X. Boyen
S/MIME Working Group L. Martin
Internet Draft Voltage Security
Expires: December 2006 June 2006
Identity-Based Cryptography Standard (IBCS) #1: Supersingular Curve
Implementations of the BF and BB1 Cryptosystems
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Abstract
This document describes the algorithms that implement Boneh-Franklin
and Boneh-Boyen Identity-based Encryption. This document is in part
based on IBCS #1 v2 of Voltage Securitys Identity-based Cryptography
Standards (IBCS) documents.
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Table of Contents
1. Introduction...................................................3
2. Notation and definitions.......................................4
2.1. Notation..................................................4
2.2. Definitions...............................................6
3. Basic elliptic curve algorithms................................7
3.1. The group action in affine coordinates....................7
3.1.1. Implementation for type-1 curves.....................7
3.2. Point multiplication......................................9
3.3. Special operations in projective coordinates.............11
3.3.1. Implementation for type-1 curves....................11
3.4. Divisors on elliptic curves..............................13
3.4.1. Implementation in F_p^2 for type-1 curves...........13
3.5. The Tate pairing.........................................15
3.5.1. The Miller algorithm for type-1 curves..............16
4. Supporting algorithms.........................................18
4.1. Integer range hashing....................................19
4.2. Pseudo-random generation by hashing......................20
4.3. Canonical encodings of extension field elements..........20
4.3.1. Type-1 curve implementation.........................21
4.4. Hashing onto a subgroup of an elliptic curve.............22
4.4.1. Type-1 curve implementation.........................22
4.5. Bilinear pairing.........................................23
4.5.1. Type-1 curve implementation.........................24
4.6. Ratio of bilinear pairings...............................25
4.6.1. Type-1 curve implementation.........................25
5. The Boneh-Franklin BF cryptosystem............................26
5.1. Setup....................................................26
5.1.1. Type-1 curve implementation.........................27
5.2. Public key derivation....................................28
5.3. Private key extraction...................................28
5.4. Encryption...............................................29
5.5. Decryption...............................................30
6. Wrapper methods for the BF system.............................32
6.1. Private key generator (PKG) setup........................32
6.2. Private key extraction by the PKG........................32
6.3. Session key encryption...................................33
7. Concrete encoding guidelines for BF...........................35
7.1. Encoding of points on a curve............................35
7.2. Public parameters blocks.................................36
7.2.1. Type-1 implementation...............................36
7.3. Master secret blocks.....................................38
7.4. Private key blocks.......................................38
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7.5. Ciphertext blocks........................................39
8. The Boneh-Boyen BB1 cryptosystem..............................40
8.1. Setup....................................................40
8.1.1. Type-1 curve implementation.........................41
8.2. Public key derivation....................................42
8.3. Private key extraction...................................43
8.4. Encryption...............................................44
8.5. Decryption...............................................46
9. Wrapper methods for the BB1 system............................48
9.1. Private key generator (PKG) setup........................48
9.2. Private key extraction by the PKG........................49
9.3. Session key encryption...................................50
10. Concrete encoding guidelines for BB1.........................51
10.1. Encoding of points on a curve...........................51
10.2. Public parameters blocks................................52
10.2.1. Type-1 implementation..............................53
10.3. Master secret blocks....................................54
10.4. Private key blocks......................................55
10.5. Ciphertext blocks.......................................56
11. ASN.1 syntax.................................................57
12. Security considerations......................................63
13. IANA considerations..........................................63
14. Acknowledgments..............................................63
15. References...................................................64
15.1. Informative references..................................64
Authors Addresses...............................................64
Intellectual Property Statement..................................64
Disclaimer of Validity...........................................65
Copyright Statement..............................................65
Acknowledgment...................................................65
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) [3], and the first IBE system proposed by Boneh and
Boyen (BB1) [2]. 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 2.3).
Implementations based on other types of curves currently fall outside
the scope of this document.
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2. Notation and definitions
2.1. Notation
This section summarizes the essential notions and definitions
regarding identity-based cryptosystems on elliptic curves. The reader
is referred to [1] for the mathematical background and to [2, 3]
regarding all notions pertaining to identity-based encryption.
Let F_p be a finite field of large prime characteristic p, and let
F_p^k denote its extension field of degree k. Denote by F*_p the
multiplicative group of F_p^k, for any k >= 1.
Let E/F_p : y^2 = x^3 + a * x + b be an elliptic curve over F_p. For
any extension degree k >= 1, the curve E/F_p defines a group
(E(F_p^k), +), which is the additive group of points of affine
coordinates (x, y) in (F_p^k)^2 satisfying the curve equation over
F_p^k, with null element, or point at infinity, denoted 0. Let
#E(F_p^k) be the size of E(F_p^k).
Let q be a prime such that E(F_p) has a cyclic subgroup G1 of order
q. Let k be the embedding degree or security multiplier of G1 in
E(F_p), or the smallest integer greater than or equal to 1 such that
q divides p^k . 1. Then E(F_p^k) contains a cyclic subgroup of order
q, denoted G1, and F*_p^k contains a cyclic subgroup of order p,
denoted G2.
