Identity-Based Cryptography Standard (IBCS) #1: Supersingular Curve Implementations of the BF and BB1 Cryptosystems
draft-martin-ibcs-07
The information below is for an old version of the document that is already published as an RFC.
| Document | Type |
This is an older version of an Internet-Draft that was ultimately published as RFC 5091.
|
|
|---|---|---|---|
| Authors | Mark J. Schertler , Xavier Boyen | ||
| Last updated | 2020-01-21 (Latest revision 2007-09-26) | ||
| RFC stream | Internet Engineering Task Force (IETF) | ||
| Intended RFC status | Informational | ||
| Formats | |||
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| Stream | WG state | (None) | |
| Document shepherd | (None) | ||
| IESG | IESG state | Became RFC 5091 (Informational) | |
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|
||
| Consensus boilerplate | Unknown | ||
| Telechat date | (None) | ||
| Responsible AD | Tim Polk | ||
| Send notices to | mark@voltage.com |
draft-martin-ibcs-07
X. Boyen
L. Martin
Internet Draft Voltage Security
Expires: March 2008 September 2007
Identity-Based Cryptography Standard (IBCS) #1: Supersingular
Curve Implementations of the BF and BB1 Cryptosystems
<draft-martin-ibcs-07.txt>
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Abstract
This document describes the algorithms that implement Boneh-
Franklin and Boneh-Boyen Identity-based Encryption. This
document is in part based on IBCS #1 v2 of Voltage Security's
Identity-based Cryptography Standards (IBCS) documents, from
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which some irrelevant sections have been removed to create the
content of this document.
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Table of Contents
1. Introduction..............................................5
1.1. Sending a Message that is Encrypted Using IBE........6
1.1.1. Sender Obtains Recipient's Public Parameters....7
1.1.2. Construct and Send IBE-encrypted Message........8
1.2. Receiving and Viewing an IBE-encrypted Message.......8
1.2.1. Recipient Obtains Public Parameters from PPS....9
1.2.2. Recipient Obtains IBE Private Key from PKG.....10
1.2.3. Recipient Decrypts IBE-encrypted Message.......10
2. Notation and definitions.................................11
2.1. Notation............................................11
2.2. Definitions.........................................13
3. Basic elliptic curve algorithms..........................14
3.1. The group action in affine coordinates..............14
3.1.1. Implementation for type-1 curves...............14
3.2. Point multiplication................................16
3.3. Operations in Jacobian projective coordinates.......18
3.3.1. Implementation for type-1 curves...............18
3.4. Divisors on elliptic curves.........................20
3.4.1. Implementation in F_p^2 for type-1 curves......20
3.5. The Tate pairing....................................23
3.5.1. Tate pairing calculation.......................23
3.5.2. The Miller algorithm for type-1 curves.........23
4. Supporting algorithms....................................26
4.1. Integer range hashing...............................26
4.1.1. Hashing to an integer range....................26
4.2. Pseudo-random byte generation by hashing............27
4.2.1. Keyed pseudo-random bytes generator............27
4.3. Canonical encodings of extension field elements.....28
4.3.1. Encoding an extension element as a string......28
4.3.2. Type-1 curve implementation....................29
4.4. Hashing onto a subgroup of an elliptic curve........30
4.4.1. Hashing a string onto a subgroup of an elliptic
curve.................................................30
4.4.2. Type-1 curve implementation....................30
4.5. Bilinear mapping....................................31
4.5.1. Regular or modified Tate pairing...............31
4.5.2. Type-1 curve implementation....................32
4.6. Ratio of bilinear pairings..........................33
4.6.1. Ratio of regular or modified Tate pairings.....33
4.6.2. Type-1 curve implementation....................34
5. The Boneh-Franklin BF cryptosystem.......................34
5.1. Setup...............................................34
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5.1.1. Master secret and public parameter generation..34
5.1.2. Type-1 curve implementation....................35
5.2. Public key derivation...............................37
5.2.1. Public key derivation from an identity and public
parameters............................................37
5.3. Private key extraction..............................37
5.3.1. Private key extraction from an identity, a set of
public parameters and a master secret.................37
5.4. Encryption..........................................38
5.4.1. Encrypt a session key using an identity and
public parameters.....................................38
5.5. Decryption..........................................40
5.5.1. Decrypt an encrypted session key using public
parameters, a private key.............................40
6. The Boneh-Boyen BB1 cryptosystem.........................41
6.1. Setup...............................................41
6.1.1. Generate a master secret and public parameters.41
6.1.2. Type-1 curve implementation....................42
6.2. Public key derivation...............................43
6.2.1. Derive a public key from an identity and public
parameters............................................43
6.3. Private key extraction..............................44
6.3.1. Extract a private key from an identity, public
parameters and a master secret........................44
6.4. Encryption..........................................45
6.4.1. Encrypt a session key using an identity and
public parameters.....................................45
6.5. Decryption..........................................47
6.5.1. Decrypt using public parameters and private key47
7. Test data................................................50
7.1. Algorithm 3.2.2 (PointMultiply).....................50
7.2. Algorithm 4.1.1 (HashToRange).......................50
7.3. Algorithm 4.5.1 (Pairing)...........................51
7.4. Algorithm 5.2.1 (BFderivePubl)......................51
7.5. Algorithm 5.3.1 (BFextractPriv).....................52
7.6. Algorithm 5.4.1 (BFencrypt).........................52
7.7. Algorithm 6.3.1 (BBextractPriv).....................53
7.8. Algorithm 6.4.1 (BBencrypt).........................54
8. ASN.1 module.............................................55
9. Security considerations..................................60
10. IANA considerations.....................................63
11. Acknowledgments.........................................63
12. References..............................................64
12.1. Normative references...............................64
12.2. Informative references.............................64
Authors' Addresses..........................................65
Intellectual Property Statement.............................65
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Disclaimer of Validity......................................66
Copyright Statement.........................................66
Acknowledgment..............................................66
1. Introduction
This document provides a set of specifications for
implementing identity-based encryption (IBE) systems based on
bilinear pairings. Two cryptosystems are described: the IBE
system proposed by Boneh and Franklin (BF) [BF], and the IBE
system proposed by Boneh and Boyen (BB1) [BB1]. Fully secure
and practical implementations are described for each system,
comprising the core IBE algorithms as well as ancillary hybrid
components used to achieve security against active attacks.
These specifications are restricted to a family of
supersingular elliptic curves over finite fields of large
prime characteristic, referred to as "type-1" curves (see
Section 2.1). Implementations based on other types of curves
currently fall outside the scope of this document.
IBE is a public-key technology, but one which varies from
other public-key technologies is a slight yet significant way.
In particular, IBE keys are calculated instead of being
generated randomly, which leads to a different architecture
for a system using IBE than for a system using other public-
key technologies. An overview of these differences and how a
system using IBE works are given in [IBEARCH].
Identity-based encryption (IBE) is a public-key encryption
technology that allows a public key to be calculated from an
identity and the corresponding private key to be calculated
from the public key. Calculation of both the public and
private keys in an IBE-based system can occur as needed,
resulting in just-in-time key material. This contrasts with
other public-key systems [P1363], in which keys are generated
randomly and distributed prior to secure communication
commencing. The ability to calculate a recipient's public key,
in particular, eliminates the need for the sender and receiver
in an IBE-based messaging system to interact with each other,
either directly or through a proxy such as a directory server,
before sending secure messages.
This document describes an IBE-based messaging system and how
the components of the system work together. The components
required for a complete IBE messaging system are the
following:
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o A Private-key Generator (PKG). The PKG contains the
cryptographic material, known as a master secret, for
generating an individual's IBE private key. A PKG
accepts an IBE user's private key request and after
successfully authenticating them in some way returns
the IBE private key.
o A Public Parameter Server (PPS). IBE System
Parameters include publicly sharable cryptographic
material, known as IBE public parameters, and policy
information for the PKG. A PPS provides a well-known
location for secure distribution of IBE public
parameters and policy information for the IBE PKG.
A logical architecture would be to have a PKG/PPS per a name
space, such as a DNS zone. The organization that controls the
DNS zone would also control the PKG/PPS and thus the
determination of which PKG/PSS to use when creating public and
private keys for the organization's members. In this case the
PPS URI can be uniquely created by the form of the identity
that it supports. This architecture would make it clear which
set of public parameters to use and where to retrieve them for
a given identity.
IBE encrypted messages can use standard message formats, such
as the Cryptographic Message Syntax [CMS]. How to use IBE with
CMS is defined in [IBECMS].
Note that IBE algorithms are used only for encryption, so if
digital signatures are required they will need to be provided
by an additional mechanism.
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 [KEYWORDS].
1.1. Sending a Message that is Encrypted Using IBE
In order to send an encrypted message, an IBE user must
perform the following steps:
1. Obtain the recipient's public parameters
The recipient's IBE public parameters allow the creation
of unique public and private keys. A user of an IBE
system is capable of calculating the public key of a
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recipient after he obtains the public parameters for
their IBE system. Once the public parameters are
obtained, IBE-encrypted messages can be sent.
2. Construct and Send IBE-encrypted Message
All that is needed, in addition to the IBE public
parameters, is the recipient's identity in order to
generate their public key for use in encrypting messages
to them. When this identity is the same as the identity
that a message would be addressed to, then no more
information is needed from a user to send someone a
secure message then is needed to send them an unsecured
message. This is one of the major benefits of an IBE-
based secure messaging system. Examples of identities
can be an individual, group, or role identifiers.
1.1.1. Sender Obtains Recipient's Public Parameters
The sender of a message obtains the IBE public parameters that
he needs for calculating the IBE public key of the recipient
from a PPS that is hosted at a well-known URI. The IBE public
parameters contain all of the information that the sender
needs to create an IBE-encrypted message except for the
identity of the recipient. [IBEARCH] describes the URI where a
PPS is located, the format of IBE public parameters, and how
to obtain them. The URI from which users obtain IBE public
parameters MUST be authenticated in some way; PPS servers MUST
support TLS 1.1 [TLS] to satisfy this requirement and MUST
verify that the subject name in the server certificate matches
the URI of the PPS. [IBEARCH] also describes the way in which
identity formats are defined and a minimum interoperable
format that all PPSs and PKGs MUST support. This step is shown
below in Figure 1.
IBE Public Parameter Request
----------------------------->
Sender PPS
<-----------------------------
IBE Public Parameters
Figure 1 Requesting IBE Public Parameters
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The sender of an IBE-encrypted message selects the PPS and
corresponding PKG based on his local security policy.
Different PPSs may provide public parameters that specify
different IBE algorithms or different key strengths, for
example, or require the use of PKGs that require different
levels of authentication before granting IBE private keys.
