3.1 Primary entropy source. 18
8.3.2 Physical entropy sources. 20
8.3.3 Non-physical entropy sources . 21
8.3.4 Additional entropy sources . 21
8.3.5 Hybrid non-deterministic random bit generators.22
8.4 Additional inputs. 23
8.4.1 Overview. 23
8.4.2 Mandatory requirements. 23
8.5 Internal state. 23
8.5.1 Overview. 23
8.5.2 Mandatory requirements. 24
8.5.3 Optional requirements. 24
8.6 Internal state transition functions. 25
8.6.1 Overview. 25
8.6.2 Mandatory requirements. 26
8.6.3 Optional requirements. 26
8.7 Output generation function . 26
8.7.1 Overview. 26
8.7.2 Mandatory requirements. 26
© ISO/IEC 2005 – All rights reserved iii

8.7.3 Optional requirement. 27
8.8 Health tests. 27
8.8.1 Overview. 27
8.8.2 General health test requirements. 27
8.8.3 Health test on deterministic components . 28
8.8.4 Health tests on entropy sources . 28
8.8.5 Health tests on random output. 29
8.9 Component interaction. 31
8.9.1 Overview. 31
8.9.2 Mandatory requirements. 31
8.9.3 Optional requirements. 32
9 Overview and requirements for a deterministic random bit generator . 32
9.1 Overview. 32
9.2 Functional model of DRBG . 33
9.2.1 Overview of the model. 33
9.3 Entropy source. 35
9.3.1 Primary entropy source. 35
9.3.2 Generating seed values. 37
9.3.3 Additional entropy sources. 37
9.3.4 Hybrid deterministic random bit generator. 38
9.4 Additional inputs. 38
9.5 Internal state. 38
9.6 Internal state transition function . 39
9.7 Output generation function. 40
9.7.1 Overview. 40
9.8 Support functions . 40
9.8.1 Overview. 40
9.8.2 Self test. 40
9.8.3 Deterministic algorithm test. 41
9.8.4 Software/Firmware integrity test . 41
9.8.5 Critical functions test . 41
9.8.6 Software/Firmware load test . 41
9.8.7 Manual key entry test. 41
9.8.8 Continuous random bit generator test . 42
9.9 Additional DRBG functional requirements. 42
9.9.1 Keys. 42
Annex A (normative) Combining random bit generators . 44
Annex B (normative) Conversion methods . 45
B.1 Random number generation . 45
B.1.1 The simple discard method. 45
B.1.2 The complex discard method . 45
B.1.3 The simple modular method . 46
B.1.4 The complex modular method. 46
B.2 Extracting bits in the Dual_EC_DRBG . 47
B.2.1 Potential bias in an elliptic curve over a prime field F . 47
p
B.2.2 Adjusting for the missing bit(s) of entropy in the x coordinates . 48
B.2.3 Values for E. 49
B.2.4 Observations. 51
Annex C (normative) Deterministic random bit generators . 52
C.1 Introduction . 52
C.2 Deterministic RBGs based on a hash-function. 52
C.2.1 Hash-function DRBG (Hash_DRBG) . 52
C.3 DRBG based on block ciphers. 60
C.3.1 CTR_DRBG. 61
C.3.2 OFB_DRBG (…). 70
C.4 Deterministic RBGs based on number theoretic problems. 72
C.4.1 Dual Elliptic Curve DRBG (Dual_EC_DRBG). 72
C.4.2 Micali Schnorr DRBG (MS_DRBG) . 81
iv © ISO/IEC 2005 – All rights reserved