Under these conditions, two mathematical constructions known as the
Weil pairing and the Tate pairing, each provide an efficiently
computable map e : G1 x G1 -> G2 that is linear in both arguments
and believed hard to invert. If an efficiently computable isomorphism
phi : G1 -> G1 is available for the selected elliptic curve on
which the Tate pairing is computed, then one 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 1536-bit prime, being the order of the finite field
F_p.
F_p - the base finite field of size p over which the elliptic curve
of interest E/F_p is defined.
#G - the size of G, where G is a finite group.
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G* - the multiplicative group of the invertible elements in G; 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 conventional 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 160bit to 256-bit prime, being the order of the cyclic
subgroup of interest in E(F_p).
k - the embedding degree, or security multiplier, of the cyclic
subgroup of order q in E(F_p).
F_p^k - the extension field of the base field F_p of degree equal to
the security multiplier k.
E(F_p^k ) - the group of points of affine coordinates in F_p^k
satisfying the curve equation E/F_p, including the point at infinity
0.
The following conventions are assumed for curve operations.
Point addition If A and B are two points on a curve E, their sum is
denoted A + B.
Point multiplication If A is a point on a curve, and n an integer,
the result of adding A to itself a total of n times is denoted [n]A.
The following class of elliptic curves is exclusively considered for
pairing operations in the present version of the IBCS#1 standard,
referred to as type-1.
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, so that the p pairs of (x, y) coordinates corresponding to the p
non-zero points E(F_p) \ {0} satisfy a useful bijective relation x <-
> y, with x = (y^2 . 1)^(1/3) (mod p) and y = (x^3 + 1)^(1/2) (mod
p). Type-1 curves always lead to a security multiplier k = 2, where
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f(x) = (x^2 + 1) is always irreducible, allowing the uniform
representation of F_p^2 = F[x] / (x^2 + 1). 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 8.1.2 (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^k elements of the extension field
F_p^k are represented as k-tuples of elements of F_p. A k-tuple (a_0,
..., a_(k.1) is interpreted as the polynomial a_(k . 1) * x^(k . 1) +
... +a_1 * x + a_0 in F_p[x] / f(x), where f(x) is an irreducible
monic polynomial of order k. The actual polynomial f(x) chosen
depends on p and k.
Type-1 curves For type-1 curves, which are supersingular curves of
equation E/F_p : y^2 = x^3 + 1 with p congruent to 11 modulo 12, the
extension degree k is always 2 and elements of F_p^2 are represented
as polynomials a_1 * x + a_0 in F_p[x] / (x^2 + 1).
Elliptic curve points Points in E(F_p^k) for k >= 1 with the point
P = (x, y) in F_p^k x F_p^k 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^k.
2.2. Definitions
The following terminology is used to describe an IBE system.
Public parameters The public parameters are set of common
systemwide parameters generated and published by the private key
server (PKG).
Master secret The master secret is the master key generated and
privately kept by the key server, and used to generate the private
keys of the users.
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Identity An identity an arbitrary string, usually a human-readable
unambiguous designator of a system user, possibly augmented with a
time stamp and other attributes.
Public key A public key is a string that is algorithmically derived
from an identity. The derivation may be performed by anyone,
autonomously.
Private key A private key is issued by the key server to correspond
to a given identity (and the public key that derives from it), under
the published set of public parameters.
Plaintext A plaintext is an unencrypted representation, or in the
clear, of any block of data to be transmitted securely. For the
present purposes, plaintexts are typically session keys, or sets of
session keys, for further symmetric encryption and authentication
purposes.
Ciphertext A ciphertext is an encrypted representation of any block
of data, including a plaintext, to be transmitted securely.
3. Basic elliptic curve algorithms
This section describes algorithms for performing all needed basic
arithmetic operations on elliptic curves. The presentation is
specialized to the type of curves under consideration for simplicity
of implementation. General algorithms may be found in [1].
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:
a point A in E(F_p^k), with A = (x, y) or 0.
an elliptic curve E/F_p : y^2 = x^3 + 1.
Output:
the point [2]A = A + A.
Method:
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1. If A = 0 or y = 0, then return 0.
2. lambda = (3 * x^2) / (2 * y).
3. x = lambda^2 2 * x.
4. y = (x x) * lambda y.
5. Return (x, y).
Algorithm 3.1.2 (PointAdd1): adds two points on a type-1 elliptic
curve.
Input:
a point A in E(F_p^k), with A = (x_A, y_A) or 0,
a point B in E(F_p^k), with B = (x_B, y_B) or 0,
an elliptic curve E/F_p : y^2 = x^3 + 1.
Output:
the point A + B.
Method:
1. If A = 0, return B.
2. If B = 0, return A.
3. If x_A = x_B:
(a) If y_A = .y_B, return 0.
(b) Else return [2]A computed using Algorithm 2.1.1
(PointDouble1).
4. Otherwise:
(a) lambda = (y_B . y_A) / (x_B . x_A).