1.1.2. Construct and Send IBE-encrypted Message
To IBE-encrypt a message, the sender chooses a content
encryption key (CEK) and uses it to encrypt his message and
then encrypts the CEK with the recipient's IBE public key (for
example, as described in [CMS]). This operation is shown below
in Figure 2. This document describes the algorithms needed to
implement two forms of IBE. [IBECMS] describes how to use the
Cryptographic Message Syntax (CMS) to encapsulate the
encrypted message along with the IBE information that the
recipient needs to decrypt the message.
CEK ----> Sender ----> IBE-encrypted CEK
^
|
|
Recipient's Identity
and IBE Public Parameters
Figure 2 Using an IBE Public-key Algorithm to Encrypt
1.2. Receiving and Viewing an IBE-encrypted Message
In order to read an encrypted message, a recipient of an IBE-
encrypted message parses the message (for example, as
described in [IBECMS]). This gives him the URI he needs to
obtain the IBE public parameters required to perform IBE
calculations as well as the identity that was used to encrypt
the message. Next the recipient must carry out the following
steps:
1. Obtain the recipient's public parameters
An IBE system's public parameters allow it to uniquely
create public and private keys. The recipient of an IBE-
encrypted message can decrypt an IBE-encrypted message
if he has both the IBE public parameters and the
necessary IBE private key. The PPS can also provide the
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URI of the PKG where the recipient of an IBE-encrypted
message can obtain the IBE private keys.
2. Obtain the IBE private key from the PKG
To decrypt an IBE-encrypted message, in addition to the
IBE public parameters the recipient needs to obtain the
private key that corresponds to the public key that the
sender used. The IBE private key is obtained after
successfully authenticating to a private key generator
(PKG), a trusted third party that calculates private
keys for users. The recipient receives the IBE private
key over an HTTPS connection. The URI of a PKG MUST be
authenticated in some way; PKG servers MUST support TLS
1.1 [TLS] to satisfy this requirement.
3. Decrypt IBE-encrypted message
The IBE private key decrypts the CEK, which is then used
to decrypt encrypted message.
The PKG may allow users other than the intended recipient to
receive some IBE private keys. Giving a mail filtering
appliance permission to obtain IBE private keys on behalf of
users, for example, can allow the appliance to decrypt and
scan encrypted messages for viruses or other malicious
features.
1.2.1. Recipient Obtains Public Parameters from PPS
Before he can perform any IBE calculations related to the
message that he has received, the recipient of an IBE-
encrypted message needs to obtain the IBE public parameters
that were used in the encryption operation. This operation is
shown below in Figure 3.
IBE Public Parameter Request
----------------------------->
Recipient PPS
<-----------------------------
IBE Public Parameters
Figure 3 Requesting IBE Public Parameters
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1.2.2. Recipient Obtains IBE Private Key from PKG
To obtain an IBE private key, the recipient of an IBE-
encrypted message provides the IBE public key used to encrypt
the message and their authentication credentials to a PKG and
requests the private key that corresponds to the IBE public
key. Section 4 of this document defines the protocol for
communicating with a PKG as well as a minimum interoperable
way to authenticate to a PKG that all IBE implementations MUST
support. Because the security of IBE private keys is vital to
the overall security of an IBE system, IBE private keys MUST
be transported to recipients over a secure protocol. PKGs MUST
support TLS 1.1 [TLS] for transport of IBE private keys. This
operation is shown below in Figure 4.
IBE Private Key Request
---------------------------->
Recipient PKG
<----------------------------
IBE Private Key
Figure 4 Obtaining an IBE Private Key
1.2.3. Recipient Decrypts IBE-encrypted Message
After obtaining the necessary IBE private key, the recipient
uses that IBE private key and the corresponding IBE public
parameters to decrypt the CEK. This operation is shown below
in Figure 5. He then uses the CEK to decrypt the encrypted
message content (for example, as specified in [IBECMS]).
IBE-encrypted CEK ----> Recipient ----> CEK
^
|
|
IBE Private Key
and IBE Public Parameters
Figure 5 Using an IBE Public-key Algorithm to Decrypt
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2. Notation and definitions
2.1. Notation
This section summarizes the notions and definitions regarding
identity-based cryptosystems on elliptic curves. The reader is
referred to [ECC] for the mathematical background and to [2,
3] regarding all notions pertaining to identity-based
encryption.
F_p denotes finite field of prime characteristic p; F_p^2
denote its extension field of degree 2.
Let E/F_p: y^2 = x^3 + a * x + b be an elliptic curve over
F_p. For an extension of degree 2, the curve E/F_p defines a
group (E(F_p^2), +), which is the additive group of points of
affine coordinates (x, y) in (F_p^2)^2 satisfying the curve
equation over F_p^2, with null element, or point at infinity,
denoted 0.
Let q be a prime such that E(F_p) has a cyclic subgroup G1' of
order q.
Let G1'' be a cyclic subgroup of E(F_p^2) of order q, and G2
be a cyclic subgroup of (F_p^2)* of order p.
Under these conditions, a mathematical construction known as
the Tate pairing provides an efficiently computable map e: G1'
x G1'' -> G2 that is linear in both arguments and believed
hard to invert [BF]. If an efficiently computable non-rational
endomorphism phi: G1' -> G1'' is available for the selected
elliptic curve on which the Tate pairing is computed, then we
can construct a function e': G1' x G1'' -> G2, defined as
e'(A, B) = e(A, phi(B)), called the modified Tate pairing. We
generically call a pairing either the Tate pairing e or the
modified Tate pairing e', depending on the chosen elliptic
curve used in a particular implementation.
The following additional notation is used throughout this
document.
p - A 512-bit to 7680-bit prime which is the order of the
finite field F_p.
F_p - The base finite field of order p over which the elliptic
curve of interest E/F_p is defined.
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#G - The size of the set G.
F* - The multiplicative group of the non-zero elements in the
field F; e.g., (F_p)* is the multiplicative group of the
finite field F_p.
E/F_p - The equation of an elliptic curve over the field F_p,
which, when p is neither 2 nor 3, is of the form E/F_p: y^2 =
x^3 + a * x + b, for specified a, b in F_p.
0 - The null element of any additive group of points on an
elliptic curve, also called the point at infinity.
E(F_p) - The additive group of points of affine coordinates
(x, y), with x, y in F_p, that satisfy the curve equation
E/F_p, including the point at infinity 0.
q - A 160-bit to 512-bit prime that is the order of the cyclic
subgroup of interest in E(F_p).
k - The embedding degree of the cyclic subgroup of order q in
E(F_p). For type-1 curves this is always equal to 2.
F_p^2 - The extension field of degree 2 of the field F_p.
E(F_p^2) - The group of points of affine coordinates in F_p^2
satisfying the curve equation E/F_p, including the point at
infinity 0.
Z_p - The additive group of integers modulo p.
lg - The base 2 logarithm function, so that 2^lg(x) = x.
The term "object identifier" will be abbreviated "OID."
A Solinas prime is a prime of the form 2^a (+/-) 2^b (+/-) 1.
The following conventions are assumed for curve operations.
Point addition - If A and B are two points on a curve E, their
sum is denoted A + B.
Point multiplication - If A is a point on a curve, and n an
integer, the result of adding A to itself a total of n times
is denoted [n]A.
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The following class of elliptic curves is exclusively
considered for pairing operations in the present version of
this document, which are referred to as "type-1" curves.
Type-1 curves - The class of curves of type 1 is defined as
the class of all elliptic curves of equation E/F_p: y^2 = x^3
+ 1 for all primes p congruent to 11 modulo 12. This class
forms a subclass of the class of supersingular curves. These
curves satisfy #E(F_p) = p + 1, and the p points (x, y) in
E(F_p) \ {0} have the property that x = (y^2 - 1)^(1/3) (mod
p). Type-1 curves always have an embedding degree k = 2.
Groups of points on type-1 curves are plentiful and easy to
construct by random selection of a prime p of the appropriate
form. Therefore, rather than to standardize upon a small set
of common values of p, it is henceforth assumed that all type-
1 curves are freshly generated at random for the given
cryptographic application (an example of such generation will
be given in Algorithm 5.1.2 (BFsetup1) or Algorithm 6.1.2
(BBsetup1)). Implementations based on different classes of
curves are currently unsupported.
We assume that the following concrete representations of
mathematical objects are used.
Base field elements - The p elements of the base field F_p are
represented directly using the integers from 0 to p - 1.
Extension field elements - The p^2 elements of the extension
field F_p^2 are represented as ordered pairs of elements of
F_p. An ordered pair (a_0, a_1) is interpreted as the complex
number a_0 + a_1 * i, where i^2 = -1. This allows operations
on elements of F_p^2 to be implemented as follows. Suppose
that a = (a_0, a_1) and b = (b_0, b_1) are elements of F_p^2.
Then a + b = ((a_0 + b_0)(mod p), (a_1 + b_1)(mod p)) and a *
b = ((a_1 * b_1 - a_0 * b_0)(mod p), (a_1 * b_0 + a_0 *
b_1)(mod p)).
Elliptic curve points - Points in E(F_p^2) with the point P =
(x, y) in F_p^2 x F_p^2 satisfying the curve equation E/F_p.
Points not equal to 0 are internally represented using the
affine coordinates (x, y), where x and y are elements of
F_p^2.
2.2. Definitions
The following terminology is used to describe an IBE system.
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Public parameters - The public parameters are set of common
system-wide parameters generated and published by the private
key server (PKG).
Master secret - The master secret is the master key generated
and privately kept by the key server, and used to generate the
private keys of the users.
Identity - An identity an arbitrary string, usually a human-
readable unambiguous designator of a system user, possibly
augmented with a time stamp and other attributes.
Public key - A public key is a string that is algorithmically
derived from an identity. The derivation may be performed by
anyone, autonomously.
Private key - A private key is issued by the key server to
correspond to a given identity (and the public key that
derives from it), under the published set of public
parameters.
Plaintext - A plaintext is an unencrypted representation, or
in the clear, of any block of data to be transmitted securely.
For the present purposes, plaintexts are typically session
keys, or sets of session keys, for further symmetric
encryption and authentication purposes.
Ciphertext - A ciphertext is an encrypted representation of
any block of data, including a plaintext, to be transmitted
securely.
3. Basic elliptic curve algorithms
This section describes algorithms for performing all needed
basic arithmetic operations on elliptic curves. The
presentation is specialized to the type of curves under
consideration for simplicity of implementation. General
algorithms may be found in [ECC].