Annex D (normative) Application specific constants. 91
D.1 Constants for the Dual_EC_DRBG. 91
D.1.1 Curves over Prime Fields. 91
D.1.2 Curves over binary fields. 94
D.2 Default moduli for the MS_DRBG (…). 103
D.2.1 Default modulus n of size 1024 bits. 103
D.2.2 Default modulus n of size 2048 bits. 103
D.2.3 Default modulus n of size 3072 bits. 104
D.2.4 Default modulus n of size 7680 bits. 104
D.2.5 Default modulus n of size 15360 bits. 105
Annex E (informative) Non-deterministic random bit generator examples. 107
E.1 Canonical coin tossing example. 107
E.1.1 Overview. 107
E.1.2 Description of basic process. 107
E.1.3 Relation to standard NRBG components. 107
E.1.4 Optional variations . 108
E.1.5 Peres unbiasing procedure . 108
E.2 Hypothetical noisy diode example. 109
E.2.1 Overview. 109
E.2.2 General structure. 109
E.2.3 Details of operation . 110
E.2.4 Failsafe design consequences. 114
E.2.5 Modified example. 114
E.3 Mouse movement example . 115
Annex F (informative) Security considerations. 116
F.1 Attack model. 116
F.2 The security of hash-functions . 116
F.3 Algorithm and key size selection . 116
F.3.1 Equivalent algorithm strengths. 117
F.3.2 Selection of appropriate DRBGs. 118
F.4 The security of block cipher DRBGS. 119
F.5 Conditioned entropy sources and the derivation function . 119
Annex G (informative) Discussion on the estimation of entropy . 120
Annex H (informative) Random bit generator assurance. 121
Annex I (informative) Random bit generator boundaries. 122
Bibliography . 124

© ISO/IEC 2005 – All rights reserved v

Foreword
ISO (the International Organization for Standardization) and IEC (International Electrotechnical Commission)
form the specialized system for worldwide standardization. National bodies that are members of ISO and IEC
participate in the development of International Standards through technical committees established by the
respective organization to deal with particular fields of technical activity. ISO and IEC technical committees
collaborate in fields of mutual interest. Other international organizations, governmental and non-governmental,
in liaison with ISO and IEC, also take part in the work. In the field of information technology, ISO and IEC have
established a joint technical committee, ISO/IEC JTC 1.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of the joint technical committee is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as
an International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/IEC 18031 was prepared by Joint Technical Committee ISO/IEC JTC 1, Information technology,
Subcommittee SC 27, IT Security techniques.
vi © ISO/IEC 2005 – All rights reserved

Introduction
This International Standard sets out specific requirements that when met will result in the development of a
random bit generator that may be applicable to cryptographic applications.
Numerous cryptographic applications require the use of random bits. These cryptographic applications include
the following:
• random keys and initialisation values (IVs) for encryption;
• random keys for keyed MAC algorithms;
• random private keys for digital signature algorithms;
• random values to be used in entity authentication mechanisms;
• random values to be used in key establishment protocols;
• random PIN and password generation;
• nonces.
The purpose of this International Standard is to establish a conceptual model, terminology, and requirements
related to the building blocks and properties of systems used for random bit generation in or for cryptographic
applications.
In general terms, it is possible to categorize random bit generators into two types depending on whether their
source of entropy varies or is fixed. This International Standard identifies the two types as non-deterministic
and deterministic random bit generators.
A non-deterministic random bit generator can be defined as a random bit generating mechanism that uses a
source of entropy to generate a random bit stream.
A deterministic random bit generator can be defined as a bit generating mechanism that uses deterministic
mechanisms, such as cryptographic algorithms on a source of entropy, to generate a random bit stream. In
this type of bit stream generation, there is a specific input (normally called a seed) and perhaps some optional
input, which, depending on its application may or may not be publicly available. The seed is processed by a
function which provides an output.

© ISO/IEC 2005 – All rights reserved vii

INTERNATIONAL STANDARD ISO/IEC 18031:2005(E)

Information technology — Security techniques — Random bit
generation
1 Scope
This International Standard specifies a conceptual model for a random bit generator for cryptographic
purposes, together with the elements of this model.
This International Standard also includes the following:
• the description of the main elements required for a non-deterministic random bit generator;
• the description of the main elements required for a deterministic random bit generator;
• their characteristics;
• their security requirements.
Where there is a requirement to produce sequences of random numbers from random bit strings, Annex B
provides guidance on how this can be performed.
Techniques for statistical testing of random bit generators for the purposes of independent verification or
validation, and detailed designs for such generators, are outside the scope of this International Standard.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO/IEC 10116, Information technology — Security techniques — Modes of operation for an n-bit block cipher
ISO/IEC 10118-3:2004, Information technology — Security techniques — Hash-functions — Part 3: Dedicated
hash-functions
ISO/IEC 18032:2004, Information technology — Security techniques — Prime number generation
ISO/IEC 18033-3:2005, Information technology — Security techniques — Encryption algorithms — Part 3:
Block ciphers
ISO/IEC 19790, Information technology — Security techniques — Security requirements for cryptographic
1)
modules
1) To be published.
© ISO/IEC 2005 – All rights reserved 1

