ISO/IEC 15946-5:2022

INTERNATIONAL ISO/IEC
STANDARD 15946-5
Third edition
2022-02
Information security — Cryptographic
techniques based on elliptic curves —
Part 5:
Elliptic curve generation
Sécurité de l'information — Techniques cryptographiques fondées sur
les courbes elliptiques —
Partie 5: Génération de courbes elliptiques
Reference number
© ISO/IEC 2022
© ISO/IEC 2022
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ii
© ISO/IEC 2022 – All rights reserved

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 Conventions for elliptic curves . 3
5.1 Definitions of elliptic curves . 3
m
5.1.1 Elliptic curves over F(p ) . 3
m
5.1.2 Elliptic curves over F(2 ) . 4
m
5.1.3 Elliptic curves over F(3 ) . 4
5.2 Group law on elliptic curves . 4
6 Framework for elliptic curve generation . 5
6.1 Trust in elliptic curve . 5
6.2 Overview of elliptic curve generation. 5
7 Verifiably pseudo-random elliptic curve generation . 5
7.1 General . 5
7.2 Constructing verifiably pseudo-random elliptic curves (prime case) . 5
7.2.1 Construction algorithm . 5
7.2.2 Test for near primality . 7
7.2.3 Finding a point of large prime order . 7
7.2.4 Verification of elliptic curve pseudo-randomness . 7
7.3 Constructing verifiably pseudo-random elliptic curves (binary case) . 8
7.3.1 Construction algorithm . 8
7.3.2 Verification of elliptic curve pseudo-randomness . 9
8 Constructing elliptic curves by complex multiplication .10
8.1 General . 10
8.2 Barreto-Naehrig (BN) curve . 10
8.3 Barreto-Lynn-Scott (BLS) curve . . 11
9 Constructing elliptic curves by lifting .12
Annex A (informative) Background information on elliptic curves .14
Annex B (informative) Background information on elliptic curve cryptosystems .16
Annex C (informative) Background information on constructing elliptic curves by complex
multiplication .19
Annex D (informative) Numerical examples .24
Annex E (informative) Summary of properties of elliptic curves generated by the complex
multiplication method .32
Bibliography .33
iii
© ISO/IEC 2022 – All rights reserved

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
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work.
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 or
www.iec.ch/members_experts/refdocs).
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) or the IEC
list of patent declarations received (see patents.iec.ch).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of 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
www.iso.org/iso/foreword.html. In the IEC, see www.iec.ch/understanding-standards.
This document was prepared by Joint Technical Committee ISO/IEC JTC 1, Information technology,
Subcommittee ISO/IEC JTC 1/SC 27, Information security, cybersecurity and privacy protection.
This third edition cancels and replaces the second edition (ISO/IEC 15946-5:2017), which has been
technically revised.
The main changes compared to the previous edition are as follows:
— BLS curves have been added to Clause 7;
— security background for pairing-friendly curves has been added to Annex B, including the exTNFS
attack that affects the security of numerical examples of BN curves;
— except for BN curves, all other curves have been moved to Annex C;
— associated numerical examples (Annex D) and properties (Annex E) have been updated.
A list of all parts in the ISO/IEC 15946 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards
body. A complete listing of these bodies can be found at www.iso.org/members.html and
www.iec.ch/national-committees.
iv
© ISO/IEC 2022 – All rights reserved

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. With current knowledge, this problem is 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.
The purpose of this document is to meet the increasing interest in elliptic curve based public-key
technology by describing elliptic curve generation methods to support key management, encryption
and digital signatures based on an elliptic curve.
v
© ISO/IEC 2022 – All rights reserved

INTERNATIONAL STANDARD ISO/IEC 15946-5:2022(E)
Information security — 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,
ISO/IEC 18033-2 and ISO/IEC 18033-5.
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).
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 terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www. iso. org/o bp;
— IEC Electropedia: available at https:// www.e lectropedia. org/.
3.1
cryptographic hash function
function that maps octet strings of any length to octet strings of fixed length, such that it is
computationally infeasible to find correlations between inputs and outputs, and such that given one
part of the output, but not the input, it is computationally infeasible to predict any bit of the remaining
output
[SOURCE: ISO/IEC 18033-2:2006, 3.11, modified — Deleted the last phrase, "The precise security
requirements depend on the application.]
3.2
definition field of an elliptic curve
field that includes all the coefficients of the formula describing an elliptic curve
© ISO/IEC 2022 – All rights reserved

