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Keywords: Encryption, Cryptography, Security, Elliptic Curves, Elliptic Curve Cryptography, Pairing-based Cryptography, Identity-based Cryptography, Identity-based Encryption, Boneh-Franklin Encryption Scheme, Boneh-Boyen Encryption Scheme
Network Working Group X. Boyen
Request for Comments: 5091 L. Martin
Category: Informational Voltage Security
December 2007
Identity-Based Cryptography Standard (IBCS) #1:
Supersingular Curve Implementations of the BF and BB1 Cryptosystems
Status of This Memo
This memo provides information for the Internet community. It does
not specify an Internet standard of any kind. Distribution of this
memo is unlimited.
IESG Note
This document specifies two mathematical algorithms for identity
based encryption (IBE). Due to its specialized nature, this document
experienced limited review within the IETF. Readers of this RFC
should carefully evaluate its value for implementation and
deployment.
Abstract
This document describes the algorithms that implement Boneh-Franklin
(BF) and Boneh-Boyen (BB1) Identity-based Encryption. This document
is in part based on IBCS #1 v2 of Voltage Security's Identity-based
Cryptography Standards (IBCS) documents, from which some irrelevant
sections have been removed to create the content of this document.
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RFC 5091 IBCS #1 December 2007
Table of Contents
1. Introduction ....................................................4
1.1. Sending a Message That Is Encrypted Using IBE ..............5
1.1.1. Sender Obtains Recipient's Public Parameters ........6
1.1.2. Construct and Send an IBE-Encrypted Message .........6
1.2. Receiving and Viewing an IBE-Encrypted Message .............7
1.2.1. Recipient Obtains Public Parameters from PPS ........8
1.2.2. Recipient Obtains IBE Private Key from PKG ..........8
1.2.3. Recipient Decrypts IBE-Encrypted Message ............9
2. Notation and Definitions ........................................9
2.1. Notation ...................................................9
2.2. Definitions ...............................................12
3. Basic Elliptic Curve Algorithms ................................12
3.1. The Group Action in Affine Coordinates ....................13
3.1.1. Implementation for Type-1 Curves ...................13
3.2. Point Multiplication ......................................14
3.3. Operations in Jacobian Projective Coordinates .............17
3.3.1. Implementation for Type-1 Curves ...................17
3.4. Divisors on Elliptic Curves ...............................19
3.4.1. Implementation in F_p^2 for Type-1 Curves ..........19
3.5. The Tate Pairing ..........................................21
3.5.1. Tate Pairing Calculation ...........................21
3.5.2. The Miller Algorithm for Type-1 Curves .............21
4. Supporting Algorithms ..........................................24
4.1. Integer Range Hashing .....................................24
4.1.1. Hashing to an Integer Range ........................24
4.2. Pseudo-Random Byte Generation by Hashing ..................25
4.2.1. Keyed Pseudo-Random Bytes Generator ................25
4.3. Canonical Encodings of Extension Field Elements ...........26
4.3.1. Encoding an Extension Element as a String ..........26
4.3.2. Type-1 Curve Implementation ........................27
4.4. Hashing onto a Subgroup of an Elliptic Curve ..............28
4.4.1. Hashing a String onto a Subgroup of an
Elliptic Curve .....................................28
4.4.2. Type-1 Curve Implementation ........................29
4.5. Bilinear Mapping ..........................................29
4.5.1. Regular or Modified Tate Pairing ...................29
4.5.2. Type-1 Curve Implementation ........................30
4.6. Ratio of Bilinear Pairings ................................31
4.6.1. Ratio of Regular or Modified Tate Pairings .........31
4.6.2. Type-1 Curve Implementation ........................32
5. The Boneh-Franklin BF Cryptosystem .............................32
5.1. Setup .....................................................32
5.1.1. Master Secret and Public Parameter Generation ......32
5.1.2. Type-1 Curve Implementation ........................33
5.2. Public Key Derivation .....................................34
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5.2.1. Public Key Derivation from an Identity and
Public Parameters ..................................34
5.3. Private Key Extraction ....................................35
5.3.1. Private Key Extraction from an Identity, a
Set of Public ......................................35
5.4. Encryption ................................................36
5.4.1. Encrypt a Session Key Using an Identity and
Public Parameters ..................................36
5.5. Decryption ................................................37
5.5.1. Decrypt an Encrypted Session Key Using
Public Parameters, a Private Key ...................37
6. The Boneh-Boyen BB1 Cryptosystem ...............................38
6.1. Setup .....................................................38
6.1.1. Generate a Master Secret and Public Parameters .....38
6.1.2. Type-1 Curve Implementation ........................39
6.2. Public Key Derivation .....................................41
6.2.1. Derive a Public Key from an Identity and
Public Parameters ..................................41
6.3. Private Key Extraction ....................................41
6.3.1. Extract a Private Key from an Identity,
Public Parameters and a Master Secret ..............41
6.4. Encryption ................................................42
6.4.1. Encrypt a Session Key Using an Identity and
Public Parameters ..................................42
6.5. Decryption ................................................45
6.5.1. Decrypt Using Public Parameters and Private Key ....45
7. Test Data ......................................................47
7.1. Algorithm 3.2.2 (PointMultiply) ...........................47
7.2. Algorithm 4.1.1 (HashToRange) .............................48
7.3. Algorithm 4.5.1 (Pairing) .................................48
7.4. Algorithm 5.2.1 (BFderivePubl) ............................49
7.5. Algorithm 5.3.1 (BFextractPriv) ...........................49
7.6. Algorithm 5.4.1 (BFencrypt) ...............................50
7.7. Algorithm 6.3.1 (BBextractPriv) ...........................51
7.8. Algorithm 6.4.1 (BBencrypt) ...............................52
8. ASN.1 Module ...................................................53
9. Security Considerations ........................................58
10. Acknowledgments ...............................................60
11. References ....................................................60