3.1. The group action in affine coordinates
3.1.1. Implementation for type-1 curves
Algorithm 3.1.1 (PointDouble1): adds a point to itself on a
type-1 elliptic curve.
Input:
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o A point A in E(F_p^2), with A = (x, y) or 0
o An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
o The point [2]A = A + A
Method:
1. If A = 0 or y = 0, then return 0
2. Let lambda = (3 * x^2) / (2 * y)
3. Let x' = lambda^2 - 2 * x
4. Let y' = (x - x') * lambda - y
5. Return (x', y')
Algorithm 3.1.2 (PointAdd1): adds two points on a type-1
elliptic curve.
Input:
o A point A in E(F_p^2), with A = (x_A, y_A) or 0
o A point B in E(F_p^2), with B = (x_B, y_B) or 0
o An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
o The point A + B
Method:
1. If A = 0, return B
2. If B = 0, return A
3. If x_A = x_B:
(a) If y_A = -y_B, return 0
(b) Else return [2]A computed using Algorithm 3.1.1
(PointDouble1)
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4. Otherwise:
(a) Let lambda = (y_B - y_A) / (x_B - x_A)
(b) Let x' = lambda^2 - x_A - x_B
(c) Let y' = (x_A - x') * lambda - y_A
(d) Return (x', y')
3.2. Point multiplication
Algorithm 3.2.1 (SignedWindowDecomposition): computes the
signed m-ary window representation of a positive integer
[ECC].
Input:
o An integer k > 0, where k has the binary representation k =
{Sum(k_j * 2^j, for j = 0 to l} where each k_j is either 0
or 1 and k_l = 0
o An integer window bit-size r > 0
Output:
o An integer d and the unique d-element sequence {(b_i, e_i),
for i = 0 to d - 1} such that k = {Sum(b_i * 2^(e_i), for i
= 0 to d - 1}, each b_i = +/- 2^j for some 0 < j <= r - 1
and each e_i is a non-negative integer
Method:
1. Let d = 0
2. Let j = 0
3. While j <= l, do:
(a) If k_j = 0 then:
i. Let j = j + 1
(b) Else:
i. Let t = min{l, j + r - 1}
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ii. Let h_d = (k_t, k_(t - 1), ..., k_j) (base 2)
iii. If h_d > 2^(r - 1) then:
A. Let b_d = h_d - 2^r
B. Increment the number (k_l, k_(l-1),...,k_j) (base
2) by 1
iv. Else:
A. Let b_d = h_d
v. Let e_d = j
vi. Let d = d + 1
vii. Let j = t + 1
4. Return d and the sequence {(b_0, e_0), ..., (b_(d - 1),
e_(d - 1))}
Algorithm 3.2.2 (PointMultiply): scalar multiplication on an
elliptic curve using the signed m-ary window method.
Input:
o A point A in E(F_p^2)
o An integer l > 0
o An elliptic curve E/F_p: y^2 = x^3 + a * x + b
Output:
o The point [l]A
Method:
1. (Window decomposition)
(a) Let r > 0 be an integer (fixed) bit-wise window size,
e.g., r = 5
(b) Let l' = l where l = {Sum(l_j * 2^j), for j = 0 to
len_l} is the binary expansion of l, where len_l =
Ceiling(lg(l))
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(c) Compute (d, {(b_i, e_i), for i = 0 to d - 1} =
SignedWindowDecomposition(l, r), the signed 2^r-ary window
representation of l using Algorithm 3.2.1
(SignedWindowDecomposition)
2. (Precomputation)
(a) Let A_1 = A
(b) Let A_2 = [2]A, using Algorithm 3.1.1 (PointDouble1)
(c) For i = 1 to 2^(r - 2) - 1, do:
i. Let A_(2 * i + 1) = A_(2 * i - 1) + A_2 using
Algorithm 3.1.2 (PointAdd1)
(d) Let Q = A_(b_(d - 1))
3. Main loop
(a) For i = d - 2 to 0 by -1, do:
i. Let Q = [2^(e_(i + 1) - e_i)]Q, using repeated
applications of Algorithm 3.1.1 (PointDouble1) e_(i + 1) - e_i
times
ii. If b_i > 0 then:
A. Let Q = Q + A_(b_i) using Algorithm 3.1.2
(PointAdd1)
iii. Else:
A. Let Q = Q - A_(-(b_i)) using Algorithm 3.1.2
(PointAdd1)
(b) Calculate Q = [2^(e_0)]Q using repeated applications of
Algorithm 3.1.1 (PointDouble1) e_0 times
4. Return Q.
3.3. Operations in Jacobian projective coordinates
3.3.1. Implementation for type-1 curves
Algorithm 3.3.1 (ProjectivePointDouble1): adds a point to
itself in Jacobian projective coordinates for type-1 curves.
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Input:
o A point (x, y, z) = A in E(F_p^2) in Jacobian projective
coordinates
o An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
o The point [2]A in Jacobian projective coordinates
Method:
1. If z = 0 or y = 0, return (0, 1, 0) = 0, otherwise:
2. Let lambda_1 = 3 * x^2
3. Let z' = 2 * y * z
4. Let lambda_2 = y^2
5. Let lambda_3 = 4 * lambda_2 * x
6. Let x' = lambda_1^2 - 2 * lambda_3
7. Let lambda_4 = 8 * lambda_2^2
8. Let y' = lambda_1 * (lambda_3 - x') - lambda_4
9. Return (x', y', z')
Algorithm 3.3.2 (ProjectivePointAccumulate1): adds a point in
affine coordinates to an accumulator in Jacobian projective
coordinates, for type-1 curves.
Input:
o A point (x_A, y_A, z_A) = A in E(F_p^2) in Jacobian
projective coordinates
o A point (x_B, y_B) = B in E(F_p^2) \ {0} in affine
coordinates
o An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
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o The point A + B in Jacobian projective coordinates
Method:
1. If z_A = 0 return (x_B, y_B, 1) = B, otherwise:
2. Let lambda_1 = z_A^2
3. Let lambda_2 = lambda_1 * x_B
4. Let lambda_3 = x_A - lambda_2
5. If lambda_3 = 0 then return (0, 1, 0), otherwise:
6. Let lambda_4 = lambda_3^2
7. Let lambda_5 = lambda_1 * y_B * z_A
8. Let lambda_6 = lambda_4 - lambda_5
9. Let lambda_7 = x_A + lambda_2
10. Let lambda_8 = y_A + lambda_5
11. Let x' = lambda_6^2 - lambda_7 * lambda_4
12. Let lambda_9 = lambda_7 * lambda_4 - 2 * x'
13. Let y' = (lambda_9 * lambda_6 -
lambda_8 * lambda_3 * lambda_4) / 2
14. Let z' = lambda_3 * z_A
15. Return (x', y', z')
3.4. Divisors on elliptic curves
3.4.1. Implementation in F_p^2 for type-1 curves
Algorithm 3.4.1 (EvalVertical1): evaluates the divisor of a
vertical line on a type-1 elliptic curve.
Input:
o A point B in E(F_p^2) with B != 0
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o A point A in E(F_p)
o A description of a type-1 elliptic curve E/F_p
Output:
o An element of F_p^2 that is the divisor of the vertical
line going through A evaluated at B
Method:
1. Let r = x_B - x_A
2. Return r
Algorithm 3.4.2 (EvalTangent1): evaluates the divisor of a
tangent on a type-1 elliptic curve.
Input:
o A point B in E(F_p^2) with B != 0
o A point A in E(F_p)
o A description of a type-1 elliptic curve E/F_p
Output:
o An element of F_p^2 that is the divisor of the line tangent
to A evaluated at B
Method:
1. (Special cases)
(a) If A = 0 return 1
(b) If y_A = 0 return EvalVertical1(B, A) using Algorithm
3.4.1 (EvalVertical1)
2. (Line computation)
(a) Let a = -3 * (x_A)^2
(b) Let b = 2 * y_A
(c) Let c = -b * y_A - a * x_A
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3. (Evaluation at B)
(a) Let r = a * x_B + b * y_B + c
4. Return r
Algorithm 3.4.3 (EvalLine1): evaluates the divisor of a line
on a type-1 elliptic curve.
Input:
o A point B in E(F_p^2) with B != 0
o Two points A', A'' in E(F_p)
o A description of a type-1 elliptic curve E/F_p
Output:
o An element of F_p^2 that is the divisor of the line going
through A' and A'' evaluated at B
Method:
1. (Special cases)
(a) If A' = 0 return EvalVertical1(B, A'') using Algorithm
3.4.1 (EvalVertical1)
(b) If A'' = 0 return EvalVertical1(B, A') using Algorithm
3.4.1 (EvalVertical1)
(c) If A' = -A'' return EvalVertical1(B, A') using
Algorithm 3.4.1 (EvalVertical1)
(d) If A' = A'' return EvalTangent1(B, A') using Algorithm
3.4.2 (EvalTangent1)
2. (Line computation)
(a) Let a = y_A' - y_A''
(b) Let b = x_A'' - x_A'
(c) Let c = -b * y_A' - a * x_A'
3. (Evaluation at B)
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(a) Let r = a * x_B + b * y_B + c
4. Return r
3.5. The Tate pairing
3.5.1. Tate pairing calculation
Algorithm 3.5.1 (Tate): computes the Tate pairing on an
elliptic curve.
Input:
o A point A of order q in E(F_p)
o A point B of order q in E(F_p^2)
o A description of an elliptic curve E/F_p such that E(F_p)
and E(F_p^2) have a subgroup of order q
Output:
o The value e(A, B) in F_p^2, computed using the Miller
algorithm
Method:
1. For a type-1 curve E, execute Algorithm 3.5.2
(TateMillerSolinas)
3.5.2. The Miller algorithm for type-1 curves
Algorithm 3.5.2 (TateMillerSolinas): computes the Tate pairing
on a type-1 elliptic curve.
