ISO/IEC 15946-5:2017

Information technology — Security techniques — Cryptographic techniques based on elliptic curves — Part 5: Elliptic curve generation

The ISO/IEC 15946 series specifies public-key cryptographic techniques based on elliptic curves described in ISO/IEC 15946‑1. ISO/IEC 15946-5:2017 defines elliptic curve generation techniques useful for implementing the elliptic curve based mechanisms defined in ISO/IEC 29192‑4, ISO/IEC 9796‑3, ISO/IEC 11770‑3, ISO/IEC 14888‑3 and ISO/IEC 18033‑2. ISO/IEC 15946-5:2017 is applicable to cryptographic techniques based on elliptic curves defined over finite fields of prime power order (including the special cases of prime order and characteristic two). This document is not applicable to the representation of elements of the underlying finite field (i.e. which basis is used). The ISO/IEC 15946 series does not specify the implementation of the techniques it defines. Interoperability of products complying with the ISO/IEC 15946 series will not be guaranteed.

Technologies de l'information — Techniques de sécurité — Techniques cryptographiques fondées sur les courbes elliptiques — Partie 5: Génération de courbes elliptiques

General Information

• Status: Withdrawn
• Publication Date: 10-Aug-2017
• ICS: 35.030 - IT Security
• ICS: 35.040 - Information coding
• Technical Committee: ISO/IEC JTC 1/SC 27 - Information security, cybersecurity and privacy protection
• Drafting Committee: ISO/IEC JTC 1/SC 27/WG 2 - Cryptography and security mechanisms
• Current Stage: 9599 - Withdrawal of International Standard
• Completion Date: 28-Feb-2022

Relations

• Revised: ISO/IEC 15946-5:2022 - Information security — Cryptographic techniques based on elliptic curves — Part 5: Elliptic curve generation
• Effective Date: 23-Apr-2020
• Revises: ISO/IEC 15946-5:2009 - Information technology — Security techniques — Cryptographic techniques based on elliptic curves — Part 5: Elliptic curve generation
• Effective Date: 05-Nov-2015

Standards Content (Sample)

INTERNATIONAL ISO/IEC
STANDARD 15946-5
Second edition
2017-08
Information technology — Security
techniques — Cryptographic
techniques based on elliptic curves —
Part 5:
Elliptic curve generation
Technologies de l’information — Techniques de sécurité —
Techniques cryptographiques fondées sur les courbes elliptiques —
Partie 5: Génération de courbes elliptiques
Reference number
ISO/IEC 15946-5:2017(E)
©
ISO/IEC 2017

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ISO/IEC 15946-5:2017(E)

COPYRIGHT PROTECTED DOCUMENT
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ISO/IEC 15946-5:2017(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and conversion functions . 2
4.1 Symbols . 2
4.2 Conversion functions . 3
5 Framework for elliptic curve generation . 3
5.1 Types of trusted elliptic curve . 3
5.2 Overview of elliptic curve generation . 3
6 Verifiably pseudo-random elliptic curve generation . 4
6.1 General . 4
6.2 Constructing verifiably pseudo-random elliptic curves (prime case) . 4
6.2.1 Construction algorithm . 4
6.2.2 Test for near primality . 5
6.2.3 Finding a point of large prime order . 5
6.2.4 Verification of elliptic curve pseudo-randomness . 6
6.3 Constructing verifiably pseudo-random elliptic curves (binary case) . 7
6.3.1 Construction algorithm . 7
6.3.2 Verification of elliptic curve pseudo-randomness . 8
7 Constructing elliptic curves by complex multiplication . 8
7.1 General construction (prime case) . 8
7.2 Miyaji-Nakabayashi-Takano (MNT) curve . 9
7.3 Barreto-Naehrig (BN) curve .10
7.4 Freeman curve (F curve) .11
7.5 Cocks-Pinch (CP) curve . .13
8 Constructing elliptic curves by lifting .13
Annex A (informative) Background information on elliptic curves .15
Annex B (informative) Background information on elliptic curve cryptosystems .17
Annex C (informative) Numerical examples .20
Annex D (informative) Summary of properties of elliptic curves generated by the complex
multiplication method .28
Bibliography .29
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ISO/IEC 15946-5:2017(E)

