Request for Comments: 5091
Category: Informational
Supersingular Curve Implementations of the BF and BB1 Cryptosystems
L. Martin
Voltage Security
December 2007
Identity-Based Cryptography Standard (IBCS) #1:
Status of This Memo
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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
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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
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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.
Table of Contents
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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 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
1. Introduction
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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:
- 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.
- 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
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In order to send an encrypted message, an IBE user must perform the following steps:
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- Obtain the recipient's public parameters.
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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.
- Construct and send an IBE-encrypted message.
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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 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.
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1.1.1. Sender Obtains Recipient's Public Parameters
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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-
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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.
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1.1.2. Construct and Send an IBE-Encrypted Message
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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
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^
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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
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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:
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- Obtain the recipient's public parameters.
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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.
- Obtain the IBE private key from the PKG.
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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.
- Decrypt the IBE-encrypted message.
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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.
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1.2.1. Recipient Obtains Public Parameters from PPS
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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-
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Figure 3. Requesting IBE Public Parameters
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1.2.2. Recipient Obtains IBE Private Key from PKG
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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-
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Figure 4. Obtaining an IBE Private Key
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1.2.3. Recipient Decrypts IBE-Encrypted Message
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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
^
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IBE Private Key
and IBE Public Parameters
Figure 5. Using an IBE Public-Key Algorithm to Decrypt
2. Notation and Definitions
2.1. Notation
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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' ->
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."
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)).
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
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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
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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
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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
- 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
- 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
- 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
- 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)- 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
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Algorithm 3.2.1 (SignedWindowDecomposition): computes the signed m-ary window representation of a positive integer [ECC].
Input:
- 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
- An integer window bit-size r > 0
Output:
- 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
- While j <= l, do:
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(a) If k_j = 0, then:
i. Let j = j + 1
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(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:
- 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:
- The point [l]A
Method:
- (Window decomposition)
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(a) Let r > 0 be an integer (fixed) bit-wise window size,
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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)) (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)- (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))- Main loop
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(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
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ii. If b_i > 0, then:
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A. Let Q = Q + A_(b_i) using Algorithm 3.1.2
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(PointAdd1)
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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- Return Q.
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3.3. Operations in Jacobian Projective Coordinates
3.3.1. Implementation for Type-1 Curves
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Algorithm 3.3.1 (ProjectivePointDouble1): adds a point to itself in Jacobian projective coordinates for type-1 curves.
Input:
- A point (x, y, z) = A in E(F_p^2) in Jacobian projective coordinates
- An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
- The point [2]A in Jacobian projective coordinates
Method:
- 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
- Let lambda_2 = y^2
5. Let lambda_3 = 4 * lambda_2 * x
- Let x' = lambda_1^2 - 2 * lambda_3
- 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:
- A point (x_A, y_A, z_A) = A in E(F_p^2) in Jacobian projective coordinates
- A point (x_B, y_B) = B in E(F_p^2) \ {0} in affine coordinates
- An elliptic curve E/F_p: y^2 = x^3 + 1
Output:
- The point A + B in Jacobian projective coordinates
Method:
- If z_A = 0, return (x_B, y_B, 1) = B, otherwise:
- Let lambda_1 = z_A^2
- Let lambda_2 = lambda_1 * x_B
- Let lambda_3 = x_A - lambda_2
- If lambda_3 = 0, then return (0, 1, 0), otherwise:
- Let lambda_4 = lambda_3^2
- Let lambda_5 = lambda_1 * y_B * z_A
- Let lambda_6 = lambda_4 - lambda_5
- Let lambda_7 = x_A + lambda_2
- Let lambda_8 = y_A + lambda_5
- Let x' = lambda_6^2 - lambda_7 * lambda_4
- 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 o A point A in E(F_p)
- A description of a type-1 elliptic curve E/F_p
Output:
- 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
- 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)
- A description of a type-1 elliptic curve E/F_p
Output:
- An element of F_p^2 that is the divisor of the line tangent to A evaluated at B
Method:
- (Special cases)
(a) If A = 0, return 1 (b) If y_A = 0, return EvalVertical1(B, A) using Algorithm 3.4.1 (EvalVertical1)- (Line computation)
(a) Let a = -3 * (x_A)^2 (b) Let b = 2 * y_A (c) Let c = -b * y_A - a * x_A- (Evaluation at B)
(a) Let r = a * x_B + b * y_B + c
- 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
- Two points A', A'' in E(F_p)
- A description of a type-1 elliptic curve E/F_p
Output:
- An element of F_p^2 that is the divisor of the line going through A' and A'' evaluated at B
Method:
- (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)- (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'- (Evaluation at B)
(a) Let r = a * x_B + b * y_B + c
- 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)
- 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:
- 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)
- 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:
- The value e(A, B) in F_p^2, computed using the Miller algorithm
Method:
- (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:
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
-
- (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) (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)
- (Correcting exponent)
(a) Let eta = (p^2 - 1) / q
- (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.