3.3
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;
— 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]
3.4
nearly prime number
positive integer, n =m⋅r, where m is a large prime number and r is a small smooth integer (3.6)
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.
3.5
order of an elliptic curve E(F)
number of points on an elliptic curve, E, defined over a finite field, F
3.6
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
0 zero element in a field F
F
a, b elliptic curve parameters
B
Β embedding degree, the smallest B such that number #E(F(q)) is a divisor of q -1
2 3 m
Ε elliptic curve, given by an equation of the form Y = X + aX + b over the field F(p ) for p > 3,
2 3 2 m
by an equation of the form Y + XY = X + aX + b over the field F(2 ), or by an equation of the
2 3 2 m
form Y = X + aX + b over the field F(3 ), together with an extra point O referred to as the
E
m m m
point at infinity; the curve is denoted by E/F(p ), E/F(2 ), or E/F(3 ), respectively
F finite field
F(p)* multiplicative group of F(p)
L, m positive integers
mod modulo, a mod b means a residue of a modulo b for integers a and b (b is positive)
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
© ISO/IEC 2022 – All rights reserved

m
q prime power, p for some prime p and some integer m ≥ 1
r cofactor, that is #E(F(q)) = rn
u positive integer
v positive integer
λ positive integer
v(i) polynomial of i
#E(F(q)) order of an elliptic curve 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
∪ or
‖ concatenation
∈ element is included in a set
4.2 Conversion functions
BS2IP bit string to integer conversion primitive
BS2OSP bit string to octet string conversion primitive
I2BSP integer to bit string conversion primitive
OS2FEP octet string to finite field element conversion primitive
NOTE Refer to ISO/IEC 15946-1:2016, Clause 7 for detailed conversion functions.
5 Conventions for elliptic curves
5.1 Definitions of elliptic curves
m
5.1.1 Elliptic curves over F(p )
m m
Let F(p ) be a definition field of an elliptic curve, where F(p ) is a finite field with a prime p > 3 and a
positive integer m. In this document, it is assumed that E is described by a “short (affine) Weierstrass
equation”, that is an equation of type
2 3 m
Y = X + aX + b with a, b ∈ F(p )
3 2 m
such that 4a + 27b ≠ 0 holds in F(p ).
F
3 2
NOTE 1 The above curve with 4a + 27b = 0 is called a singular curve, which is not an elliptic curve.
F
© ISO/IEC 2022 – All rights reserved

m
The set of F(p )-valued points of E is given by Formula (1):
m m m 2 3
E(F(p )) = {Q = (x , y ) ∈ F(p ) × F(p )|y = x + ax + b} ∪ {O} (1)
Q Q Q Q Q E
where O is an extra point referred to as the point at infinity of E.
E
m
NOTE 2 In applications not based on a pairing, E/F(p) or E/F(2 ) is preferable from an efficiency point of view.
In applications that use a pairing, E/F(p) is preferable from an efficiency point of view.
m
5.1.2 Elliptic curves over F(2 )
m
Let F(2 ) for m ≥ 1 be a definition field of an elliptic curve. In this document, it is assumed that E is
described by an equation of the type
2 3 2 m
Y + XY = X + aX + b with a, b ∈ F(2 )
m
such that b ≠ 0 holds in F(2 ).
F
For cryptographic use, m shall be a prime to prevent certain kinds of attacks on the cryptosystem.
NOTE The above curve with b = 0 is called a singular curve, which is not an elliptic curve.
F
m
The set of F(2 )-valued points of E is given by Formula (2):
m m m 2 3 2
E(F(2 )) = {Q = (x , y ) ∈ F(2 ) × F(2 )|y + x y = x + ax + b} ∪ {O} (2)
Q Q Q Q Q Q Q E
where O is an extra point referred to as the point at infinity of E.
E
m
5.1.3 Elliptic curves over F(3 )
m m
Let F(3 ) be a definition field of an elliptic curve, where F(3 ) is a finite field with a positive integer m.
In this document, it is assumed that E is described by an equation of the type
2 3 2 m
Y = X + aX + b with a, b ∈ F(3 )
m
such that a, b ≠ 0 holds in F(3 ).
F
NOTE The above curve with a or b = 0 is called a singular curve, which is not an elliptic curve.
F
m
The set of F(3 )-valued points of E is given by Formula (3):
m m m 2 3 2
E(F(3 )) = {Q = (x , y ) ∈ F(3 ) × F(3 )|y = x + ax + b} ∪ {O} (3)
Q Q Q Q Q E
where O is an extra point referred to as the point at infinity of E.
E
5.2 Group law on elliptic curves
Elliptic curves are endowed with the addition operation +: E × E → E, defining for each pair (Q , Q )
1 2
of points on E a third point Q + Q . With respect to this addition, E is an abelian group with identity
1 2
element O . The kth multiple of Q is given as kQ, where kQ = Q + …+ Q (k summands) if k > 0, kQ = (−k)
E
(−Q) if k < 0, and kQ = O if k = 0. The smallest positive k with kQ = O is called the order of Q.
E E
In order to use an elliptic curve for a cryptosystem, it is necessary to compute group law on an elliptic
curve. ISO/IEC 15946-1 shall be referred to for methods of group law on elliptic curves.
© ISO/IEC 2022 – All rights reserved