Input:
o A point A of order q in E(F_p)
o A point B of order q in E(F_p^2)
o A description of a type-1 supersingular elliptic curve
E/F_p such that E(F_p) and E(F_p^2) have a subgroup of
Solinas prime order q where q = 2^a + s * 2^b + c, where c
and s are limited to the values +/-1
Output:
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o The value e(A, B) in F_p^2, computed using the Miller
algorithm
Method:
1. (Initialization)
(a) Let v_num = 1 in F_p^2
(b) Let v_den = 1 in F_p^2
(c) Let V = (x_V , y_V , z_V ) = (x_A, y_A, 1) in (F_p)^3,
being the representation of (x_A, y_A) = A using Jacobian
projective coordinates
(d) Let t_num = 1 in F_p^2
(e) Let t_den = 1 in F_p^2
2. (Calculation of the (s * 2^b) contribution)
(a) (Repeated doublings) For n = 0 to b - 1:
i. Let t_num = t_num^2
ii. Let t_den = t_den^2
iii. Let t_num = t_num * EvalTangent1(B, (x_V / z_V^2,
y_V / z_V^3)) using Algorithm 3.4.2 (EvalTangent1)
iv. Let V = (x_V , y_V , z_V ) = [2]V using Algorithm
3.3.1 (ProjectivePointDouble1)
v. Let t_den = t_den * EvalVertical1(B, (x_V / z_V^2,
y_V / z_V^3)using Algorithm 3.4.1 (EvalVertical1)
(b) (Normalization)
i. Let V_b = (x_(V_b) , y_(V_b))
= (x_V / z_V^2, s * y_V / z_V^3) in (F_p)^2,
resulting in a point V_b in E(F_p)
(c) (Accumulation) Selecting on s:
i. If s = -1:
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A. Let v_num = v_num * t_den
B. Let v_den = v_den * t_num * EvalVertical1(B, (x_V
/ z_V^2, y_V / z_V^3))) using Algorithm 3.4.1 (EvalVertical1)
ii. If s = 1:
A. Let v_num = v_num * t_num
B. Let v_den = v_den * t_den
3. (Calculation of the 2^a contribution)
(a) (Repeated doublings) For n = b to a - 1:
i. Let t_num = t_num^2
ii. Let t_den = t_den^2
iii. Let t_num = t_num * EvalTangent1(B, (x_V / z_V^2,
y_V / z_V^3))) using Algorithm 3.4.2 (EvalTangent1)
iv. Let V = (x_V , y_V , z_V) = [2]V using Algorithm
3.3.1 (ProjectivePointDouble1)
v. Let t_den = t_den * EvalVertical1(B, (x_V / z_V^2,
y_V / z_V^3))) using Algorithm 3.4.1 (EvalVertical1)
(b) (Normalization)
i. Let V_a = (x_(V_a) , y_(V_a)) =
(x_V /z_V^2, s * x_V / z_V^3) in (F_p)^2,
resulting in a point V_a in E(F_p)
(c) (Accumulation)
i. Let v_num = v_num * t_num
ii. Let v_den = v_den * t_den
4. (Correction for the (s * 2^b) and (c) contributions)
(a) Let v_num = v_num * EvalLine1(B, V_a, V_b) using
Algorithm 3.4.3 (EvalLine1)
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(b) Let v_den = v_den * EvalVertical1(B, V_a + V_b) using
Algorithm 3.4.1 (EvalVertical1)
(c) If c = -1 then:
i. Let v_den = v_den * EvalVertical1(B, A) using
Algorithm 3.4.1 (EvalVertical1)
5. (Correcting exponent)
(a) Let eta = (p^2 - 1) / q
6. (Final result)
(a) Return (v_num / v_den)^eta
4. Supporting algorithms
This section describes a number of supporting algorithms for
encoding and hashing.
4.1. Integer range hashing
4.1.1. Hashing to an integer range
HashToRange(s, n, hashfcn) takes a string s, an integer n and
a cryptographic hash function hashfcn as input, and returns an
integer in the range 0 to n - 1 by cryptographic hashing. The
input n MUST be less than 2^(hashlen) where hashlen is the
number of octets comprising the output of the hash function
hashfcn. Based on Merkle's method for hashing [MERKLE], which
is provably as secure as the underlying hash function hashfcn.
Algorithm 4.1.1 (HashToRange): cryptographically hashes
strings to integers in a range.
Input:
o A string s of length |s| octets
o A positive integer n represented as Ceiling(lg(n) / 8)
octets.
o A cryptographic hash function hashfcn
Output:
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o A positive integer v in the range 0 to n - 1
Method:
1. Let hashlen be the number of octets comprising the output
of hashfcn
2. Let v_0 = 0
3. Let h_0 = 0x00...00, a string of null octets with a length
of hashlen
4. For i = 1 to 2, do:
(a) Let t_i = h_(i - 1) || s, which is the (|s| + hashlen)-
octet string concatenation of the strings h_(i - 1) and s
(b) Let h_i = hashfcn(t_i), which is a hashlen-octet string
resulting from the hash algorithm hashfcn on the input t_i
(c) Let a_i = Value(h_i) be the integer in the range 0 to
256^hashlen - 1 denoted by the raw octet string h_i
interpreted in the unsigned big endian convention
(d) Let v_i = 256^hashlen * v_(i - 1) + a_i
5. Let v = v_l (mod n)
4.2. Pseudo-random byte generation by hashing
4.2.1. Keyed pseudo-random bytes generator
HashBytes(b, p, hashfcn) takes an integer b, a string p and a
cryptographic hash function hashfcn as input, and returns a b-
octet pseudo-random string r as output. The value of b MUST be
less than or equal to the number of bytes in the output of
hashfcn. Based on Merkle's method for hashing [MERKLE], which
is provably as secure as the underlying hash function hashfcn.
Algorithm 4.2.1 (HashBytes): keyed cryptographic pseudo-random
bytes generator.
Input:
o An integer b
o A string p
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o A cryptographic hash function hashfcn
Output:
o A string r comprising b octets
Method:
1. Let hashlen be the number of octets comprising the output
of hashfcn
2. Let K = hashfcn(p)
2. Let h_0 = 0x00...00, a string of null octets with a length
of hashlen
3. Let l = Ceiling(b / hashlen)
4. For each i in 1 to l do:
(a) Let h_i = hashfcn(h_(i - 1))
(b) Let r_i = hashfcn(h_i || K), where h_i || K is the (2 *
hashlen)-octet concatenation of h_i and K
5. Let r = LeftmostOctets(b, r_1 || ... || r_l), i.e., r is
formed as the concatenation of the r_i, truncated to the
desired number of octets
4.3. Canonical encodings of extension field elements
4.3.1. Encoding an extension element as a string
Canonical(p, k, o, v) takes an element v in F_p^k, and returns
a canonical octet-string of fixed length representing v. The
parameter o MUST be either 0 or 1, and specifies the ordering
of the encoding.
Algorithm 4.3.1 (Canonical): encodes elements of an extension
field F_p^2 as strings.
Input:
o An element v in F_p^2
o A description of F_p^2
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o A ordering parameter o, either 0 or 1
Output:
o A fixed-length string s representing v
Method:
1. For a type-1 curve, execute Algorithm 4.3.2 (Canonical1)
4.3.2. Type-1 curve implementation
Canonical1(p, o, v) takes an element v in F_p^2 and returns a
canonical representation of v as a octet-string s of fixed
size. The parameter o MUST be either 0 or 1, and specifies the
ordering of the encoding.
Algorithm 4.3.2 (Canonical1): canonically represents elements
of an extension field F_p^2.
Input:
o An element v in F_p^2
o A description of p, where p is congruent to 3 modulo 4
o A ordering parameter o, either 0 or 1
Output:
o A string s of size 2 * Ceiling(lg(p) / 8) octets
Method:
1. Let l = Ceiling(lg(p) / 8), the number of octets needed to
represent integers in Z_p
2. Let v = a + b * i, where i^2 = -1.
3. Let a_(256^l) be the big-endian zero-padded fixed-length
octet-string representation of a in Z_p
4. Let b_(256^l) be the big-endian zero-padded fixed-length
octet-string representation of b in Z_p
5. Depending on the choice of ordering o:
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(a) If o = 0, then let s = a_(256^l) || b_(256^l), which is
the concatenation of a_(256^l) followed by b_(256^l)
(b) If o = 1, then let s = b_(256^l) || a_(256^l), which is
the concatenation of b_(256^l) followed by a_(256^l)
6. Return s
4.4. Hashing onto a subgroup of an elliptic curve
4.4.1. Hashing a string onto a subgroup of an elliptic curve
HashToPoint(E, p, q, id, hashfcn) takes an identity string id
and the description of a subgroup of prime order q in E(F_p)
or E(F_p^2) and a cryptographic hash function hashfcn and
returns a point Q_id of order q in E(F_p) or E(F_p^2).
Algorithm 4.4.1 (HashToPoint): cryptographically hashes
strings to points on elliptic curves.
Input:
o An elliptic curve E
o A prime p
o A prime q
o A string id
o A cryptographic hash function hashfcn
Output:
o A point Q_id = (x, y) of order q n E(F_p)
Method:
1. For a type-1 curve E, execute Algorithm 4.4.2
(HashToPoint1)
4.4.2. Type-1 curve implementation
HashToPoint1(p, q, id, hashfcn) takes an identity string id
and the description of a subgroup of order q in E(F_p) where
E: y^2 = x^3 + 1 with p congruent to 11 modulo 12, and returns
a point Q_id of order q in E(F_p) that is calculated using the
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cryptgraphic has function hashfcn. The parameters p, q and
hashfcn MUST be part of a valid set of public parameters as
defined in section 5.1.2 or section 6.1.2.
Algorithm 4.4.2 (HashToPoint1). Cryptographically hashes
strings to points on type-1 curves.
Input:
o A prime p
o A prime q
o A string id
o A cryptographic hash function hashfcn
Output:
o A point Q_id of order q in E(F_p)
Method:
1. Let y = HashToRange(id, p, hashfcn), using Algorithm 4.1.1
(HashToRange), an element of F_p
2. Let x = (y^2 - 1)^((2 * p - 1) / 3) modulo p, an element of
F_p
3. Let Q' = (x, y), a non-zero point in E(F_p)
4. Let Q = [(p + 1) / q ]Q', a point of order q in E(F_p)
4.5. Bilinear mapping
4.5.1. Regular or modified Tate pairing
Pairing(E, p, q, A, B) takes two points A and B, both of order
q, and, in the type-1 case, returns the modified pairing e'(A,
phi(B)) in F_p^2 where A and B are both in E(F_p).
Algorithm 4.5.1 (Pairing): computes the regular or modified
Tate pairing depending on the curve type.
Input:
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o A description of an elliptic curve E/F_p such that E(F_p)
and E(F_p^2) have a subgroup of order q
o Two points A and B of order q in E(F_p) or E(F_p^2)
Output:
o On supersingular curves, the value of e'(A, B) in F_p^2
where A and B are both in E(F_p)
Method:
1. If E is a type-1 curve, execute Algorithm 4.5.2 (Pairing1)
4.5.2. Type-1 curve implementation
Algorithm 4.5.2 (Pairing1): computes the modified Tate pairing
on type-1 curves. The values of p and q MUST be part of a
valid set of public parameters as defined in section 5.1.2 or
section 6.1.2.