Foreword
ISO (the International Organization for Standardization) and IEC (the International Electrotechnical
Commission) form the specialized system for worldwide standardization. National bodies that are
members of ISO or 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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for
the different types of document should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject
of patent rights. ISO and IEC shall not be held responsible for identifying any or all such patent
rights. Details of any patent rights identified during the development of the document will be in the
Introduction and/or on the ISO list of patent declarations received (see www .iso .org/ patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO’s adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following
URL: w w w . i s o .org/ iso/ foreword .html.
This document was prepared by Technical Committee ISO/IEC JTC 1, Information technology,
Subcommittee SC 27, IT Security techniques.
This second edition cancels and replaces the first edition (ISO/IEC 15946-5:2009), which has been
technically revised.
It also incorporates the Technical Corrigendum ISO/IEC 15946-5:2009/Cor.1:2012.
The main technical changes between the first edition and this second edition are as follows:
— the terms and definitions given in ISO/IEC 15946-1 are used;
— the scope of verifiably pseudo-random elliptic curve generation has been added;
— the numerical examples in C.4.2 and C.4.3 have been modified.
A list of all parts in the ISO/IEC 15946 series can be found on the ISO website.
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Introduction
Some of the most interesting alternatives to the RSA and F(p) based systems are cryptosystems
based on elliptic curves defined over finite fields. The concept of an elliptic curve based public-key
cryptosystem is rather simple.
— Every elliptic curve over a finite field is endowed with an addition operation “+”, under which it
forms a finite abelian group.
— The group law on elliptic curves extends in a natural way to a “discrete exponentiation” on the point
group of the elliptic curve.
— Based on the discrete exponentiation on an elliptic curve, one can easily derive elliptic curve
analogues of the well-known public-key schemes of Diffie-Hellman and ElGamal type.
The security of such a public-key system depends on the difficulty of determining discrete logarithms in
the group of points of an elliptic curve. This problem is, with current knowledge, much harder than the
factorization of integers or the computation of discrete logarithms in a finite field. Indeed, since Miller
and Koblitz independently suggested the use of elliptic curves for public-key cryptographic systems
in 1985, the elliptic curve discrete logarithm problem has only been shown to be solvable in certain
specific and easily recognizable cases. There has been no substantial progress in finding an efficient
method for solving the elliptic curve discrete logarithm problem on arbitrary elliptic curves. Thus, it
is possible for elliptic curve based public-key systems to use much shorter parameters than the RSA
system or the classical discrete logarithm-based systems that make use of the multiplicative group of a
finite field. This yields significantly shorter digital signatures and system parameters.
This document describes elliptic curve generation techniques useful for implementing the elliptic curve
based mechanisms defined in ISO/IEC 29192-4, ISO/IEC 9796-3, ISO/IEC 11770-3, ISO/IEC 14888-3, and
ISO/IEC 18033-2.
It is the purpose of this document to meet the increasing interest in elliptic curve based public-key
technology by describing elliptic curve generation methods to support key-exchange, key-transport
and digital signatures based on an elliptic curve.
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INTERNATIONAL STANDARD ISO/IEC 15946-5:2017(E)
Information technology — Security techniques —
Cryptographic techniques based on elliptic curves —
Part 5:
Elliptic curve generation