6 Framework for elliptic curve generation
6.1 Trust in elliptic curve
If an elliptic curve is poorly or maliciously generated, this can enable an adversary to break a
cryptosystem that uses it. 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 can be obtained from an impartial trusted source (e.g. an international or national
standard).
— The curve can be generated and/or verified by a trusted third party.
— The curve can 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 cryptography.
6.2 Overview of elliptic curve generation
There are three main ways to generate elliptic curves:
— by applying the order of counting algorithms to a (pseudo-)randomly chosen elliptic curve as
specified in Clause 7;
— by applying a complex multiplication method as specified in Clause 8;
— by lifting an elliptic curve over a small finite field over a reasonably large field as specified in
Clause 9.
NOTE 1 Refer to Annex A for background information on elliptic curves.
NOTE 2 Refer to Annex B for background information on elliptic curve cryptography.
7 Verifiably pseudo-random elliptic curve generation
7.1 General
The generation of verifiably pseudo-random elliptic curves focuses on curves over prime and binary
fields.
7.2 Constructing verifiably pseudo-random elliptic curves (prime case)
7.2.1 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 IEEE P1363-2000.
NOTE 2 Methods of choosing a prime number p (pseudo) randomly are described in ISO/IEC 18032.
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
© ISO/IEC 2022 – All rights reserved

— the bit length, L, of inputs to H, satisfying L ≥ L .
Hash
The following notation is adopted below:
— v = ⎾log p⏋;
— 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.
d) Let Z = BS2IP(X).
e) For i from 1 to s, do the following:
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
F F
guarantees the conditions hold, and this choice is recommended.
NOTE 3 It is possible that choosing a = b = c is not optimal from a performance perspective.
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 7.2.2. If so, the
output of the algorithm specified in 7.2.2 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 7.2.3.
p) Output X, a, b, n, G.
NOTE 7 Methods to compute the order #E(F(p)) are described in References [12], [31] and [32].
© ISO/IEC 2022 – All rights reserved

7.2.2 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 the following:
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 ISO/IEC 18032 and Reference [11].
7.2.3 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.
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.
7.2.4 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 7.2.1. The quantities L , L, v, s, and w, and the hash-function H, are as in 7.2.1.
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).
© ISO/IEC 2022 – All rights reserved

b) Let W be the bit string obtained by taking the w rightmost bits of h.
c) Let Z = BS2IP(X).
d) For i from 1 to s, do the following:
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”.
7.3 Constructing verifiably pseudo-random elliptic curves (binary case)
7.3.1 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 D for additional information.
NOTE 1 The algorithm is consistent with IEEE P1363-2000.
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
© ISO/IEC 2022 – All rights reserved