11.1. Normative References .....................................60
11.2. Informative References ...................................60
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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 in 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 is
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:
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.
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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 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) [CMS]. How to use IBE with CMS is
described 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 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 an 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
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someone a secure message than 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 Transport Layer Security (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
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 an 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.
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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 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 the IBE-encrypted message.
The IBE private key decrypts the CEK, which is then used to
decrypt encrypted message.
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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
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
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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
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 [BF],
[IBEARCH] regarding all notions pertaining to identity-based
encryption.
F_p denotes finite field of prime characteristic p; F_p^2 denotes 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 as 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' ->
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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.
#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."
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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 as 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.
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)).
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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.
Public parameters - The public parameters are a set of common,
system-wide parameters generated and published by the private key
generator (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 is 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 - 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 - Ciphertext is an encrypted representation of any block
of data, including 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].
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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:
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
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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)
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:
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i. Let j = j + 1
(b) Else:
i. Let t = min{l, j + r - 1}
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
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(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))
(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.
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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.
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
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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:
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')
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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
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
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(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
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)
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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)
(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)
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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:
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)
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(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:
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)
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(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)
(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. HashToRange is
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.
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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:
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. HashBytes
is based on Merkle's method for hashing [MERKLE], which is provably
as secure as the underlying hash function hashfcn.
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Algorithm 4.2.1 (HashBytes): keyed cryptographic pseudo-random bytes
generator.
Input:
o An integer b
o A string p
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)
3. Let h_0 = 0x00...00, a string of null octets with a length of
hashlen
4. Let l = Ceiling(b / hashlen)
5. 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
6. 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.
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Input:
o An element v in F_p^2
o A description of F_p^2
o An 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 an 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
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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:
(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, 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)
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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 cryptographic hash 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.
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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 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:
1. If E is a type-1 curve, execute Algorithm 4.6.2 (PairingRatio1)
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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 An integer version number
o A security parameter n (MUST take values either 1024, 2048, 3072,
7680, 15360)
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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.