Input:
o A curve E/F_p: y^2 = x^3 + 1 where p is congruent to 11
modulo 12 and E(F_p) has a subgroup of order q
o Two points A and B of order q in E(F_p)
Output:
o The value of e'(A, B) = e(A, phi(B)) in F_p^2
Method:
1. Compute B' = phi(B), as follows:
(a) Let (x, y) in F_p x F_p be the coordinates of B in
E(F_p)
(b) Let zeta = (a_zeta , b_zeta), where a_zeta = (p - 1) /
2 and b_zeta = 3^((p + 1) / 4) (mod p), an element of F_p^2
(c) Let x' = x * zeta in F_p^2
(d) Let B' = (x', y) in F_p^2 x F_p
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2. Compute the Tate pairing e(A, B') = e(A, phi(B)) in F_p^2
using the Miller method, as in Algorithm 3.5.1 (Tate)
described in Section 3.5
4.6. Ratio of bilinear pairings
4.6.1. Ratio of regular or modified Tate pairings
PairingRatio(E, p, q, A, B, C, D) takes four points as input,
and computes the ratio of the two bilinear pairings,
Pairing(E, p, q, A, B) / Pairing(E, p, q, C, D), or,
equivalently, the product, Pairing(E, p, q, A, B) * Pairing(E,
p, q, C, -D).
On type-1 curves, all four points are of order q in E(F_p),
and the result is an element of order q in the extension field
F_p^2 .
The motivation for this algorithm is that the ratio of two
pairings can be calculated more efficiently than by computing
each pairing separately and dividing one into the other, since
certain calculations that would normally appear in each of the
two pairings can be combined and carried out at once. Such
calculations include the repeated doublings in steps 2(a)i,
2(a)ii, 3(a)i, and 3(a)ii of Algorithm 3.5.2
(TateMillerSolinas), as well as the final exponentiation in
step 6(a) of Algorithm 3.5.2 (TateMillerSolinas).
Algorithm 4.6.1 (PairingRatio): computes the ratio of two
regular or modified Tate pairings depending on the curve type.
Input:
o A description of an elliptic curve E/F_p such that E(F_p)
and E(F_p^2) have a subgroup of order q
o Four points A, B, C, and D, of order q in E(F_p) or
E(F_p^2)
Output:
o On supersingular curves, the value of e'(A, B) / e'(C, D)
in F_p^2 where A, B, C, D are all in E(F_p)
Method:
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1. If E is a type-1 curve, execute Algorithm 4.6.2
(PairingRatio1)
4.6.2. Type-1 curve implementation
Algorithm 4.6.2 (PairingRatio1). Computes the ratio of two
modified Tate pairings on type-1 curves. The values of p and q
MUST be part of a valid set of public parameters as defined in
section 5.1.2 or section 6.1.2.
Input:
o A curve E/F_p: y^2 = x^3 + 1, where p is congruent to 11
modulo 12 and E(F_p) has a subgroup of order q
o Four points A, B, C, and D, of order q in E(F_p)
Output:
o The value of e'(A, B) / e'(C, D) = e(A, phi(B)) / e(C,
phi(D)) = e(A, phi(B)) * e(-C, phi(D)), in F_p^2
Method:
1. The step-by-step description of the optimized algorithm is
omitted in this normative specification
The correct result can always be obtained, although more
slowly, by computing the product of pairings Pairing1(E, p, q,
A, B) * Pairing1(E, p, q, -C, D) by using two invocations of
Algorithm 4.5.2 (Pairing1).
5. The Boneh-Franklin BF cryptosystem
This chapter describes the algorithms constituting the Boneh-
Franklin identity-based cryptosystem as described in [BF].
5.1. Setup
5.1.1. Master secret and public parameter generation
Algorithm 5.1.1 (BFsetup): randomly selects a master secret
and the associated public parameters.
Input:
o A integer version number
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o A security parameter n (MUST take values either 1024, 2048,
3072, 7680, 15360)
Output:
o A set of public parameters (version, E, p, q, P, P_pub,
hashfcn)
o A corresponding master secret s
Method:
1. Depending on the selected type t:
(a) If version = 2, then execute Algorithm 5.1.2 (BFsetup1)
2. The resulting master secret and public parameters are
separately encoded as per the application protocol
requirements
5.1.2. Type-1 curve implementation
BFsetup1 takes a security parameter n as input. For type-1
curves, the scale of n corresponds to the modulus bit-size
believed [BF] of comparable security in the classical Diffie-
Hellman or RSA public-key cryptosystems.
Algorithm 5.1.2 (BFsetup1): establishes a master secret and
public parameters for type-1 curves.
Input:
o A security parameter n which MUST be either 1024, 2048,
3072, 7680 or 15360
Output:
o A set of common public parameters (version, p, q, P, Ppub,
hashfcn)
o A corresponding master secret s
Method:
1. Set the version to version = 2.
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2. Determine the subordinate security parameters n_p and n_q
as follows:
(a) If n = 1024 then let n_p = 512, n_q = 160, hashfcn =
1.3.14.3.2.26 (SHA-1 [SHA].
(b) If n = 2048 then let n_p = 1024, n_q = 224, hashfcn =
2.16.840.1.101.3.4.2.4 (SHA-224 [SHA]).
(c) If n = 3072 then let n_p = 1536, n_q = 256, hashfcn =
2.16.840.1.101.3.4.2.1 (SHA-256 [SHA]).
(d) If n = 7680 then let n_p = 3840, n_q = 384, hashfcn =
2.16.840.1.101.3.4.2.2 (SHA-384 [SHA]).
(e) If n = 15360 then let n_p = 7680, n_q = 512, hashfcn =
2.16.840.1.101.3.4.2.3 (SHA-512 [SHA]).
3. Construct the elliptic curve and its subgroup of interest,
as follows:
(a) Select an arbitrary n_q-bit Solinas prime q
(b) Select a random integer r such that p = 12 * r * q - 1
is an n_p-bit prime
4. Select a point P of order q in E(F_p), as follows:
(a) Select a random point P' of coordinates (x', y') on the
curve E/F_p: y^2 = x^3 + 1 (mod p)
(b) Let P = [12 * r]P'
(c) If P = 0, then start over in step 3a
5. Determine the master secret and the public parameters as
follows:
(a) Select a random integer s in the range 2 to q - 1
(b) Let P_pub = [s]P
6. (version, E, p, q, P, P_pub) are the public parameters
where E: y^2 = x^3 + 1 is represented by the OID
2.16.840.1.114334.1.1.1.1.
7. The integer s is the master secret
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5.2. Public key derivation
5.2.1. Public key derivation from an identity and public
parameters
BFderivePubl takes an identity string id and a set of public
parameters, and returns a point Q_id. The public parameters
used MUST be a valid set of public parameters as defined by
section 5.1.2.
Algorithm 5.2.1 (BFderivePubl): derives the public key
corresponding to an identity string.
Input:
o An identity string id
o A set of public parameters (version, E, p, q, P, P_pub,
hashfcn)
Output:
o A point Q_id of order q in E(F_p) or E(F_p^2)
Method:
1. Q_id = HashToPoint(E, p, q, id, hashfcn), using Algorithm
4.4.1 (HashToPoint)
5.3. Private key extraction
5.3.1. Private key extraction from an identity, a set of public
parameters and a master secret
BFextractPriv takes an identity string id, and a set of public
parameters and corresponding master secret, and returns a
point S_id. The public parameters used MUST be a valid set of
public parameters as defined by section 5.1.2.
Algorithm 5.3.1 (BFextractPriv): extracts the private key
corresponding to an identity string.
Input:
o An identity string id
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o A set of public parameters (version, E, p, q, P, P_pub,
hashfcn)
Output:
o A point S_id of order q in E(F_p).
Method:
1. Let Q_id = HashToPoint(E, p, q, id, hashfcn) using
Algorithm 4.4.1 (HashToPoint)
2. Let S_id = [s]Q_id
5.4. Encryption
5.4.1. Encrypt a session key using an identity and public
parameters
BFencrypt takes three inputs: a public parameter block, an
identity id, and a plaintext m. The plaintext MUST be a random
symmetric session key. The public parameters used MUST be a
valid set of public parameters as defined by section 5.1.2.
Algorithm 5.4.1 (BFencrypt): encrypts a random session key for
an identity string.
Input:
o A plaintext string m of size |m| octets
o A recipient identity string id
o A set of public parameters (version, E, p, q, P, P_pub,
hashfcn)
Output:
o A ciphertext tuple (U, V, W) in E(F_p) x {0, ... ,
255}^hashlen x {0, ... , 255}^|m|
Method:
1. Let hashlen be the length of the output of the
cryptographic hash function hashfcn from the public
parameters.
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2. Q_id = HashToPoint(E, p, q, id, hashfcn), using Algorithm
4.4.1 (HashToPoint), which results in a point of order q in
E(F_p).
3. Select a random hashlen-bit vector rho, represented as
(hashlen / 8)-octet string in big-endian convention
4. Let t = hashfcn(m), a hashlen-octet string resulting from
applying the hashfcn algorithm to the input m
5. Let l = HashToRange(rho || t, q, hashfcn), an integer in
the range 0 to q - 1 resulting from applying Algorithm 4.1.1
(HashToRange) to the (2 * hashlen)-octet concatenation of rho
and t
6. Let U = [l]P, which is a point of order q in E(F_p)
7. Let theta = Pairing(E, p, q, P_pub, Q_id), which is an
element of the extension field F_p^2 obtained using the
modified Tate pairing of Algorithm 4.5.1 (Pairing)
8. Let theta' = theta^l, which is theta raised to the power of
l in F_p^2
9. Let z = Canonical(p, k, 0, theta'), using Algorithm 4.3.1
(Canonical), the result of which is a canonical string
representation of theta'
10. Let w = hashfcn(z) using the hashfcn hashing algorithm,
the result of which is a hashlen-octet string
11. Let V = w XOR rho, which is the hashlen-octet long bit-
wise XOR of w and rho
12. Let W = HashBytes(|m|, rho, hashfcn) XOR m, which is the
bit-wise XOR of m with the first |m| octets of the pseudo-
random bytes produced by Algorithm 4.2.1 (HashBytes) with seed
rho
13. The ciphertext is the triple (U, V, W)
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5.5. Decryption
5.5.1. Decrypt an encrypted session key using public parameters,
a private key
BFdecrypt takes three inputs: a public parameter block, a
private key block key, and a ciphertext parsed as (U', V',
W'). The public parameters used MUST be a valid set of public
parameters as defined by section 5.1.2.
Algorithm 5.5.1 (BFdecrypt): decrypts an encrypted session key
using a private key.