3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.1
algorithm
clearly specified mathematical process for computation; a set of rules that, if followed, will give a prescribed
result.
3.2
backward secrecy
assurance that previous values cannot be determined from the current value or subsequent values.
3.3
biased source
source of bit strings (or numbers) from a sample space is said to be biased if some bit string(s) (or number(s))
are more likely than some other bit string(s) (or number(s)) to be chosen. Equivalently, if the sample space
consists of r elements, some elements will occur with probability different from 1/r.
cf. unbiased source
3.4
bit stream
continuous output of bits from a device or mechanism
3.5
bit string
finite sequence of ones and zeros.
3.6
black box
idealized mechanism that accepts inputs and produces outputs, but is designed such that an observer cannot
see inside the box or determine exactly what is happening inside that box.
cf. glass box
3.7
block cipher
symmetric encipherment system with the property that the encryption operates on a block of plaintext, i.e., a
string of bits of a defined length, to yield a block of ciphertext. [ISO/IEC 18033-1]
3.8
cryptographic boundary
explicitly defined continuous perimeter that establishes the physical bounds of a cryptographic module and
contains all the hardware, software and/or firmware components of a cryptographic module. [ISO/IEC 19790]
3.9
deterministic algorithm
characteristic of an algorithm that states that given the same input, the same output is always produced.
3.10
deterministic random bit generator
DRBG
random bit generator that produces a random-appearing sequence of bits by applying a deterministic
algorithm to a suitably random initial value called a seed and, possibly, some secondary inputs upon which the
security of the random bit generator does not depend. In particular, non-deterministic sources may also form
part of these secondary inputs.
2 © ISO/IEC 2005 – All rights reserved

3.11
entropy
measure of the disorder, randomness or variability in a closed system. The entropy of X is a mathematical
measure of the amount of information provided by an observation of X.
3.12
entropy source
component, device or event which produces outputs which, when captured and processed in some way,
produces a bit string containing entropy.
3.13
forward secrecy
assurance that subsequent (future) values cannot be determined from current or previous values.
3.14
glass box
idealized mechanism that accepts inputs and produces outputs and is designed such that an observer can
see inside and determine exactly what is going on.
cf. black box
3.15
hash-function
function, which maps strings of bits to fixed-length, strings of bits, satisfying the following two properties.
• It is computationally infeasible to find for a given output, an input that maps to this output.
• It is computationally infeasible to find for a given input, a second input, which maps to the same output.
NOTE Computational feasibility depends on the specific security requirements and environment. [ISO/IEC 10118-1]
3.16
human entropy source
entropy source that has some kind of random human component.
3.17
hybrid deterministic random bit generator (Hybrid DRBG)
DRBG is said to be a "hybrid" DRBG if it uses a non-deterministic entropy source as an additional entropy
source.
3.18
hybrid non-deterministic random bit generator
Hybrid NRBG
(physical or non-physical) NRBG is said to be a "hybrid" NRBG if it takes a seed value as an additional
entropy source.
3.19
initialisation value
value used in defining the starting point of a cryptographic algorithm (e.g., a hash-function or an encryption
algorithm).
3.20
Kerckhoffs box
idealized cryptosystem where the design and public keys are known to an adversary, but in which there are
secret keys and/or other private information that is not known to an adversary. A Kerckhoffs Box lies between
a black box and a glass box in terms of the knowledge of an adversary.
© ISO/IEC 2005 – All rights reserved 3