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]
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(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
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 it returns a point Q_id. The public parameters used
MUST be a valid set of public parameters as defined by Section 5.1.2.
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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, a set of public
parameters, and corresponding master secret, and it 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
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
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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.
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)
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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)
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
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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-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.
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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:
(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 or 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:
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(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])
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 4.5.1 (Pairing)
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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 6.1.2.
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 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, a set of public
parameters, and corresponding master secrets, and it 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 6.1.2.
Algorithm 6.3.1 (BBextractPriv): extracts the private key
corresponding to an identity string.
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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)
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 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
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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|
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
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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
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 4.2.1 (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)
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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 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 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
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(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
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 the 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)
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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
: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)
Input:
q = 0xfffffffffffffffffffffffffffbffff
p = 0xbffffffffffffffffffffffffffcffff3
E/F_p: y^2 = x^3 + 1
A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
0x510c6972d795ec0c2b081b81de767f808)
B = (0x40e98b9382e0b1fa6747dcb1655f54f75,
0xb497a6a02e7611511d0db2ff133b32a3f)
Output:
e'(A, B) = (0x8b2cac13cbd422658f9e5757b85493818,
0xbc6af59f54d0a5d83c8efd8f5214fad3c)
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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)
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)
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7.6. Algorithm 5.4.1 (BFencrypt)
Note: 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)
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
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RFC 5091 IBCS #1 December 2007
7.7. Algorithm 6.3.1 (BBextractPriv)
Input:
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)
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)
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RFC 5091 IBCS #1 December 2007
7.8. Algorithm 6.4.1 (BBencrypt)
Note: 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)
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)
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RFC 5091 IBCS #1 December 2007
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.
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)
}
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RFC 5091 IBCS #1 December 2007
--
-- 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.
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)
-- }
--
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RFC 5091 IBCS #1 December 2007
-- 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)
-- }
--
--
-- 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
}
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RFC 5091 IBCS #1 December 2007
--
-- A BF master secret is an integer.
-- The only version supported is version 2.
--
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
}
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RFC 5091 IBCS #1 December 2007
--
-- BB1 master secret block
-- The only version supported is version 2.
--
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
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RFC 5091 IBCS #1 December 2007
9. Security Considerations
This document describes cryptographic algorithms. We assume that the
security provided by such algorithms depends entirely on the secrecy
of the relevant private key, and for an adversary to defeat the
security provided by the algorithms, he will need to perform
computationally-intensive cryptanalytic attacks to recover the
private 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 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 Advanced Encryption
Standard (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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RFC 5091 IBCS #1 December 2007
Note that this document specifies the use of the National Institute
of Standards and Technology (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 that 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 the 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.
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
Boyen & Martin Informational [Page 59]
RFC 5091 IBCS #1 December 2007
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. 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.
11. References
11.1. Normative References
[KEYWORDS] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[TLS] Dierks, T. and E. Rescorla, "The Transport Layer
Security (TLS) Protocol Version 1.1", RFC 4346, April
2006.
11.2. 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] Housley, R., "Cryptographic Message Syntax (CMS)", RFC
3852, July 2004.
Boyen & Martin Informational [Page 60]
RFC 5091 IBCS #1 December 2007
[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", Work in Progress.
[IBECMS] L. Martin and M. Schertler, "Using the Boneh-Franklin
and Boneh-Boyen identity-based encryption algorithms
with the Cryptographic Message Syntax (CMS)", Work in
Progress.
[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.
[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.
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RFC 5091 IBCS #1 December 2007
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
Boyen & Martin Informational [Page 62]
RFC 5091 IBCS #1 December 2007
Full Copyright Statement
Copyright (C) The IETF Trust (2007).
This document is subject to the rights, licenses and restrictions
contained in BCP 78 and at www.rfc-editor.org/copyright.html, and
except as set forth therein, the authors retain all their rights.
This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND
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OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF
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Boyen & Martin Informational [Page 63]