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.
d) Let Z = BS2IP(x).
e) For i from 1 to s, do the following:
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 guarantees the conditions hold, and this
F
choice is recommended.
NOTE 2 It is possible that the default values are not 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 [12], [31] and [34].
m
k) Test whether #E(F(2 )) is a nearly prime number using the algorithm specified in 7.2.2. If so, the
output of the algorithm specified in 7.2.2 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).
m) Test whether the prime divisor n satisfies the condition described in B.2.4 for cryptosystems based
on ECDLP, or ECDHP 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 7.2.3.
o) Output X, a, b, n, G.
NOTE 5 The necessity of near primality is described in B.2.2.
7.3.2 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 7.3.1.
The quantities L , L, s, and w, and the hash-function H, are as in 7.3.1.
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.
c) Let Z = BS2IP(x).
© ISO/IEC 2022 – All rights reserved

d) For i from 1 to s, do the following:
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”.
8 Constructing elliptic curves by complex multiplication
8.1 General
This document defines how to construct two types of elliptic curves by complex multiplication: Barreto-
Naehrig (BN) curves and Barreto-Lynn-Scott (BLS) curves. Methods for constructing more curves are
presented in Annex C. Numerical examples and curve properties are given in Annex D and Annex E,
respectively.
8.2 Barreto-Naehrig (BN) curve
The following algorithm produces an elliptic curve E over F(p) with embedding degree B = 12, which is
useful for cryptosystems based on bilinear pairings. The embedding degree is described in B.2.2.
NOTE 1 Detailed information is given in Reference [13].
NOTE 2 This method always generates at most one curve for a given value of m.
Input: the approximate desired size m of the curve order (in bits) and upper bound (odd integer) p
max
for the definition field.
Output: prime p, curve parameters of elliptic curve E/F(p), prime divisor n and basepoint G.
4 3 2
a) Let P(u) = 36u + 36u + 24u + 6u + 1.
m/4
b) Compute the smallest u ≈ 2 such that ⎾log P(−u)⏋ = m.
c) While p ≤ p :
max
1) t = 6u + 1;
2) p = P(−u) and N = p + 1 − t;
3) if p and N are prime, then go to step e);
© ISO/IEC 2022 – All rights reserved

4) p = P(u) and N = p + 1 − t;
5) if p and N are prime, then go to step e);
6) u = u + 1 and go to step 1).
d) Stop and output “fail”.
e) 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).
2 3
f) Set an elliptic curve E: y = x + b with b = 1.
1) Test whether E has order n. If not, then b = b +1 (mod p).
g) x = 1:
1) if x + b does not have square roots, then x = x + 1 and go to 1);
0 0 0
2) compute y = √(x + b) mod p:
0 0
i) Set the basepoint G = (x , y ) ∈ E;
0 0
ii) If nG ≠ O , then x = x + 1 and go to 1).
E 0 0
h) Output p, E, n, and G.
NOTE 3 A technique for speeding up a protocol based on a bilinear pairing is described in Reference [13].
8.3 Barreto-Lynn-Scott (BLS) curve
The following algorithm produces an elliptic curve E over F(p) with embedding degree B = 12, 24 or 48
which is useful for cryptosystems based on a bilinear pairing. The embedding degree is described in
B.2.2.
[35] [36]
NOTE 1 Detailed information is given in and .
NOTE 2 This method always generates at most one curve for a given value of m.
Input: the approximate desired size m of the curve order (in bits) and upper bound (odd integer) p
max
for the definition field.
Output: prime p, curve parameter of elliptic curve E/F(p), prime divisor n and base point G.
a) Set B = 12, 24 or 48.
4 2 m/4
1) If B = 12, then L(u) = (u – u + 1) and the smallest u = 2 for ⎾ log L(−u)⏋ = m.
8 4 m/8
2) If B = 24, then L(u) = (u – u + 1) and the smallest u = 2 for ⎾ log L(−u)⏋ = m.
16 8 m/16
3) If B = 48, then L(u) = (u – u + 1) and the smallest u = 2 for⎾ log L(−u)⏋ = m.
2 2
b) Set P(u) = (u - 1) L(u)/3+ u and R(u) = (u-1) /3.
c) While p ≤ p :
max
1) t = u + 1;
2) p = P(−u), n = L(−u) and r = R(−u);
3) if p and n are primes then go to step e);
4) p = P(u), n = L(u), and r = R(u);
5) if p and n are primes, then go to step e);
© ISO/IEC 2022 – All rights reserved