Input:
o A private key point S_id of order q in E(F_p)
o A ciphertext triple (U, V, W) in E(F_p) x {0, ... ,
255}^hashlen x {0, ... , 255}*
o A set of public parameters (version, E, p, q, P, P_pub,
hashfcn)
Output:
o A decrypted plaintext m, or an invalid ciphertext flag
Method:
1. Let hashlen be the length of the output of the hash
function hashlen measured in octets
2. Let theta = Pairing(E, p ,q, U, S_id) by applying the
modified Tate pairing of Algorithm 4.5.1 (Pairing)
3. Let z = Canonical(p, k, 0, theta) using Algorithm 4.3.1
(Canonical), the result of which is a canonical string
representation of theta
4. Let w = hashfcn(z), using the hashfcn hashing algorithm,
the result of which is a hashlen-octet string
5. Let rho = w XOR V, the bit-wise XOR of w and V
6. Let m = HashBytes(|W|, rho, hashfcn) XOR W, which is the
bit-wise XOR of m with the first |W| octets of the pseudo-
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random bytes produced by Algorithm 4.2.1 (HashBytes) with seed
rho
7. Let t = hashfcn(m) using the hashfcn algorithm
8. Let l = HashToRange(rho || t, q, hashfcn) using Algorithm
4.1.1 (HashToRange) on the (2 * hashlen)-octet concatenation
of rho and t
9. Verify that U = [l]P:
(a) If this is the case, then the decrypted plaintext m is
returned
(b) Otherwise, the ciphertext is rejected and no plaintext
is returned
6. The Boneh-Boyen BB1 cryptosystem
This section describes the algorithms constituting the first
of the two Boneh-Boyen identity-based cryptosystems proposed
in [BB1]. The description follows the practical implementation
given in [BB1].
6.1. Setup
6.1.1. Generate a master secret and public parameters
Algorithm 6.1.1 (BBsetup). Randomly selects a set of master
secrets and the associated public parameters.
Input:
o An integer version number
o An integer security parameter n (MUST take values either
1024, 2048, 3072, 7680, or 15360.
Output:
o A set of public parameters
o A corresponding master secret
Method:
1. Depending on the version:
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(a) If version = 2, then execute Algorithm 6.1.2 (BBsetup1)
6.1.2. Type-1 curve implementation
BBsetup1 takes a security parameter n as input. For type-1
curves, n corresponds to the modulus bit-size believed [BF] of
comparable security in the classical Diffie-Hellman or RSA
public-key cryptosystems. For this implementation n MUST be
one of 1024, 2048, 3072, 7680 and 15360, which correspond to
the equivalent bit security levels of 80, 112, 128, 192 and
256 bits respectively.
Algorithm 6.1.2 (BBsetup1): randomly establishes a master
secret and public parameters for type-1 curves.
Input:
o A security parameter n, either 1024, 2048, 3072, 7680, or
15360
Output:
o A set of public parameters (version, k, E, p, q, P, P_1,
P_2, P_3, v, hashfcn)
o A corresponding triple of master secrets (alpha, beta,
gamma)
Method:
1. Determine the subordinate security parameters n_p and n_q
as follows:
(a) If n = 1024 then let n_p = 512, n_q = 160, hashfcn =
1.3.14.3.2.26 (SHA-1 [SHA]
(b) If n = 2048 then let n_p = 1024, n_q = 224, hashfcn =
2.16.840.1.101.3.4.2.4 (SHA-224 [SHA])
(c) If n = 3072 then let n_p = 1536, n_q = 256, hashfcn =
2.16.840.1.101.3.4.2.1 (SHA-256 [SHA])
(d) If n = 7680 then let n_p = 3840, n_q = 384, hashfcn =
2.16.840.1.101.3.4.2.2 (SHA-384 [SHA])
(e) If n = 15360 then let n_p = 7680, n_q = 512, hashfcn =
2.16.840.1.101.3.4.2.3 (SHA-512 [SHA])
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2. Construct the elliptic curve and its subgroup of interest
as follows:
(a) Select a random n_q-bit Solinas prime q
(b) Select a random integer r such that p = 12 * r * q - 1
is an n_p-bit prime
3. Select a point P of order q in E(F_p), as follows:
(a) Select a random point P' of coordinates (x', y') on the
curve E/F_p: y^2 = x^3 + 1 (mod p)
(b) Let P = [12 * r]P'
(c) If P = 0, then start over in step 3a
4. Determine the master secret and the public parameters as
follows:
(a) Select three random integers alpha, beta, gamma, each
of them in the range 1 to q - 1
(b) Let P_1 = [alpha]P
(c) Let P_2 = [beta]P
(d) Let P_3 = [gamma]P
(e) Let v = Pairing(E, p, q, P_1, P_2), which is an element
of the extension field F_p^2 obtained using the modified Tate
pairing of Algorithm 3.5.1 (Pairing)
5. (version, E, p, q, P, P_1, P_2, P_3, v, hashfcn) are the
public parameters
6. (alpha, beta, gamma) constitute the master secret
6.2. Public key derivation
6.2.1. Derive a public key from an identity and public parameters
Takes an identity string id and a set of public parameters,
and returns an integer h_id. The public parameters used MUST
be a valid set of public parameters as defined by section
section 6.1.2.
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Algorithm 6.2.1 (BBderivePubl): derives the public key
corresponding to an identity string. The public parameters
used MUST be a valid set of public parameters as defined by
section section 6.1.2.
Input:
o An identity string id
o A set of common public parameters (version, k, E, p, q, P,
P_1, P_2, P_3, v, hashfcn)
Output:
o An integer h_id modulo q
Method:
1. Let h_id = HashToRange(id, q, hashfcn), using Algorithm
4.1.1 (HashToRange)
6.3. Private key extraction
6.3.1. Extract a private key from an identity, public parameters
and a master secret
BBextractPriv takes an identity string id, and a set of public
parameters and corresponding master secrets, and returns a
private key consisting of two points D_0 and D_1. The public
parameters used MUST be a valid set of public parameters as
defined by section section 6.1.2.
Algorithm 6.3.1 (BBextractPriv): extracts the private key
corresponding to an identity string.
Input:
o An identity string id
o A set of public parameters (version, k, E, p, q, P, P_1,
P_2, P_3, v, hashfcn)
Output:
o A pair of points (D_0, D_1), each of which has order q in
E(F_p)
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Method:
1. Select a random integer r in the range 1 to q - 1
2. Calculate the point D_0 as follows:
(a) Let hid = HashToRange(id, q, hashfcn), using Algorithm
4.1.1 (HashToRange)
(b) Let y = alpha * beta + r * (alpha * h_id + gamma) in
F_q
(c) Let D_0 = [y]P
3. Calculate the point D_1 as follows:
(a) Let D_1 = [r]P
4. The pair of points (D_0, D_1) constitutes the private key
for id
6.4. Encryption
6.4.1. Encrypt a session key using an identity and public
parameters
BBencrypt takes three inputs: a set of public parameters, an
identity id, and a plaintext m. The plaintext MUST be a random
session key. The public parameters used MUST be a valid set of
public parameters as defined by section section 6.1.2.
Algorithm 6.4.1 (BBencrypt): encrypts a session key for an
identity string.
Input:
o A plaintext string m of size |m| octets
o A recipient identity string id
o A set of public parameters (version, k, E, p, q, P, P_1,
P_2, P_3, v, hashfcn)
Output:
o A ciphertext tuple (u, C_0, C_1, y) in F_q x E(F_p) x
E(F_p) x {0, ... , 255}^|m|
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Method:
1. Select a random integer s in the range 1 to q - 1
2. Let w = v^s, which is v raised to the power of s in F_p^2,
the result is an element of order q in F_p^2
3. Calculate the point C_0 as follows:
(a) Let C_0 = [s]P
4. Calculate the point C_1 as follows:
(a) Let _hid = HashToRange(id, q, hashfcn), using Algorithm
4.1.1 (HashToRange)
(b) Let y = s * h_id in F_q
(c) Let C_1 = [y]P_1 + [s]P_3
5. Obtain canonical string representations of certain
elements:
(a) Let psi = Canonical(p, k, 1, w) using Algorithm 4.3.1
(Canonical), the result of which is a canonical octet-string
representation of w
(b) Let l = Ceiling(lg(p) / 8), the number of octets needed
to represent integers in F_p, and represent each of these F_p
elements as a big-endian zero-padded octet-string of fixed
length l:
(x_0)_(256^l) to represent the x coordinate of C_0
(y_0)_(256^l) to represent the y coordinate of C_0
(x_1)_(256^l) to represent the x coordinate of C_1
(y_1)_(256^l) to represent the y coordinate of C_1
6. Encrypt the message m into the string y as follows:
(a) Compute an encryption key h_0 as a two-pass hash of w
via its representation psi:
i. Let zeta = hashfcn(psi), using the hashing algorithm
hashfcn
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ii. Let xi = hashfcn(zeta || psi), using the hashing
algorithm hashfcn
iii. Let h' = xi || zeta, the concatenation of the
previous two hashfcn outputs
(b) Let y = HashBytes(|m|, h', hashfcn) XOR m, which is the
bit-wise XOR of m with the first |m| octets of the pseudo-
random bytes produced by Algorithm 3.2.1 (HashBytes) with seed
h'
7. Create the integrity check tag u as follows:
(a) Compute a one-time pad h'' as a dual-pass hash of the
representation of (w, C_0, C_1, y):
i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) ||
(y_0)_(256^l) || (x_0)_(256^l) || y || psi be the
concatenation of y and the five indicated strings in the
specified order
ii. Let eta = hashfcn(sigma), using the hashing
algorithm hashfcn
iii. Let mu = hashfcn(eta || sigma), using the hashfcn
hashing algorithm
iv. Let h'' = mu || eta, the concatenation of the
previous two outputs of hashfcn
(b) Build the tag u as the encryption of the integer s with
the one-time pad h'':
i. Let rho = HashToRange(h'', q, hashfcn) to get an
integer in Z_q
ii. Let u = s + rho (mod q)
8. The complete ciphertext is given by the quadruple (u, C_0,
C_1, y)
6.5. Decryption
6.5.1. Decrypt using public parameters and private key
BBdecrypt takes three inputs: a set of public parameters
(version, k, E, p, q, P, P_1, P_2, P_3, v, hashfcn), a private
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key (D_0, D_1), and a ciphertext (u, C_0, C_1, y). It outputs
a message m, or signals an error if the ciphertext is invalid
for the given key. The public parameters used MUST be a valid
set of public parameters as defined by section section 6.1.2.