3.21
known-answer test
method of testing a deterministic mechanism where a given input is processed by the mechanism and the
resulting output is then compared to a corresponding known value.
NOTE Known Answer Testing of a deterministic mechanism may also include testing the integrity of the software
which implements the deterministic mechanism. For example, if the software implementing the deterministic mechanism is
digitally signed, the signature can be recalculated, and compared to the known signature value.
3.22
min-entropy
-k
bit string X has min-entropy k if k is the largest value such that Pr [X = x] ≤ 2 . That is, X contains k bits of
min-entropy or randomness.
NOTE Informally, this is the “best” kind of entropy.
3.23
non-deterministic random bit generator
NRBG
RBG whose security depends upon sampling an entropy source. The entropy source shall be sampled
whenever the RBG produces output, and possibly more often.
3.24
one-way function
function with the property that it is easy to compute the output for a given input but it is computationally
infeasible to find for a given output an input, which maps to this output. [ISO/IEC 11770-3]
3.25
output generation function
function in a random bit generator that outputs bits that appear to be random.
3.26
pseudorandom sequence
sequence of bits or a number is pseudorandom if it appears to be selected at random even though the
selection process is done by a deterministic algorithm.
3.27
pure deterministic random bit generator
DRBG is said to be a "pure" DRBG if all of its entropy sources are seeds.
3.28
pure non-deterministic random bit generator
(physical or non-physical) NRBG is said to be a "pure" NRBG if all of its entropy sources are non-deterministic.
3.29
random bit generator
RBG
device or algorithm that outputs a sequence of bits that appears to be statistically independent and unbiased.
3.30
reseeding
specialized internal state transition function which updates the internal state in the event that a new seed
value is supplied.
3.31
secret parameter
input to the RBG during initialisation. It provides additional entropy in the case of an entropy source failure or
compromise.
4 © ISO/IEC 2005 – All rights reserved

3.32
seed
string of bits that is used as input to a deterministic random bit generator (DRBG). The seed will determine a
portion of the state of the DRBG.
3.33
seedlife
period of time between initialising the DRBG with one seed and reseeding (fully initialising) that DRBG with
another seed.
3.34
state
state is defined as a condition of a random bit generator or any part thereof with respect to time and
circumstance.
3.35
unbiased source
source of bit strings (or numbers) from a sample space is said to be unbiased if all potential bit strings (or
numbers) have the same possibility of being chosen. Equivalently, if the sample space consists of r elements,
all elements will occur with probability 1/r.
cf. biased source
4 Symbols
For the purposes of this document, the following symbols apply.
Symbol Meaning
Pr[x] Probability[x]
IV Initialisation Value.
⎡X⎤ Ceiling: the smallest integer greater than or equal to X. For example, 5 = 5,
⎡ ⎤
⎢ ⎥
and 53. = 6.
⎡⎤
⎢⎥
X ⊕ Y Bitwise exclusive-or (also bit wise addition mod 2) of two bit strings X and Y of
the same length.
X || Y Concatenation of two strings X and Y in that order. X and Y are either both bit
strings, or both octet strings.
| a | The length in bits of string a.
x mod n The unique remainder r, 0 ≤ r ≤ n-1, when integer x is divided by n. For
example, 23 mod 7 = 2.
Used in a figure to illustrate a "switch" between sources of input.

5 Overarching objectives and requirements of a random bit generator
The properties of randomness may be demonstrated by tossing a coin in the air and observing which side is
uppermost when it lands, where one side is called “heads” (H) and the opposite is called “tails” (T). A coin also
has a rim, but the probability that a coin might land on its rim is so unlikely an occurrence that for the purpose
of this demonstration it may be ignored.
© ISO/IEC 2005 – All rights reserved 5

Flipping a coin multiple times produces an ordered series of coin flip results denoted as a series of H(s) and
T(s). For example, the sequence “HTTHT” (reading left to right) indicates a head followed by a tail, followed by
a tail, followed by a head, followed by a tail. This coin flip sequence can be translated into a binary string in a
straightforward manner by assigning H to a binary one ('1’) and T to a binary zero ('0’); the resulting example
bit string is '10010’.
5.1 Required properties of randomness
The required properties of randomness can be examined using the example of the idealized coin toss
described above. The result of each coin flip is:
1. Unpredictable: Before the flip, it is unknown whether the coin will land on heads or tails. Also, if that flip is
kept secret, it is not possible to determine what the flip was if any subsequent flip outcome is known. The
unpredictability after the flip depends on whether the observer can observe the coin flip or not. The notion
of entropy quantifies the amount of unpredictability or uncertainty relative to an observer and will be
discussed more thoroughly later in this International Standard;
2. Unbiased: That is, each potential outcome has the same chance of occurring; and
3. Independent: The coin flip is said to be uncorrelated, memoryless or historyless; whatever happened
before the current flip does not influence it.
Such a series of idealized coin flips is directly applicable to a random bit generator. The random bit generators
specified in this International Standard will try to simulate a series of idealized coin flips.
5.2 Backward and forward secrecy
As indicated above, unpredictability is a required property of a random bit generator (RBG). It should not be
possible to predict the output of a properly implemented and working random bit generator. The inability to
predict future output is known as forward secrecy. The inability to determine prior output of an RBG, given
knowledge of the current or any future output of the RBG is known as backward secrecy.
Before specifying requirements for forward and backward secrecy, the following factors should be considered.
1. In some instances, achieving backward secrecy is more important than achieving forward secrecy. For
example, if a cryptosystem is stolen, an adversary may attempt to read the old messages processed by
that system. Forward secrecy is not really a concern, since the system is no longer in use by the original
owner. Achieving backward secrecy is straightforward (for example by the appropriate use of a one-way
function in the design), although there may be a performance cost associated with providing this property,
depending on the design.
2. Trying to achieve forward secrecy may not be appropriate for some cryptosystems. For example, a smart
card may be initialised at the point of manufacture with sufficient entropy in the seed and is set to expire
after a limited time (e.g., two or three years). In this case, it may be much easier to replace the card with a
new smart card that is seeded with a different seed than it is to build forward secrecy into the RBG design.
3. In some instances, achieving forward secrecy may be more important than achieving backward secrecy.
Consider, for example, the secure generation of nonces. It is not necessary for a random bit generation
algorithm to have backward secrecy as all of the previous outputs will be known. However, forward
secrecy may be useful to prevent an adversary with knowledge of the generator from being able to predict
later outputs.
The decision whether to incorporate backward and/or forward secrecy is determined by the requirements of
the consuming application.
6 © ISO/IEC 2005 – All rights reserved