6) u = u + 1 and go to step 1).
d) Stop and output “fail”.
e) 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).
2 3
f) Set an elliptic curve E: y = x + b with b = 1.
1) Test whether E has order n. If not, then b = b +1 (mod p).
g) x = 1.
h) While x ≤ p:
1) if x + b does not have square roots, then x = x + 1 and go to 1);
0 0 0
2) compute y = √(x + b) mod p.
0 0
i) Put G = (x , y ) ∈ E.
0 0 0
ii) If rG = O then x = x + 1 and go to 1).
E 0 0
iii) Set G = rG .
iv) If nG ≠ O , then x = x + 1 and go to 1).
E 0 0
i) Stop and output “fail”.
j) Output p, E, n and G.
9 Constructing elliptic curves by lifting
m
The following algorithm produces an elliptic curve E over F(p ) by lifting an elliptic curve E over F(p).
NOTE 1 The algorithm is based on Reference [33].
Input: small finite field F(p), elliptic curve E over F(p), lower and upper bound N and N for the
min max
order of elliptic curve (in bits).
m
Output: extension degree m, order N = #E(F(p )), basepoint G, and order n of G.
m
a) Count the order of N = #E(F(p)), which is easily executed since F(p) is small.
b) Set t = p + 1 − N and compute algebraic integers α and β that satisfy t = α + β and p = αβ.
c) Set m = 1.
Find a triple of (m, N , n) as follows:
m
m m m
1) Compute N = pm + 1– (α + β ) and q = p , which is an integer.
m
2) If N < N , then m = m + 1 and go to Step 1).
m min
3) If N > N , then stop and output “fail”.
m max
4) Test whether N is a nearly prime number using the algorithm specified in 7.2.2. If so, the
m
output of 7.2.2 consists of the integers r and n. If not, then m = m + 1 and go to step 1).
5) Check whether E(F(q)) satisfies the MOV-condition, that is the smallest integer B such that n
B
divides q – 1 ensures the desirable security level. If not, then m = m + 1 and go to step 1).
NOTE 2 See the MOV-condition in B.2.3.
© ISO/IEC 2022 – All rights reserved

e) Generate a point G on E(F(q)) of order n using the algorithm specified in 7.2.3.
f) Output an extension degree m, the order N = #E(F(q)), a basepoint G and the order n.
m
© ISO/IEC 2022 – All rights reserved

Annex A
(informative)
Background information on elliptic curves
A.1 j-invariant
m
Let F(q) be a finite field with q = p , where prime p > 3. Let E be an elliptic curve over F(q) given by the
short Weierstrass equation:
2 3
Y = X + aX + b with a, b ∈ F(q),
3 2
where the inequality 4a + 27b ≠ 0 holds in F(q). Then, the j-invariant is defined as:
F
3 3 2
j = 1 728·(4a )/(4a + 27b )
m m
Let F(2 ), for some m ≥ 1, be a finite field. Let E be an elliptic curve over F(2 ) given by:
2 3 2 m
Y + XY = X + a X + b with a, b ∈ F(2 ),
where b ≠ 0 . Then, the j-invariant is defined as:
F
j = 1/b
m m
Let F(3 ), for some m ≥ 1, be a finite field. Let E be an elliptic curve over F(3 ) given by:
2 3 2 m
Y = X + aX + b with a, b ∈ F(3 )
such that a, b ≠ 0 . Then, the j-invariant is defined as:
F
j = −a /b
A.2 Hilbert class polynomial
The construction of elliptic curves by complex multiplication uses the theory of imaginary quadratic
fields Q(√−D). In the case of the imaginary quadratic field Q(√−D), the Hilbert class field K is the
extension field of Q(√−D), which is the unramified abelian extension of Q(√−D). The Hilbert class
polynomial P (X) is defined by the minimum polynomial of K over Q(√−D). In the construction of elliptic
D
curves by complex multiplication, the fact that the j-invariants of elliptic curves E/F(p) are given as a
solution of a Hilbert class polynomial P (X) mod p is used.
D
NOTE 1 These facts are described in References [14] and [17].
NOTE 2 Online databases of Hilbert class polynomials are available in Reference [22].
A.3 Cryptographic pairing
A cryptographic pairing e satisfies the conditions of non-degeneracy, bilinearity, and computability. A
n
pairing e is defined over < G > × < G > as follows:
n 1 2
e : < G > × < G > → μ
n 1 2 n
© ISO/IEC 2022 – All rights reserved