(b) x = lambda^2 . x_A . x_B.
(c) y = (x_A . x) * lambda - y_A.
(d) Return (x, y).
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3.2. Point multiplication
Algorithm 3.2.1 (SignedWindowDecomposition): computes the signed m-
ary window representation of a positive integer.
Input:
an integer l > 0,
an integer window bit-size r > 0.
Output:
The unique d-element sequence {(b_i, e_i)} for i = 0 to d - 1 such
that l = {Sum(b_i * 2^(e_i) for i = 0 to d 1} and b_i = +/- 2^j for
some 0 <= j <= r - 1.
Method:
1. d = 0.
2. j = 0.
3. While j <= l, do:
(a) If l_k = 0 then:
i. j = j + 1.
(b) Else:
i. t = min{j + r . 1, l}.
ii. h_d = (l_t, l_(t 1), ..., l_j)(base 2).
iii. If h_d > 2^(r . 1) then:
A. b_d = h_d . 2^r.
B. l = l + 2^(t + 1).
iv. Else:
A. b_d = h_d.
v. e_d = j.
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vi. d = d + 1.
vii. 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^k),
an integer l > 0,
an elliptic curve E/F_p : y^2 = x^3 + a * x + b.
Output:
the point [l]A.
Method:
1. (Window decomposition)
(a) Let r > 0 be an integer (fixed) bit-wise window size, e.g., r
= 5.
(b) Let l = l where l = {Sum(l_j * 2^j), for j = 0 to l} is the
binary expansion of 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) A_1 = A.
(b) A_2 = [2]A, using Algorithm 3.1.1 (PointDouble1).
(c) For i = 1 to 2^(r . 2) . 1, do:
i. A_(2 * i + 1) = A_(2 * i . 1) + A_2 using Algorithm 3.1.2
(PointAdd1).
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(d) Q = A_(b_(d . 1)).
3. Main loop
(a) For i = d . 2 to 0 by .1, do:
i. 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. Q = Q + A_(b_i) using Algorithm 3.1.2 (PointAdd1).
iii. Else:
A. Q = Q . A_(.b_i) using Algorithm 3.1.2 (PointAdd1).
(b) Calculate Q = [2^(e_0)]Q using repeated applications of
Algorithm 3.1.1 (PointDouble1) e_0 times.
4. Return Q.
3.3. Special operations in projective coordinates
3.3.1. Implementation for type-1 curves
Algorithm 3.3.1 (ProjectivePointDouble1): adds a point to itself in
projective coordinates for type-1 curves.
Input:
a point (x, y, z) = A in E(F_p^k ) in projective coordinates,
an elliptic curve E/F_p : y^2 = x^3 + 1.
Output:
the point [2]A in projective coordinates.
Method:
1. If z = 0 or y = 0, return (0, 1, 0) = 0. Otherwise:
2. lambda_1 = 3 * x^2.
3. z = 2 * y * z.
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4. lambda_2 = y^2.
5. lambda_3 = 4 * lambda_2 * x.
6. x = lambda_1^2 2 * lambda_3.
7. lambda_4 = 8 * lambda_2^2.
8. 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 projective coordinates, for type-1
curves.
Input:
a point (x_A, y_A, z_A) = A in E(F_p^k ) in projective coordinates,
a point (x_B, y_B) = B in E(F_p^k ) \ {0} in affine coordinates,
an elliptic curve E/F_p : y^2 = x^3 + 1.
Output:
the point A + B in projective coordinates.
Method:
1. If z_A = 0 return (x_B, y_B, 1) = B. Otherwise:
2. lambda_1 = z_A^2
3. lambda_2 = lambda_1 * x_B.
4. lambda_3 = x_A lambda_2.
5. If lambda_3 = 0 then return (0, 1, 0) = 0. Otherwise:
6. lambda_4 = lambda_3^2.
7. lambda_5 = lambda_1 * y_B * z_A.
8. lambda_6 = lambda_4 lambda_5.
9. lambda_7 = x_A + lambda_2.
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10. lambda_8 = y_A + lambda_5.
11. x = lambda_6^2 lambda_7 * lambda_4.
12. lambda_9 = lambda_7 * lambda_4 2 * x.
13. y = (lambda_9 * lambda_6 lambda_8 * lambda_3 * lambda_4) / 2.
14. 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:
a point B in E(F_p^2) with B != 0.
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. r = x_B . x_A.
2. Return r.
Algorithm 2.4.2 (EvalTangent1): evaluates the divisor of a tangent on
a type-1 elliptic curve.
Input:
a point B in E(F_p^2 ) with B != 0.
a point A in E(F_p).
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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:
1. (Special cases)
(a) If A = 0 return 1 = 1 + 0 * i.
(b) If y_A = 0 return EvalVertical1(B, A) using Algorithm 3.4.1
(EvalVertical1).
2. (Line computation)
(a) a = .3 * (x_A)^2.
(b) b = 2 * y_A.
(c) c = .b * y_A . a * x_A.