Algorithm 6.5.1 (BBdecrypt): decrypts a ciphertext using
public parameters and a private key.
Input:
o A private key given as a pair of points (D_0, D_1) of order
q in E(F_p)
o A ciphertext quadruple (u, C_0, C_1, y) in Z_q x E(F_p) x
E(F_p) x {0, ... , 255}*
o A set of public parameters (version, k, E, p, q, P, P_1,
P_2, P_3, v, hashfcn)
Output:
o A decrypted plaintext m, or an invalid ciphertext flag
Method:
1. Let w = PairingRatio(E, p, q, C_0, D_0, C_1, D_1), which
computes the ratio of two Tate pairings (modified, for type-1
curves) as specified in Algorithm 4.6.1 (PairingRatio)
2. Obtain canonical string representations of certain
elements:
(a) Let psi = Canonical(p, k, 1, w), using Algorithm 4.3.1
(Canonical); the result is a canonical octet-string
representation of w
(b) Let l = Ceiling(lg(p) / 8), the number of octets needed
to represent integers in F_p, and represent each of these F_p
elements as a big-endian zero-padded octet-string of fixed
length l:
(x_0)_(256^l) to represent the x coordinate of C_0
(y_0)_(256^l) to represent the y coordinate of C_0
(x_1)_(256^l) to represent the x coordinate of C_1
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(y_1)_(256^l) to represent the y coordinate of C_1
3. Decrypt the message m from the string y as follows:
(a) Compute the decryption key h' as a dual-pass hash of w
via its representation psi:
i. Let zeta = hashfcn(psi), using the hashing algorithm
hashfcn
ii. Let xi = hashfcn(zeta || psi), using the hashing
algorithm hashfcn
iii. Let h' = xi || zeta, the concatenation of the
previous two hashfcn outputs
(b) Let m = HashBytes(|y|, h', hashfcn)_XOR y, which is the
bit-wise XOR of y with the first |y| octets of the pseudo-
random bytes produced by Algorithm 4.2.1 (HashBytes) with
seed h'
4. Obtain the integrity check tag u as follows:
(a) Recover the one-time pad h'' as a dual-pass hash of the
representation of (w, C_0, C_1, y):
i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) ||
(y_0)_(256^l) || (x_0)_(256^l) || y || psi be the
concatenation of y and the five indicated strings in the
specified order
ii. Let eta = hashfcn(sigma) using the hashing algorithm
hashfcn
iii. Let mu = hashfcn(eta || sigma), using the hashing
algorithm hashfcn
iv. Let h'' = mu || eta, the concatenation of the
previous two hashfcn outputs
(b) Unblind the encryption randomization integer s from the
tag u using h'':
i. Let rho = HashToRange(h'', q, hashfcn) to get an
integer in Z_q
ii. Let s = u - rho (mod q)
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5. Verify the ciphertext consistency according to the
decrypted values:
(a) Test whether the equality w = v^s holds
(b) Test whether the equality C_0 = [s]P holds
6. Adjudication and final output:
(a) If either of the tests performed in step 5 fails, the
ciphertext is rejected, and no decryption is output
(b) Otherwise, i.e., when both tests performed in step 5
succeed, the decrypted message is output
7. Test data
The following data can be used to verify the correct operation
of selected algorithms that are defined in this document.
7.1. Algorithm 3.2.2 (PointMultiply)
Input:
q = 0xfffffffffffffffffffffffffffbffff
p = 0xbffffffffffffffffffffffffffcffff3
E/F_p: y^2 = x^3 + 1
A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
0x510c6972d795ec0c2b081b81de767f808)
l = 0xb8bbbc0089098f2769b32373ade8f0daf
Output:
[l]A = (0x073734b32a882cc97956b9f7e54a2d326,
0x9c4b891aab199741a44a5b6b632b949f7)
7.2. Algorithm 4.1.1 (HashToRange)
Input:
s =
54:68:69:73:20:41:53:43:49:49:20:73:74:72:69:6e:67:20:77:69:74
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:68:6f:75:74:20:6e:75:6c:6c:2d:74:65:72:6d:69:6e:61:74:6f:72
("This ASCII string without null-terminator")
n = 0xffffffffffffffffffffefffffffffffffffffff
hashfcn = 1.3.14.3.2.16 (SHA-1)
Output:
v = 0x79317c1610c1fc018e9c53d89d59c108cd518608
7.3. Algorithm 4.5.1 (Pairing)
q = 0xfffffffffffffffffffffffffffbffff
p = 0xbffffffffffffffffffffffffffcffff3
E/F_p: y^2 = x^3 + 1
A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
0x510c6972d795ec0c2b081b81de767f808)
B = (0x40e98b9382e0b1fa6747dcb1655f54f75,
0xb497a6a02e7611511d0db2ff133b32a3f)
Output:
e'(A, B) = (0x8b2cac13cbd422658f9e5757b85493818,
0xbc6af59f54d0a5d83c8efd8f5214fad3c)
7.4. Algorithm 5.2.1 (BFderivePubl)
Input:
id = 6f:42:62 ("Bob")
version = 2
p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
q = 0xffffffffffffffffffffffeffffffffffff
P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
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Output:
Q_id = (0x22fa1207e0d19e1a4825009e0e88e35eb57ba79391498f59,
0x982d29acf942127e0f01c881b5ec1b5fe23d05269f538836)
7.5. Algorithm 5.3.1 (BFextractPriv)
Input:
s = 0x749e52ddb807e0220054417e514742b05a0
version = 2
p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
q = 0xffffffffffffffffffffffeffffffffffff
P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
Output:
Q_id = (0x8212b74ea75c841a9d1accc914ca140f4032d191b5ce5501,
0x950643d940aba68099bdcb40082532b6130c88d317958657)
7.6. Algorithm 5.4.1 (BFencrypt)
(Note that the following values can also be used to test
Algorithm 5.5.1 (BFdecrypt))
Input:
m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
id = 6f:42:62 ("Bob")
version = 2
p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
q = 0xffffffffffffffffffffffeffffffffffff
P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
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P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
Output:
Using the random value rho =
0xed5397ff77b567ba5ecb644d7671d6b6f2082968, we get the
following output:
U =
(0x1b5f6c461497acdfcbb6d6613ad515430c8b3fa23b61c585e9a541b199e
2a6cb,
0x9bdfbed1ae664e51e3d4533359d733ac9a600b61048a7d899104e826a0ec
4fa4)
V =
e0:1d:ad:81:32:6c:b1:73:af:c2:8d:72:2e:7a:32:1a:7b:29:8a:aa
W = f9:04:ba:40:30:e9:ce:6e:ff
7.7. Algorithm 6.3.1 (BBextractPriv)
Inputs:
alpha = 0xa60c395285ded4d70202c8283d894bad4f0
beta = 0x48bf012da19f170b13124e5301561f45053
gamma = 0x226fba82bc38e2ce4e28e56472ccf94a499
version = 2
p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
q = 0xfffffffffbfffffffffffffffffffffffff
P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
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v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2)
id = 6f:42:62 ("Bob")
Output:
Using the random value r =
0x695024c25812112187162c08aa5f65c7a2c, we get the following
output:
D_0 = (0x3264e13feeb7c506493888132964e79ad657a952334b9e53,
0x3eeaefc14ba1277a1cd6fdea83c7c882fe6d85d957055c7b)
D_1 = (0x8d7a72ad06909bb3bb29b67676d935018183a905e7e8cb18,
0x2b346c6801c1db638f270af915a21054f16044ab67f6c40e)
7.8. Algorithm 6.4.1 (BBencrypt)
(Note that the following values can also be used to test
Algorithm 5.5.1 (BFdecrypt))
Input:
m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
id = 6f:42:62 ("Bob")
version = 2
E: y^2 = x^3 + 1
p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
q = 0xfffffffffbfffffffffffffffffffffffff
P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
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v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2)
hashfcn = 1.3.14.3.2.26 (SHA-1)
Output:
Using the random value s =
0x62759e95ce1af248040e220263fb41b965e, we get the following
output:
u = 0xad1ebfa82edf0bcb5111e9dc08ff0737c68
C_0 = (0x79f8f35904579f1aaf51897b1e8f1d84e1c927b8994e81f9,
0x1cf77bb2516606681aba2e2dc14764aa1b55a45836014c62)
C_1 = (0x410cfeb0bccf1fa4afc607316c8b12fe464097b20250d684,
0x8bb76e7195a7b1980531b0a5852ce710cab5d288b2404e90)
y = 82:a6:42:b9:bb:e9:82:c4:57
8. ASN.1 module
This section defines the ASN.1 module for the encodings
discussed in this document.
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IBCS { joint-iso-itu-t(2) country(16) us(840) organization(1)
identicrypt(114334) ibcs(1) module(5) version(1) }
DEFINITIONS IMPLICIT TAGS ::= BEGIN
--
-- Identity-based cryptography standards (IBCS):
-- supersingular curve implementations of
-- the BF and BB1 cryptosystems
--
-- This version only supports IBE using
-- type-1 curves, i.e., the curve y^2 = x^3 + 1.
--
ibcs OBJECT IDENTIFIER ::= {
joint-iso-itu-t(2) country(16) us(840) organization(1)
identicrypt(114334) ibcs(1)
}
--
-- IBCS1
--
-- IBCS1 defines the algorithms used to implement IBE
--
ibcs1 OBJECT IDENTIFIER ::= {
ibcs ibcs1(1)
}
--
-- An elliptic curve is specified by an OID.
-- A type1curve is defined by the equation y^2 = x^3 + 1.
--
type1curve OBJECT IDENTIFIER ::= {
ibcs1 curve-types(1) type1-curve(1)
}
--
-- Supporting types
--
--
-- Encoding of a point on an elliptic curve E/F_p
-- An FpPoint can either represent an element of
-- F_p^2 or an element of (F_p)^2.