1 Scope
The ISO/IEC 15946 series specifies public-key cryptographic techniques based on elliptic curves
described in ISO/IEC 15946-1.
This document defines elliptic curve generation techniques useful for implementing the elliptic curve
based mechanisms defined in ISO/IEC 29192-4, ISO/IEC 9796-3, ISO/IEC 11770-3, ISO/IEC 14888-3 and
ISO/IEC 18033-2.
This document is applicable to cryptographic techniques based on elliptic curves defined over finite
fields of prime power order (including the special cases of prime order and characteristic two). This
document is not applicable to the representation of elements of the underlying finite field (i.e. which
basis is used).
The ISO/IEC 15946 series does not specify the implementation of the techniques it defines.
Interoperability of products complying with the ISO/IEC 15946 series will not be guaranteed.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements 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 15946-1, Information technology ― Security techniques ― Cryptographic techniques based on
elliptic curves ― Part 1: General
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO/IEC 15946-1 and the
following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http:// www .electropedia .org/
— ISO Online browsing platform: available at http:// www .iso .org/ obp
3.1
definition field of an elliptic curve
field that includes all the coefficients of the formula describing an elliptic curve
3.2
hash-function
function which maps strings of bits of variable (but usually upper bounded) length to fixed-length
strings of bits, satisfying the following two properties:
— for a given output, it is computationally infeasible to find an input which maps to this output;
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— for a given input, it is computationally infeasible to find a second input which maps to the same output
Note 1 to entry: Computational feasibility depends on the specific security requirements and environment. Refer
to ISO/IEC 10118-1:2016, Annex C.
[SOURCE: ISO/IEC 10118-1:2016, 3.4]
3.3
nearly prime number
positive integer, n =m⋅r, where m is a large prime number and r is a small smooth integer (3.5)
Note 1 to entry: The meaning of the terms large and small prime numbers is dependent on the application, and is
based on bounds determined by the designer.
3.4
order of an elliptic curve
E(F)
number of points on an elliptic curve, E, defined over a finite field, F
3.5
smooth integer
integer, r, whose prime factors are all small (i.e. less than some defined bound)
4 Symbols and conversion functions
4.1 Symbols
B
Β embedding degree, the smallest B such that number #E[F(q)] | q −1
2 3 m
Ε elliptic curve, given by a formula of the form Y = X + aX + b over the field F(p ) for p > 3, by
2 3 2 m
a formula of the form Y + XY = X + aX + b over the field F(2 ), or by a formula of the form
2 3 2 m
Y = X + aX + b over the field F(3 ), together with an extra point O referred to as the point
E
m m m
at infinity. The elliptic curve is denoted by E/F(p ), E/F(2 ), or E/F(3 ) respectively.
m
NOTE 1  In applications not based on a pairing, E/F(p) or E/F(2 ) is preferable from an
m
efficiency point of view. In applications that use a pairing, E/F(p) or E/F(3 ) is preferable
from an efficiency point of view.
NOTE 2  An elliptic curve is not only the set of points on the curve, but also a group under
an operation defined on these points.
N number of points on an elliptic curve E over F(q), #E[F(q)]
n prime divisor of #E[F(q)]
O elliptic curve point at infinity
E
p prime number
m
q prime power, p for some prime p and some integer m ≥ 1
r cofactor, that is #E[F(q)] = rn
#E[F(q)] order (or cardinality) of E[F(q)]
⎾x⏋ smallest integer greater than or equal to the real number x
⎿x⏌ largest integer smaller than or equal to the real number x
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4.2 Conversion functions
BS2IP bit string to integer conversion primitive
BS2OSP bit string to octet string conversion primitive