3. (Evaluation at B)
(a) 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:
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:
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1. (Special cases)
(a) If A = 0 return EvalVertical1(B, A) using Algorithm 3.4.1
(EvalVertical1).
(b) If A = 0 return EvalVertical1(B, A) using Algorithm 3.4.1
(EvalVertical1).
(c) If A = .A return EvalVertical1(B, A) using Algorithm 3.4.1
(EvalVertical1).
( d) If A = A return EvalTangent1(B, A) using Algorithm 3.4.2
(EvalTangent1).
2. (Line computation)
(a) a = y_A . y_A.
(b) b = x_A . x_A.
(c) c = .b * y_A . a * x_A.
3. (Evaluation at B)
(a) r = a * x_B + b * y_B + c.
4. Return r.
3.5. The Tate pairing
Algorithm 3.5.1 (Tate): computes the Tate pairing on an elliptic
curve.
Input:
a point A of order q in E(F_p),
a point B of order q in E(F_p^k),
a description of an elliptic curve E/F_p such that E(F_p) and
E(F_p^k) have a subgroup of order q.
Output:
the value e(A, B) in F_p^k , computed using the Miller algorithm.
Method:
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1. For type-1 curve E, proceed with Algorithm 3.5.2
(TateMillerSolinas).
3.5.1. The Miller algorithm for type-1 curves
Algorithm 3.5.2 (TateMillerSolinas): computes the Tate pairing on a
type-1 elliptic curve.
Input:
a point A of order q in E(F_p),
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 prime order q, where q =
2^a + s * 2^b + c with c and s equal to either 1 or -1.
Output:
the value e(A, B) in F_p^2 , computed using the Miller algorithm.
The following description assumes that F_p^2 = F_p[i], where i^2 = -
1.
Elements x in F_p^2 may be explicitly represented as a + i * b, with
a, b in F_p.
Points in E(F_p) may also be represented as coordinate pairs (x, y)
with x, y in F_p.
Points in E(F_p^2) may be represented either as (x, y), with x, y in
F_p^2 or as (a + i * b, c + i * d), with a, b, c, d in F_p.
Method:
1. (Initialization)
(a) v_num = 1 in F_p^2.
(b) v_den = 1 in F_p^2.
(c) 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 projective coordinates.
(d) t_num = 1 in F_p^2.
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(e) 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. t_num = t_num^2.
ii. t_den = t_den^2.
iii. t_num = t_num * EvalTangent1(B, V ) using Algorithm 3.4.2
(EvalTangent1).
iv. V = (x_V , y_V , z_V ) = [2]V using Algorithm 3.3.1
(ProjectivePointDouble1).
v. t_den = t_den * EvalVertical1(B, V ) using Algorithm 3.4.1
(EvalVertical1).
(b) (Normalization)
i. 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. v_num = v_num * t_den.
B. v_den = v_den * t_num * EvalVertical1(B, V) using
Algorithm 3.4.1 (EvalVertical1).
ii. If s = 1:
A. v_num = v_num * t_num.
B. v_den = v_den * t_den.
3. (Calculation of the 2^a contribution)
(a) (Repeated doublings) For n = b to a . 1:
i. t_num = t_num^2.
ii. t_den = t_den^2.
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iii. t_num = t_num * EvalTangent1(B, V) using Algorithm 3.4.2
(EvalTangent1).
iv. V = (x_V , y_V , z_V) = [2]V using Algorithm 3.3.1
(ProjectivePointDouble1).
v. t_den = t_den * EvalVertical1(B, V) using Algorithm 3.4.1
(EvalVertical1).
(b) (Normalization)
i. 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. v_num = v_num * t_num.
ii. v_den = v_den * t_den.
4. (Correction for the (s * 2^b) and (c) contributions)
(a) v_num = v_num * EvalLine1(B, V_a, V_b) using Algorithm 3.4.3
(EvalLine1).
(b) v_den = v_den * EvalVertical1(B, V_a + V_b) using Algorithm
3.4.1 (EvalVertical1).
(c) If c = .1 then:
i. v_den = v_den * EvalVertical1(B,A) using Algorithm 3.4.1
(EvalVertical1).
5. (Correcting exponent)
(a) Let eta = (p^2 . 1) / q (an integer).
6. (Final result)
(a) Return (v_num / v_den)^eta in F_p^2 .
4. Supporting algorithms
This section describes a number of supporting algorithms for encoding
and hashing.
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4.1. Integer range hashing
HashToRangen(s, n) takes a string s and an integer n as input, and
returns an integer in the range 0 to n . 1 by cryptographic hashing.
The function performs a number l of SHA1 applications, with l chosen
in function of n so that, for random input, the output has an almost
uniform distribution in the entire range 0 to n . 1 with a
statistical relative non-uniformity no greater than 1/sqrt(n). I.e.,
for arbitrarily large n, for all v in 0 to n . 1, the probability
that HashToRangen(s, n) = v lies in the interval [(1 . n^(.1/2)) / n,
(1 + n^(.1/2)) / n].