where
< G > and < G > are the cyclic groups of order n;
1 2
μ is the cyclic group of the nth roots of unity.
n
[37]
A pairing e is realized by restricting the domain of the Weil, Tate pairings or optimal Ate pairings .
n
A.4 Pell equation
The Pell equation is of the form:
2 2
T − dU = ±1
where d is a fixed integer. In the construction of elliptic curves by complex multiplication, the Pell
equation with a positive integer d that is not a perfect square is used. Then, all positive integer solutions
of (T, U) are given by using the least positive solution (T , U ) with the smallest U > 0 as follows:
0 0 0
k
T + U√d = (T + U √d)
0 0
where k = 1, 2, ….
NOTE These facts are described in Reference [29].
2 2
A.5 Diophantine equation, x − dy = n
2 2
In the construction of elliptic curves by complex multiplication, the Diophantine equation, x − dy = n,
is used where n is an integer and d is a positive integer that is not a perfect square. The number of
integer solutions of this formula is zero or infinite. An infinite number of integer solutions (x, y) are
given by using the least positive solution (T , U ) with the smallest U > 0 of the related Pell equation of
0 0 0
2 2
T − dU = 1.
NOTE Details are described in Reference [29].
© ISO/IEC 2022 – All rights reserved

Annex B
(informative)
Background information on elliptic curve cryptosystems
B.1 Definition of cryptographic problems
B.1.1 Elliptic curve discrete logarithm problem (ECDLP)
For an elliptic curve E/F(q), the base point G ∈ E(F(q)) with order n, and a point P∈E(F(q)), the elliptic
curve discrete logarithm problem (with respect to the base point G) is to find the integer x ∈ (0, n−1)
such that P = xG if such an x exists.
The security of elliptic curve cryptosystems is based on the believed hardness of the elliptic curve
discrete logarithm problem.
B.1.2 Computational elliptic curve Diffie-Hellman problem (ECDHP)
For an elliptic curve E/F(q), the base point G ∈ E(F(q)) with order n, and points aG, bG ∈ E(F(q)), the
computational elliptic curve Diffie-Hellman problem is to compute abG.
The security of some elliptic curve cryptosystems is based on the believed hardness of the computational
elliptic curve Diffie-Hellman problem.
B.1.3 Decisional elliptic curve Diffie-Hellman problem (ECDDHP)
For an elliptic curve E/F(q), the base point G ∈ E(F(q)) with order n, and points aG, bG, Y ∈ E(F(q)), the
decisional elliptic curve Diffie-Hellman problem is to decide whether Y = abG or not.
The security of some elliptic curve cryptosystems is based on the believed hardness of the decisional
elliptic curve Diffie-Hellman problem.
B.1.4 Bilinear Diffie-Hellman problem (BDHP)
The bilinear Diffie-Hellman problems are described in two ways according to the corresponding
cryptographic bilinear maps.
— For two groups < G > and < G > with order n, a cryptographic bilinear map e : < G > ×
1 2 n 1
< G > → μ , aG , bG ∈ < G > , and aG , cG ∈ < G > , the bilinear Diffie-Hellman problem is to
2 n 1 1 1 2 2 2
abc
compute e (G , G ) .
n 1 2
— For a group < G > with order n, a cryptographic bilinear map e : < G > × < G > → μ , and
1 n 1 1 n
abc
aG , bG , cG ∈ < G > , the bilinear Diffie-Hellman problem is to compute e (G , G ) .
1 1 1 1 n 1 1
The security of some elliptic curve cryptosystems is based on the believed hardness of the elliptic curve
bilinear Diffie-Hellman problem.
B.1.5 Elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP with
auxiliary inputs)
The security of some cryptosystems is based on the elliptic curve discrete logarithm problem with
auxiliary inputs.
2 3 k
— ECDLP with additional inputs x G, x G, …, x G.
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2 3 k
— ECDHP with additional inputs a G, a G, …, a G.
2 3 k
— BDHP with additional inputs a G , a G , …, a G .
1 1 1
Three examples of elliptic curve problems with auxiliary inputs are as follows (the notation follows
from the original definitions of the problems in B.1.1, B.1.2, and B.1.4).
B.2 Algorithms to determine discrete logarithms on elliptic curves
B.2.1 Hardness of ECDLP
The hardness of ECDLP depends on the selected elliptic curves E/F(q) and the size n of the or
...