EC2OSP elliptic curve point to octet string conversion primitive
E
FE2IP finite field element to integer conversion primitive
F
FE2OSP finite field element to octet string conversion primitive
F
I2BSP integer to bit string conversion primitive
I2OSP integer to octet string conversion primitive
I2ECP integer to elliptic curve conversion primitive
OS2BSP octet string to bit string conversion primitive
OS2FEP octet string to finite field element conversion primitive
F
OS2ECP octet string to elliptic curve point conversion primitive
E
OS2IP octet string to integer conversion primitive
5 Framework for elliptic curve generation
5.1 Types of trusted elliptic curve
There are a number of ways in which a user can obtain trust in the provenance of an elliptic curve,
including the following.
— The curve could be obtained from an impartial trusted source (e.g. an international or national
standard).
— The curve could be generated and/or verified by a trusted third party.
— The curve could be generated and/or verified by the user.
NOTE 1 Refer to Annex A for background information on elliptic curves.
NOTE 2 Refer to Annex B for background information on elliptic curve cryptosystems.
5.2 Overview of elliptic curve generation
There are three main ways to generate elliptic curves.
— Generate an elliptic curve by applying the order counting algorithms to a (pseudo-)randomly chosen
elliptic curve. Such a technique is specified in Clause 6.
— Generate an elliptic curve by applying the complex multiplication method. Such a technique is
specified in Clause 7.
— Generate an elliptic curve by lifting an elliptic curve over a small finite field to that over a reasonably
large field. Such a technique is specified in Clause 8.
NOTE 1 Refer to Annex A for background information on elliptic curves.
NOTE 2 Refer to Annex B for background information on elliptic curve cryptosystems.
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6 Verifiably pseudo-random elliptic curve generation
6.1 General
The generation of verifiably pseudo-random elliptic curves focuses on curves over prime and binary
fields (and so, for example, does not deal with curves over fields of characteristic 3).
6.2 Constructing verifiably pseudo-random elliptic curves (prime case)
6.2.1 Construction algorithm
The following algorithm produces a set of elliptic curve parameters over a field F(p) selected (pseudo-)
randomly from the curves of appropriate order, along with sufficient information for others to verify
that the curve was indeed chosen pseudo-randomly.
NOTE 1 The algorithm is consistent with Reference [9].
NOTE 2 Methods of choosing a prime number p (pseudo) randomly are described in Reference [5].
It is assumed that the following quantities have been chosen:
— a lower bound, n , for the order of the base point;
min
— a cryptographic hash function, H, with output length L bits;
Hash
— the bit length, L, of inputs to H, satisfying L ≥ L .
Hash
The following notation is adopted below:
— v = ⎾log p⏋,
2
— s = ⎿(v − 1)/L ⏌,
Hash
— w = v − sL − 1.
Hash
Input: a prime number p; lower bound n for n; a trial division bound l .
min max
Output: a bit string X; EC parameters a, b, n, and G.
a) Choose an arbitrary bit string X of bit length L.
b) Compute h = H (X).
c) Let W be the bit string obtained by taking the w rightmost bits of h.
0
d) Let Z = BS2IP(X).
e) For i from 1 to s do:
L
1) Let X = I2BSP(Z + i mod 2 ).
i
2) Compute W = H (X ).
i i
f) Let W = W || W || … || W .
0 1 s
g) Let c = OS2FEP [BS2OSP (W)].
h) If c = 0 or 4c + 27 = 0 , then go to step a).
F F
2 3
i) Choose finite field elements a, b ∈ F(p) such that b ≠ 0 and cb − a = 0 . Choosing a = b = c will
F F
guarantee the conditions hold, and this choice is recommended.
NOTE 3 Choosing a = b = c may not be optimal from a performance perspective.
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NOTE 4 If the default values are chosen as suggested, the randomness of the generated curve is explicitly
guaranteed.
2 3
j) Compute the order #E[F(p)] of the elliptic curve E over F(p) given by y = x + ax + b.