Algorithm 4.1.1 (HashToRange): cryptographically hashes strings to
integers in a range.
Input:
a string s of length |s| bytes,
a positive integer n represented as Ceiling(8 * lg(n)) bytes.
Output:
a positive integer v in the range 0 to n . 1.
Method:
1. v_0 = 0.
2. h_0 = 0x0000000000000000000000000000000000000000, a string of 20
null bytes.
3. l = Ceiling((3 / 5) * lg(n)).
4. for each i in 1 to l, do:
(a) t_i = h_(i . 1) || s, which is the (|s| + 20)-byte string
concatenation of the strings h_(i . 1) and s.
(b) h_i = SHA1(t_i), which is a 20-byte string resulting from the
SHA1 algorithm on input t_i.
(c) Let a_i = Value(h_i) be the integer in the range 0 to 256^20 .
1 denoted by the raw byte string h_i interpreted in the unsigned big
endian convention.
(d) v_i = 256^20 * v_(i . 1) + a_i.
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5. v = v_l (mod n).
4.2. Pseudo-random generation by hashing
HashStream(b, p) takes an integer b and a string p as input, and
returns a b-byte pseudo-random string r as output. This function
relies on the SHA1 cryptographic hashing algorithm, and has a 160-bit
internal effective key space equal to the range of SHA1.
Algorithm 4.2.1 (HashStream): keyed cryptographic pseudo-random
stream generator.
Input:
an integer b,
a string p.
Output:
a string r of size b bytes.
Method:
1. K = SHA1(p).
2. h_0 = 0x0000000000000000000000000000000000000000 , a string of 20
null bytes.
3. l = Ceiling(b / 20).
4. for each i in 1 to l do:
(a) h_i = SHA1(h_(i . 1)).
(b) r_i = SHA1(h_i || K), where h_i || K is the 40-byte
concatenation of h_i and K.
5. r = LeftmostBytes(b, r_1 || ... || r_l), i.e., r is formed as the
concatenation of the r_i, truncated to the desired number of bytes.
4.3. Canonical encodings of extension field elements
Canonical(p, k, o, v) takes an element v in F_p^k, and returns a
canonical byte-string of fixed length representing v. The parameter o
must be either 0 or 1, and specifies the ordering of the encoding.
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Algorithm 4.3.1 (Canonical): encodes elements of an extension field
F_p^k as strings.
Input:
an element v in F_p^k,
a description of F_p^k ,
a 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).
4.3.1. Type-1 curve implementation
Canonical1(p, o, v) takes an element v in F_p^2 and returns a
canonical representation of v as a byte-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 Ceiling(16 * lg(p)) bytes.
Method:
1. l = 8 * Ceiling(lg(p)), the number of bytes needed to represent
integers in Zp.
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2. (a, b) = v, where (a, b) in (Z_p)^2 is the canonical
representation of v in F_p^2 = F_p / (x^2 + 1) as a polynomial a +i *
b with i^2 = .1.
3. Let a_(256^l) be the big-endian zero-padded fixed-length byte-
string representation of a in Zp.
4. Let b_(256^l) be the big-endian zero-padded fixed-length byte-
string representation of b in Zp.
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. The fixed-length encoding of v is output as the string s.
4.4. Hashing onto a subgroup of an elliptic curve
HashToPoint(E, p, q, id) takes an identity string id and the
description of a subgroup of prime order q in E(F_p) or E(F_p^k) and
returns a point Q_id of order q in E(F_p) or E(F_p^k).
Algorithm 4.4.1 (HashToPoint): cryptographically hashes strings to
points on elliptic curves.
Input:
a string id,
a description of a subgroup of prime order q on a curve E/F_p.
Output:
a point Q_id = (x, y) of order q on E.
Method:
1. For a type-1 curve E, execute Algorithm 4.4.2 (HashToPoint1).
4.4.1. Type-1 curve implementation
HashToPoint1(E, p, q, id) takes an identity string id and the
description of a subgroup of order q in E(F_p) where E : y^2 = x^3 +
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1 with p congruent to 11 modulo 12, and returns a point Q_id of order
q in E/F_p. This algorithm exploits the bijective mapping between the
x and y coordinates of non-zero points on such supersingular curves.
Algorithm 4.4.2 (HashToPoint1). Cryptographically hashes strings to
points on type-1 curves.
Input:
a string id,
a description of a subgroup of prime order q on a curve E/F_p : y^2
= x^3 + 1 where p is congruent to 11 modulo 12.
Output:
a point Q_id of order q on E(F_p).
Method:
1. n = q (compatibility mode) or p (preferred mode)
2. y = HashToRangen(n, id), using Algorithm 4.1.1 (HashToRange).
3. x = (y^2 . 1)^(1/3) = (y^2 . 1)^((2 * p . 1) / 3).
4. Let Q = (x, y), a non-zero point in E(F_p).
5. Q = [(p + 1) / q ]Q, a point of order q in E(F_p).
4.5. Bilinear 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^k) have a subgroup of order q,
two points A and B of order q in E(F_p) or E(F_p^k).