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FpPoint ::= SEQUENCE {
x INTEGER,
y INTEGER
}
--
-- The following hash functions are supported:
--
-- SHA-1
--
-- id-sha1 OBJECT IDENTIFIER ::= {
-- iso(1) identified-organization(3) oiw(14)
-- secsig(3) algorithms(2) hashAlgorithmIdentifier(26)
-- }
--
-- SHA-224
--
-- id-sha224 OBJECT IDENTIFIER ::= {
-- joint-iso-itu-t(2)country(16) us(840)
-- organization(1) gov(101)
-- csor(3) nistAlgorithm(4) hashAlgs(2) sha224(4)
-- }
--
-- SHA-256
--
-- id-sha256 OBJECT IDENTIFIER ::= {
-- joint-iso-itu-t(2)country(16) us(840)
-- organization(1) gov(101)
-- csor(3) nistAlgorithm(4) hashAlgs(2) sha256(1)
-- }
--
-- SHA-384
--
-- id-sha384 OBJECT IDENTIFIER ::= {
-- joint-iso-itu-t(2)country(16) us(840)
-- organization(1) gov(101)
-- csor(3) nistAlgorithm(4) hashAlgs(2) sha384(2)
-- }
--
-- SHA-512
--
-- id-sha512 OBJECT IDENTIFIER ::= {
-- joint-iso-itu-t(2) country(16) us(840)
-- organization(1) gov(101)
-- csor(3) nistAlgorithm(4) hashAlgs(2) sha512(3)
-- }
--
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--
-- Algorithms
--
ibe-algorithms OBJECT IDENTIFIER ::= {
ibcs1 ibe-algorithms(2)
}
---
--- Boneh-Franklin IBE
---
bf OBJECT IDENTIFIER ::= { ibe-algorithms bf(1) }
--
-- Encoding of a BF public parameters block.
-- The only version currently supported is version 2.
-- The values p and q define a subgroup of E(F_p) of order q.
--
BFPublicParameters ::= SEQUENCE {
version INTEGER { v2(2) },
curve OBJECT IDENTIFIER,
p INTEGER,
q INTEGER,
pointP FpPoint,
pointPpub FpPoint,
hashfcn OBJECT IDENTIFIER
}
--
-- A BF private key is a point on an elliptic curve,
-- which is an FpPoint.
-- The only version supported is version 2.
--
BFPrivateKeyBlock ::= SEQUENCE {
version INTEGER { v2(2) },
privateKey FpPoint
}
--
-- A BF master secret is an integer.
-- The only version supported is version 2.
--
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BFMasterSecret ::= SEQUENCE {
version INTEGER {v2(2) },
masterSecret INTEGER
}
--
-- BF ciphertext block
-- The only version supported is version 2.
--
BFCiphertextBlock ::= SEQUENCE {
version INTEGER { v2(2) },
u FpPoint,
v OCTET STRING,
w OCTET STRING
}
--
-- Boneh-Boyen (BB1) IBE
--
bb1 OBJECT IDENTIFIER ::= { ibe-algorithms bb1(2) }
--
-- Encoding of a BB1 public parameters block.
-- The version is currently fixed to 2.
--
--
BB1PublicParameters ::= SEQUENCE {
version INTEGER { v2(2) },
curve OBJECT IDENTIFIER,
p INTEGER,
q INTEGER,
pointP FpPoint,
pointP1 FpPoint,
pointP2 FpPoint,
pointP3 FpPoint,
v FpPoint,
hashfcn OBJECT IDENTIFIER
}
--
-- BB1 master secret block
-- The only version supported is version 2.
--
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BB1MasterSecret ::= SEQUENCE {
version INTEGER { v2(2) },
alpha INTEGER,
beta INTEGER,
gamma INTEGER
}
--
-- BB1 private Key block
-- The only version supported is version 2.
--
BB1PrivateKeyBlock ::= SEQUENCE {
version INTEGER { v2(2) },
pointD0 FpPoint,
pointD1 FpPoint
}
--
-- BB1 ciphertext block
-- The only version supported is version 2.
--
BB1CiphertextBlock ::= SEQUENCE {
version INTEGER {v2(2) },
pointChi0 FpPoint,
pointChi1 FpPoint,
nu INTEGER,
y OCTET STRING
}
END
9. Security considerations
This document describes cryptographic algorithms, for which we
assume that the security of the algorithm relies entirely on
the secrecy of the relevant private key, so that an adversary
will need to intercept encrypted messages and perform
computationally-intensive cryptanalytic attacks against the
ciphertext that he obtains in this way to recover either
plaintext or a secret cryptographic key.
We assume that users of the algorithms described in this
document will require one of five levels of cryptographic
strength: the equivalent of 80 bits, 112 bits, 128 bits, 192
bits or 256 bits. The 80-bit level is suitable for legacy
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applications and SHOULD NOT be used to protect information
whose useful life extends past the year 2010. The 112-bit
level is suitable for use in key transport of Triple-DES keys
and should be adequate to protect information whose useful
life extends up to the year 2030. The 128-bit levels and
higher are suitable for use in the transport of AES keys of
the corresponding length or less and are adequate to protect
information whose useful life extends past the year 2030.
Table 1 summarizes the security parameters for the BF and BB1
algorithms that will attain these levels of security. In this
table, |p| represents the number of bits in a prime number p
and |q| represents the number of bits in a subprime q. This
table assumes that a Type-1 supersingular curve is used.
Bits of Security |p| |q|
80 512 160
112 1024 224
128 1536 256
192 3840 384
256 7680 512
Table 1: Sizes of BF and BB1 parameters required to attain
standard levels of bit security [SP800-57].
If an IBE key is used to transport a symmetric key that
provides more bits of security than the bit strength of the
IBE key, users should understand that the security of the
system is then limited by the strength of the weaker IBE key.
So if an IBE key that provides 112 bits of security is used to
transport a 128-bit AES key, then the security provided is
limited by the 112 bits of security of the IBE key.
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Note that this document specifies the use of the NIST hashing
algorithms [SHA] to hash identities to either a point on an
elliptic curve or an integer. Recent attacks on SHA-1 [SHA]
have discovered ways to find collisions with less work than
the expected 2^80 hashes required based on the size of the
output of the hash function alone. If an attacker can find a
collision then they could use the colliding preimages to
create two identities which have the same IBE private key. The
practical use of such a SHA-1 [SHA] collision is extremely
unlikely, however.
Identities are typically not random strings, like the
preimages of a hash collision would be. In particular, this is
true if IBE is used as described in [IBECMS], in which
components of an identity are defined to be an e-mail address,
a validity period and a URI. In this case, the unpredictable
results of a collision are extremely unlikely to fit the
format of a valid identity, and thus are of no use to an
attacker. Any protocol using IBE MUST define an identity in a
way that makes collisions in a hash function essentially
useless to an attacker. Because random strings are rarely used
as identities, this requirement should not be unduly difficult
to fulfill.
The randomness of the random values that are required by the
cryptographic algorithms is vital to the security provided by
the algorithms. Any implementation of these algorithms MUST
use a source of random values that provides an adequate level
of security. Appropriate algorithms to generate such values
include [FIPS186-2] and [X9.62]. This will ensure that the
random values used to mask plaintext messages in sections 5.4
and 6.4 are not reused with a significant probability.
The strength of a system using the algorithms described in
this document relies on the strength of the mechanism used to
authenticate a user requesting a private key from a PKG, as
described in step 2 of section 1.2 of this document. This is
analogous to way in which the strength of a system using
digital certificates [X.509] is limited by the strength of the
authentication required of users before certificates are
granted to them. In either case, a weak mechanism for
authenticating users will result in a weak system that relies
on the technology. A system that uses the algorithms described
in this document MUST require users to authenticate in a way
that is suitably strong, particularly if IBE private keys will
be used for authentication.
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Note that IBE systems have different properties than other
asymmetric cryptographic schemes when it comes to key
recovery. If a master secret is maintained on a secure PKG
then the PKG and any administrator with the appropriate level
of access will be able to create arbitrary private keys, so
that controls around such administrators and logging of all
actions performed by such administrators SHOULD be part of a
functioning IBE system.
On the other hand, it is also possible to create IBE private
keys using a master secret and to then destroy the master
secret, making any key recovery impossible. If this property
is not desired, an administrator of an IBE system SHOULD
require that the format of the identity used by the system
contain a component that is short-lived. The format of
identity that is defined in [IBECMS], for example, contains
information about the time period of validity of the key that
will be calculated from the identity. Such an identity can
easily be changed to allow the rekeying of users if their IBE
private key is somehow compromised.
10. IANA considerations
No further action by the IANA is necessary for this document.
11. Acknowledgments
This document is based on the IBCS #1 v2 document of Voltage
Security, Inc. Any substantial use of material from this
document should acknowledge Voltage Security, Inc. as the
source of the information.
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12. References
12.1. Normative references
[KEYWORDS] S. Bradner, "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[TLS] T. Dierks and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.1," RFC 4346, April 2006.
12.2. Informative references
[BB1] D. Boneh and X. Boyen, "Efficient selective-ID secure
identity based encryption without random oracles," In Proc. of
EUROCRYPT 04, LNCS 3027, pp. 223-238, 2004.
[BF] D. Boneh and M. Franklin, "Identity-based encryption from
the Weil pairing," in Proc. of CRYPTO 01, LNCS 2139, pp. 213-
229, 2001.
[CMS] R. Housley, "Cryptographic Message Syntax," RFC 3852,
July 2004.
[ECC] I. Blake, G. Seroussi, and N. Smart, Elliptic Curves in
Cryptography, Cambridge University Press, 1999.
[FIPS186-2] National Institute of Standards and Technology,
"Digital Signature Standard," Federal Information Processing
Standard 186-2, August 2002.
[IBEARCH] G. Appenzeller, L. Martin, and M. Schertler,
"Identity-based Encryption Architecture," draft-ietf-smime-
ibearch-05.txt, April 2007.
[IBECMS] L. Martin and M. Schertler, "Using the Boneh-
Franklin and Boneh-Boyen identity-based encryption algorithms
with the Cryptographic Message Syntax (CMS)" draft-ietf-
smime-bfibecms-06.txt, June 2007.
[MERKLE] R. Merkle, "A fast software one-way hash function,"
Journal of Cryptology, Vol. 3 (1990), pp. 43-58.
[P1363] IEEE P1363-2000, "Standard Specifications for Public
Key Cryptography," 2001.
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[SP800-57] E. Barker, W. Barker, W. Burr, W. Polk and M. Smid,
"Recommendation for Key Management - Part 1: General
(Revised)," NIST Special Publication 800-57, March 2007.
[SHA] National Institute for Standards and Technology, "Secure
Hash Standard," Federal Information Processing Standards
Publication 180-2, August 2002, with Change Notice 1, February
2004.
[X9.62] American National Standards Institute, "Public Key
Cryptography for the Financial Services Industry: The Elliptic
Curve Digital Signature Algorithm (ECDSA)," American National
Standard for Financial Services X9.62-2005, November 2005.
[X.509] ITU-T Recommendation X.509 (2000) | ISO/IEC 9594-
8:2001, Information Technology - Open Systems Interconnection
- The Directory: Public-key and Attribute Certificate
Frameworks.
Authors' Addresses
Xavier Boyen
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304
Email: xavier@voltage.com
Luther Martin
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304
Email: martin@voltage.com
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