k) Test whether #E[F(p)] is a nearly prime number using the algorithm specified in 6.2.2. If so, the
output of the algorithm specified in 6.2.2 consists of integers r, n. If not, then go to step a).
NOTE 5 The necessity of near primality is described in B.2.2
l) Check if E[F(p)] satisfies the MOV-condition specified in B.2.3, that is the smallest integer B such
B
that n divides q − 1 ensures the desirable security level. If not, then go to step a).
m) If #E[F(p)] = p, then go to step a).
NOTE 6 This check is performed in order to protect against the attack specified in B.2.2.
n) Test whether the prime divisor n satisfies the condition described in B.2.4 for cryptosystems based
on ECDLP, ECDHP, or BDHP with auxiliary inputs as in B.1.5. If not, then go to step a).
o) Generate a point G on E of order n using the algorithm specified in 6.2.3.
p) Output X, a, b, n, G.
NOTE 7 Methods to compute the order #E[F(p)] are described in References [11], [30] and [31].
6.2.2 Test for near primality
Given a lower bound n and a trial division bound l , the following procedures test N = #E[F(p)] for
min max
near primality.
Input: positive integers N, l , and n .
max min
Output: if N is nearly prime, output a prime n with n ≤ n and a smooth integer r such that N = rn. If N
min
is not nearly prime, output the message “not nearly prime”.
a) Set n = N, r = 1.
b) For l from 2 to l do:
max
1) If l is composite, then go to step 3).
2) While (l divides n)
i) Set n = n/l and r = rl.
ii) If n < n , then output “not nearly prime” and stop.
min
3) Next l.
c) Test n for primality.
d) If n is prime, then output r and n and stop.
e) Output “not nearly prime”.
NOTE Methods to test for primality are described in References [5] and [10]
6.2.3 Finding a point of large prime order
If the order #E[F(q)] of an elliptic curve E is nearly prime, the following algorithm efficiently produces a
random point in E[F(q)] whose order is the large prime factor n of #E[F(q)] = rn.
Input: an elliptic curve E over the field F(q), a prime n, and a positive integer r not divisible by n.
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Output: if #E[F(q)] = rn, a point G on E of order n; if not, the message “wrong order.”
a) Generate a random point P (not O ) on E.
E
b) Set G = rP.
c) If G = O , then go to step a).
E
d) Set Q = nG.
e) If Q ≠ O , then output “wrong order” and stop.
E
f) Output G.
6.2.4 Verification of elliptic curve pseudo-randomness
The following algorithm determines whether or not an elliptic curve over F(p) was generated using the
method of 6.2.1. The quantities L , L, v, s, and w, and the hash function H, are as in 6.2.1.
Hash
Input: a bit string X of length L, EC parameters q = p, a, b, n, and G = (x , y ), and a positive integer n .
G G min
Output: “True” or “False”.
a) Compute h = H (X).
b) Let W be the bit string obtained by taking the w rightmost bits of h.
0
c) Let Z = BS2IP(X).
d) For i from 1 to s do:
L
1) Let X = I2BSP(Z + i mod 2 ).
i
2) Compute W = H (X ).
i i
e) Let W = W || W || … || W .
0 1 s
f) Convert W to a finite field element c = OS2FEP [BS2OSP (W)].
g) Verify the following conditions.
1) n ≥ n .
min
2) n is a prime.
3) c ≠ 0 .
F
4) 4c + 27 ≠ 0 .
F
5) b ≠ 0 .
F
2 3
6) cb − a = 0 .
F
7) G ≠ O .
E
2 3
8) y = x + ax + b.
G G G
9) nG = O .
E
h) If all the conditions in step g) hold, then output “True”; otherwise output “False”.
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6.3 Constructing verifiably pseudo-random elliptic curves (binary case)
6.3.1 Construction algorithm
The following algorithm produces a set of elliptic curve parameters for a pseudo-random curve over
m
a field F(2 ), along with sufficient information for others to verify that the curve was indeed chosen
pseudo-randomly. See Annex C for additional information.
NOTE 1 The algorithm is consistent with Reference [9].
It is assumed that the following quantities have been chosen:
m
— a field F(2 );
— a lower bound n for the order of the base point;
min
— a cryptographic hash function H with output length L bits;
Hash