Output:
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on supersingular curves, the value of e(A, B) in F_p^k 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).
4.5.1. Type-1 curve implementation
Algorithm 4.5.2 (Pairing1): computes the modified Tate pairing on
type-1 curves.
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,
two points A and B of order q in E(F_p),
Output:
the value of e(A, B) = e(A, phi(B)) in F_p^k = 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 = 1^(1/3) in F_p^2 , with zeta != 1. Specifically, as
p is congruent to 3 modulo 4, and representing the elements of F_p^2
= F_p[x] / (x^2 + 1) as polynomials a + bx with x = (.1)^(1/2), the
representation of zeta = (a_zeta , b_zeta) is obtained as:
i. a_zeta = (p . 1) / 2.
ii. b_zeta = 3^((p + 1) / 4) (mod p).
(c) x = x * x_zeta in F_p^2 ,
(d) 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 4.5.1 (Tate) described in Section 4.5.
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4.6. Ratio of bilinear 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 4.5.2 (TateMillerSolinas), as well as the final
exponentiation in step 6(a) of Algorithm 4.5.2 (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^k) have a subgroup of order q,
four points A, B, C, and D, of order q in E(F_p) or E(F_p^k).
Output:
on supersingular curves, the value of e(A, B) / e(C, D) in F_p^k
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).
4.6.1. Type-1 curve implementation
Algorithm 4.6.2 (PairingRatio1). Computes the ratio of two modified
Tate pairings on type-1 curves.
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,
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four points A, B, C, and D, of order q in E(F_p),
Output:
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^k = 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, albeit 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 [3].
5.1. Setup
Algorithm 5.1.1 (BFsetup): randomly selects a master secret and the
associated public parameters.
Input:
a curve type t (currently required to be fixed to t = 1),
a security parameter n (currently required to take values n >=
1024).
Output:
a set of common public parameters,
a corresponding master secret.
Method:
1. Depending on the selected type t:
(a) If t = 1, then Algorithm 5.1.2 (BFsetup1) is executed.
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2. The resulting master secret and public parameters are separately
encoded as per the application protocol requirements.
5.1.1. 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 of
comparable security in the classical Diffie-Hellman or RSA public-key
cryptosystems. For this implementation, the allowed value of n is
limited to 1024, which corresponds to 80 bits of symmetric key
security.
Algorithm 5.1.2 (BFsetup1): randomly establishes a master secret and
public parameters for type-1 curves.
Input:
a security parameter n, assumed to be equal to 1024.
Output:
a set of common public parameters (t, p, q, P, Ppub),
a corresponding master secret s.
Method:
1. Determine the subordinate security parameters n_p and n_q as
follows:
(a) n_p = 512, which will determine the size of the field F_p.
(b) n_q = 160, which will determine the size of the subgroup order
q.
2. Construct the elliptic curve and its subgroup of interest, as
follows:
(a) Select an arbitrary n_q-bit prime q, i.e., such that
Ceiling(lg(q)) = n_q. For better performance, q is chosen as a
Solinas prime, i.e., a prime of the form q = 2^a +/- 2^b +/- 1 where
0 < b < a.
(b) Select a random integer r such that p = 12 * r * q . 1 is an
n_p-bit prime, i.e., such that Floor(lg(p)) = n_p.
3. Select a point P of order q in E(F_p), as follows:
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(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 a random integer s in the range 2 to q . 1.
(b) Let P_pub = [s]P.
5. (t, E, p, q, P, P_pub) are the common public parameters, where E:
y^2 = x^3 + 1.
6. s is the master secret.
5.2. Public key derivation
BFderivePubl takes an identity string id and a set of public
parameters, and returns a point Q_id.
Algorithm 5.2.1 (BFderivePubl): derives the public key corresponding
to an identity string.
Input:
an identity string id,
a set of common public parameters (t, E, p, q, P, P_pub).
Output:
a point Q_id of order q in E(F_p) or E(F_p^k).
Method:
1. Q_id = HashToPoint(E, p, q, id), using Algorithm 4.4.1
(HashToPoint).
5.3. Private key extraction
BFextractPriv takes an identity string id, and a set of public
parameters and corresponding master secret, and returns a point S_id.
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Algorithm 4.3.1 (BFextractPriv): extracts the private key
corresponding to an identity string.
Input:
an identity string id,
a set of common public parameters (t, E, p, q, P, P_pub).
Output:
a point S_id or order q in E(F_p).
Method:
1. Q_id HashToPoint(E, p, q, id) using Algorithm 4.4.1
(HashToPoint).
2. S_id = [s]Q_id.
5.4. Encryption
BFencrypt takes three inputs: a public parameter block, an identity
id, and a plaintext m. The plaintext is intended to be a symmetric
session key, although variable-sized short messages are allowed.
Algorithm 5.4.1 (BFencrypt): encrypts a short message or session key
for an identity string.
Input:
a plaintext string m of size |m| bytes,
a recipient identity string id,
a set of public parameters.