5.3 Top-level objectives and requirements for a random bit generator (RBG) output
The top-level objectives and requirements provided below are fundamental to the security of cryptographic
mechanisms that require random input.
The objectives and requirements treat the RBG as a black box, and therefore, apply to any random bit
generator, either deterministic or non-deterministic. The requirements are, basically, variations on what it
means for the output bit stream to be sufficiently random.
The threshold between feasible and infeasible shall be determined by the overall requirement for the minimum
acceptable strength of cryptographic security required by the application.
The top-level objectives and requirements for RBG output streams are as follows.
1. Under reasonable assumptions, it shall not be feasible to distinguish the output of the RBG from true
random bits that are uniformly distributed. Informally, all possible outputs occur with equal probability and
a series of outputs appears to conform to the uniform distribution.
2. Given a sequence of output bits, it shall not be known to be feasible to compute or predict any other
output bit, either past or future.
3. The output stream shall not repeat in the RBG life time except strictly by chance.
4. The RBG output shall not leak secret information, such as the internal state, from the perspective of an
adversary.
5.4 Top-level objectives and requirements for RBG operation
The top-level objectives and requirements for the operation of an RBG are as follows.
1. The RBG shall not generate bits unless the generator has been assessed to possess sufficient entropy.
The criteria for sufficiency shall be the greater of the requirements of this International Standard and the
requirements of the consuming application.
2. On detection of an error, the RBG shall either (a) enter a permanent error state, or (b) be able to recover
from a loss or compromise of entropy if the permanent error state is deemed unacceptable for the
application requirements. These requirements may be satisfied procedurally or innately in the design.
3. The design and implementation of an RBG shall have a defined protection boundary, for example, an
ISO/IEC 19790 cryptographic module boundary (see Annex I).
4. The RBG output shall not leak secret information (e.g., internal state).
5. The probability that the RBG can “misbehave” in some pathological way that violates the output
requirements (e.g., constant output or small cycles, i.e., looping such that the same output is repeated)
shall be sufficiently small. That means that the probability of error should be consistent with the overall
confidence in correct operation that is required of the RBG, which need not be the same as the required
strength of cryptographic security.
6. The RBG design shall include methods to prohibit predictable influence, manipulation, or predicting the
output of the RBG by observing the generator's physical characteristics (e.g., power consumption, timing
or emissions).
Possible optional features for the operation of an RBG are as follows.
1. If the RBG is capable of operating in more than one mode, the RBG should return information about the
mode in which it is operating, upon the request.
© ISO/IEC 2005 – All rights reserved 7

2. A consuming application may require that the RBG be run in a test mode, for example, when using a
seed that is non-secret and therefore is capable of reconstructing the bits it generates. When an RBG is
in the test mode, the RBG shall not be capable of being used to generate secret bits. When using a
secret seed, the RBG shall not operate in the test mode.
3. Considered as a glass box, an RBG should have backward secrecy. If th
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