Input:
- A string s of length |s| octets
- A positive integer n represented as Ceiling(lg(n) / 8) octets.
- A cryptographic hash function hashfcn
Output:
o A positive integer v in the range 0 to n - 1
Method:
- Let hashlen be the number of octets comprising the output of hashfcn
2. Let v_0 = 0
- Let h_0 = 0x00...00, a string of null octets with a length of hashlen
- 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.
Algorithm 4.2.1 (HashBytes): keyed cryptographic pseudo-random bytes generator.
Input:
- An integer b
- A string p
- A cryptographic hash function hashfcn
Output:
- A string r comprising b octets
Method:
- Let hashlen be the number of octets comprising the output of hashfcn
2. Let K = hashfcn(p)
- Let h_0 = 0x00...00, a string of null octets with a length of hashlen
4. Let l = Ceiling(b / hashlen)
- 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.
Input:
- An element v in F_p^2
- A description of F_p^2
- An ordering parameter o, either 0 or 1
Output:
- 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:
- An element v in F_p^2
- A description of p, where p is congruent to 3 modulo 4
- A ordering parameter o, either 0 or 1
Output:
- A string s of size 2 * Ceiling(lg(p) / 8) octets
Method:
- 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- 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)- 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:
- An elliptic curve E
- A prime p
- A prime q
- A string id
- 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 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:
- A prime p
- A prime q
- A string id
- A cryptographic hash function hashfcn
Output:
- 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:
- 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
- Two points A and B of order q in E(F_p) or E(F_p^2)
Output:
- 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:
- 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:
- 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 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:
- 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
- Four points A, B, C, and D, of order q in E(F_p) or E(F_p^2)
Output:
- 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)
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:
- 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^2Method:
- 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:
- An integer version number
- 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)
- A corresponding master secret s
Method:
- Depending on the selected type t:
-
(a) If version = 2, then execute Algorithm 5.1.2 (BFsetup1)
- 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:
- A security parameter n, which MUST be either 1024, 2048, 3072, 7680 or 15360
Output:
- A set of common public parameters (version, p, q, P, Ppub, hashfcn)
- A corresponding master secret s
Method:
- Set the version to version = 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])- 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- 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- 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- (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.
- 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.
Algorithm 5.2.1 (BFderivePubl): derives the public key corresponding to an identity string.
Input:
- An identity string id
o A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
Output:
- 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:
- An identity string id
o A set of public parameters (version, E, p, q, P, P_pub, hashfcn)
Output:
- 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:
- A plaintext string m of size |m| octets
- 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:
- 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- 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)- 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'- Let w = hashfcn(z) using the hashfcn hashing algorithm, the result of which is a hashlen-octet string
- 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:
- 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:
- A decrypted plaintext m, or an invalid ciphertext flag
Method:
- 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- 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- 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- 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:
- An integer version number
- An integer security parameter n (MUST take values either 1024, 2048, 3072, 7680, or 15360)
Output:
- A set of public parameters
- A corresponding master secret
Method:
- 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:
- 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)- A corresponding triple of master secrets (alpha, beta, gamma)
Method:
- 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])- 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- 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- 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) 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:
- An identity string id
- A set of common public parameters (version, k, E, p, q, P, P_1, P_2, P_3, v, hashfcn)
Output:
- 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.
Input:
- 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:
- 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
- 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- Calculate the point D_1 as follows:
(a) Let D_1 = [r]P
- 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:
- A plaintext string m of size |m| octets
- 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:
- 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- Calculate the point C_0 as follows:
(a) Let C_0 = [s]P
- 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- 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
-
- 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'- 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)- 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 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:
- A private key given as a pair of points (D_0, D_1) of order q in E(F_p)
- 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:
- 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)- 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
(y_1)_(256^l) to represent the y coordinate of C_1
-
- 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'- 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)- 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
- 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)
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)
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)
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
-
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)
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)
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) } -- -- 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) -- } -- -- 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 } -- -- 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 } -- -- 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
9. Security Considerations
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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.
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 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
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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
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[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
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[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. [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.
Authors' Addresses
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Xavier Boyen
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304EMail:
xavier@voltage.com
Luther Martin
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304EMail:
martin@voltage.com
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