Output:
a ciphertext tuple (U, V, W) in E(F_p) x {0, ... , 255}^20 x {0,
... , 255}^|m|.
Method:
1. Let the public parameter set be comprised of a prime p, a curve
E/F_p, the order q of a large prime subgroup of E(F_p), and two
points P and P_pub of order q in E(F_p).
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2. Q_id = HashToPoint(E, p, q, id), using Algorithm 4.4.1
(HashToPoint), which results in a point of order q in E(F_p) or
E(F_p^k).
3. Select s random 160-bit vector rho, represented as 20-byte string
in big-endian convention.
4. t = SHA1(m), a 20-byte string resulting from the SHA1 algorithm.
5. l = HashToRangeq(rho || t), an integer in the range 0 to q . 1
resulting from applying Algorithm 4.1.1 (HashToRange) to the 40-byte
concatenation of rho and t.
6. U = [l]P, which is a point of order q in E(F_p).
7. Theta = Pairing(E, p, q, P_pub, Q_id), which is an element of the
extension field F_p^k obtained using the modified Tate pairing of
Algorithm 4.5.1 (Pairing).
8. Let theta = theta^l, which is theta raised to the power of l in
F_p^k .
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 = SHA1(z) using the SHA1 hashing algorithm, the result of
which is a 20-byte string.
11. Let V = w XOR rho, which is the 20-byte long bit-wise exclusive-
OR of w and rho.
12. Let W = HashStream(|m|, rho XOR m), which is the bit-wise
exclusive-OR of m with the first |m| bytes of the pseudo-random
stream produced by Algorithm 4.2.1 (HashStream) with seed rho.
13. The ciphertext is the triple (U, V, W).
5.5. Decryption
BFdecrypt takes three inputs: a public parameter block, a private key
block key, and a ciphertext parsed as (U , V , W).
Algorithm 5.5.1 (BFdecrypt): decrypts a short message or session key
using a private key.
Input:
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a private key point S_id of order q in E(F_p),
a ciphertext triple (U, V, W) in E(F_p) x {0, . . . , 255}^20 x
{0, . . . , 255}*.
a set of public parameters.
Output:
a decrypted plaintext m, or an invalid ciphertext flag.
Method:
1. Let the public parameter set be comprised of a prime p, a curve
E/F_p, the order q of a large prime subgroup of E(F_p), and two
points P and P_pub of order q in E(F_p).
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 = SHA1(z), using the SHA1 hashing algorithm, the result of
which is a 20-byte string.
5. Let rho = w XOR V, the bit-wise XOR of w and V.
6. Let m = HashStream(|W|, rho) XOR W, which is the bit-wise
exclusive-OR of m with the first |W| bytes of the pseudo-random
stream produced by Algorithm 4.2.1 (HashStream) with seed rho.
7. Let t = SHA1(m) using the SHA1 algorithm.
8. Let l = HashToRange(q, rho || t) using Algorithm 4.1.1
(HashToRange) on the 40-byte 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.
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6. Wrapper methods for the BF system
This chapter describes a number of wrapper methods providing the
identity-based cryptosystem functionalities using concrete encodings.
The following functions are presently given based on the Boneh-
Franklin algorithms.
6.1. Private key generator (PKG) setup
Algorithm 6.1.1 (BFwrapperPKGSetup): randomly selects a PKG master
secret and a set of public parameters.
Input:
a curve type t,
a security parameter n.
Output:
a common public parameter block pi,
a corresponding master secret block sigma.
Method:
1. Perform Algorithm 5.1.1 (BFsetup) on parameters t and n, producing
a public parameter set and a master secret.
2. Apply Algorithm 7.2.1 (BFencodeParams) on the public parameter set
obtained in step 1 to get the public parameter block pi.
3. Apply Algorithm 7.3.1 (BFencodeMaster) on the master secret
obtained in step 1 to get the master secret block sigma.
6.2. Private key extraction by the PKG
Algorithm 5.2.1 (BFwrapperPrivateKeyExtract): extraction by the PKG
of a private key corresponding to an identity.
Input:
a master secret block sigma,
a corresponding public parameter block pi,
an identity string id.
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Output:
a private key block kappa_id
Method:
1. Apply Algorithm 7.2.2 (BFdecodeParams) to the public parameter
block pi to obtain the public parameters, comprising a prime p, a
curve E/F_p, the order q of a large prime subgroup of E(F_p), and two
points P and P_pub of order q in E(F_p).
2. Apply Algorithm 7.3.2 (BFdecodeMaster) on the master secret block
sigma to obtain the master secret s.
3. Perform Algorithm 5.3.1 (BFextractPriv) on the identity id, using
the decoded parameters and secret, to produce a private key point
S_id.
4. Apply Algorithm 7.4.1 (BFencodePrivate) to S_id to produce a
private key block kid.
6.3. Session key encryption
Algorithm 5.3.1 (BFwrapperSessionKeyEncrypt): encrypts a short
message or session key for an identity.
Input:
a public parameter block pi,
a recipient identity string id,
a plaintext string m (possibly comprising the concatenation of a
pair of random session keys for symmetric encryption and message
authentication purposes on a larger plaintext).