— the bit length L of inputs to H, satisfying L ≥ L .
Hash
The following notation is adopted below:
— s = ⎿(m − 1)/ L ⏌,
Hash
— w = m − sL .
Hash
m
Input: a field F(2 ); a lower bound n for n; a trial division bound l .
min max
Output: a bit string X; EC parameters a, b, n, and G.
a) Choose an arbitrary bit string X of bit length L.
b) Compute h = H (X).
c) Let W be the bit string obtained by taking the w rightmost bits of h.
0
d) Let Z= BS2IP(x).
e) For i from 1 to s, do:
L
1) Let X = I2BSP(Z + i mod 2 ).
i
2) Compute W = H (X ).
i i
f) Let W = W || W || … || W .
0 1 s
g) Let b = OS2FEP [BS2OSP(W)].
h) If b = 0 , then go to step a).
F
m
i) Let a be an arbitrary element in F(2 ). Choosing a = 0 will guarantee the conditions hold, and this
F
choice is recommended.
NOTE 2 The default values may not be chosen because of performance reasons.
NOTE 3 If the default values are chosen as suggested, the randomness is explicitly guaranteed.
m m 2 3 2
j) Compute the order #E[F(2 )] of the elliptic curve E over F(2 ) given by y + xy = x + ax + b.
m
NOTE 4 Methods of computing the order #E[F(2 )] are described in References [11], [30] and [33].
m
k) Test whether #E[F(2 )] is a nearly prime number using the algorithm specified in 6.2.2. If so, the
output of the algorithm specified in 6.2.2 consists of integers r, n. If not, then go to step a).
m
l) Check that E[F(2 )] satisfies the MOV-condition specified in B.2.3. If not, then go to step a).
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m) Test whether the prime divisor n satisfies the condition described in B.2.4 for cryptosystems based
on ECDLP, ECDHP, or BDHP with auxiliary inputs as in B.1.5. If not, then go to step a).
n) Generate a point G on E of order n using the algorithm specified in 6.2.3.
o) Output X, a, b, n, G.
NOTE 5 The necessity of near primality is described in B.2.2.
6.3.2 Verification of elliptic curve pseudo-randomness
The following algorithm verifies the validity of a set of elliptic curve parameters. In addition, it
m
determines whether an elliptic curve over F(2 ) was generated using the method of 6.3.1.
The quantities L , L, s, and w, and the hash function H, are as in 6.3.1.
Hash
m
Input: a bit string X of length L, EC parameters q = 2 , a, b, n, and G = (x , y ), and a positive integer n .
G G min
Output: “True” or “False”.
a) Compute h = H (X).
b) Let W be the bit string obtained by taking the w rightmost bits of h.
0
c) Let Z = BS2IP(x).
d) For i from 1 to s, do:
L
1) Let X = I2BSP(Z + i mod 2 ).
i
2) Compute W = H (X ).
i i
e) Let W = W || W || … || W
0 1 s.
f) Let b′ = OS2FEP [BS2OSP(W)].
g) Verify the following conditions.
1) n ≥ n
min
2) n is a prime.
3) b ≠ 0
F
4) b = b′
5) G ≠ O
E
2 3 2
6) y + x y = x + ax + b
G G G G G
7) nG = O
E
h) If all the conditions in step g) hold, then output “True”; otherwise output “False”.
7 Constructing elliptic curves by complex multiplication
7.1 General construction (prime case)
The following algorithm produces an elliptic curve E over F(p) with the given number of rational points N.
[10]
NOTE 1 The algorithm is based on Reference [17] which is applied to the primality proving .
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Input: the definition field F(p) and the number of points N = rn, where n is the largest prime divisor of N
and r is a cofactor.
Output: curve parameters of elliptic curve E with #E[F(p)] = N and base point G.
a) Test whether the prime divisor n satisfies the condition described in B.2.4 for cryptosystems based
on ECDLP, ECDHP, or BDHP with auxiliary inputs as in B.1.5. If not, then execute a new input.
b) Set t = p + 1 − N.
2 2
c) Choose a pair of integers (D,V) such that 4p − t = DV .
d) Construct the Hilbert class polynomial P (X).
D
e) Find a solution j in F(p) of P (X) = 0 modulo p.
0 D
f) Choose c ∈ F(p)* and construct an elliptic curve over F(p) with the j-invariant j .
0
2 3 2 3
1) E : y = x + [3c j / (1 728 − j )]x + 2c j / (1 728 − j ) (if j ≠ 0 , 1 728).
0 0 0 0 0 F
Dj,,c
0
2 3
2) E : y = x + c (if j = 0 ).
0 F
Dj,,c
0
2 3
3) E : y = x + cx (if j = 1 728).
0
D
...