Output:
a ciphertext block
Method:
1. Apply Algorithm 7.2.2 (BFdecodeParams) on the public parameter
block pi to obtain a set of public parameters, comprising a prime p,
a curve E/F_p, the order q of a large prime subgroup of E(F_p), and
two points P and P_pub of order q in E(F_p).
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2. Perform Algorithm 5.4.1 (BFencrypt) on the plaintext m for
identity id using the decoded set of parameters, to obtain a
ciphertext tuple (U, V, W).
3. Apply Algorithm 7.5.1 (BFencodeCiphertext) on (U, V, W) to obtain
a serialized ciphertext string
Algorithm 6.3.2 (BFwrapperSessionKeyDecrypt): decrypts a short
message or session key using a private key.
Input:
a public parameter block pi,
a private key block kappa,
a ciphertext block gamma.
Output:
a decrypted plaintext string m, or an error flag signaling an
invalid ciphertext.
Method:
1. Apply Algorithm 7.2.2 (BFdecodeParams) on the public parameter
block pi to obtain the public parameters, comprising a prime p, a
curve E/F_p, the order q of a large prime subgroup of E(F_p), and two
points P and P_pub of order q in E(F_p).
2. Apply Algorithm 7.4.2 (BFdecodePrivate) to kappa to obtain a
private key point S_id.
3. Apply Algorithm 7.5.2 (BFdecodeCiphertext) to gamma to obtain a
ciphertext triple (U, V, W).
4. Perform Algorithm 5.5.1 (BFdecrypt) on (U, V, W) using the
private key S_id and the decoded set of public parameters, to obtain
decrypted plaintext m, or an invalid ciphertext flag.
(a) If the decryption was successful, return the plaintext m.
(b) Otherwise, raise an error condition.
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7. Concrete encoding guidelines for BF
This section specifies a set of concrete encoding schemes for the
inputs and outputs of the previously described algorithms. ASN.1
encodings are specified in Section 11 of this document.
7.1. Encoding of points on a curve
Algorithm 7.1.1 (EncodePoint): encodes a point in E(F_p) in an
exportable format.
Input:
a non-zero point Q in E(F_p).
Output:
a fixed-length (for given p) byte-string encoding of Q.
Method:
1. Let (x, y) in F_p x F_p be the coordinates of P, where (x, y)
satisfy the equation of E.
2. The point P is then encoded as a FpPoint using the ASN.1 rules
given in the ASN.1 module given in Section 11 of this document.
Algorithm 6.1.2 (DecodePoint): decodes a point in E(F_p) from an
exportable format.
Input:
a byte-string encoding of a non-zero point Q in E(F_p).
Output:
Q = (x, y).
Method:
1. The string is parsed and decoded as a pair (x, y), where x and y
are integers in Z_p.
2. Q is reconstructed as (x, y).
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7.2. Public parameters blocks
Algorithm 7.2.1 (BFencodeParams): encodes a BF public parameter set
in an exportable format.
Input:
a set of public parameters (t, E, p, q, P, P_pub).
Output:
a public parameter block pi, represented as a byte string.
Method:
1. Separate encodings for E, p, q, P, P_pub are obtained as follows:
(a) If t = 1, execute Algorithm 7.2.3 (BFencodeParams1).
2. The separate encodings as well as a type indicator flag for t are
then serialized in any suitable manner as dictated by the
application.
Algorithm 7.2.2 (BFdecodeParams): imports a BF public parameter block
from a serialized format.
Input:
a public parameter block pi, represented as a byte string.
Output:
a set of public parameters (t, E, p, q, P, P_pub).
Method:
1. Identify from the appropriate flag the type t of curve upon which
the parameter block is based.
2. Then:
(a) If t = 1, execute Algorithm 7.2.4 (BFdecodeParams1).
7.2.1. Type-1 implementation
Algorithm 7.2.3 (BFencodeParams1): encodes a BF type-1 public
parameter set in an exportable format.
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Input:
a set of public parameters (t, E, p, q, P, P_pub) with t = 1.
Output:
separate encodings for each of the E, p, q, P, P_pub components.
Method:
1. E : y^2 = x^3 + a * x + b is represented as a constant string,
such as the empty string, since a and b are invariant for type-1
curves.
2. p = 12 * r * q . 1 is represented as the smaller integer r,
encoded, e.g., using a big-endian byte-string representation.
3. q = 2^a + s * 2^b + c, where a, b are small and c and s are either
1 or -1, is compactly represented as the 4-tuple (a, b, c, s).
4. P = (x_P , y_P ) in F_p x F_p is represented using the point
compression technique of Algorithm 7.1.1 (EncodePoint).
5. P_pub is similarly encoded using Algorithm 7.1.1 (EncodePoint).
Algorithm 7.2.4 (BFdecodeParams1): decodes the components of a BF
type-1 public parameter block.
Input:
separate encodings for each one of E, p, q, P, P_pub.
Output:
a set of public parameters (t, E, p, q, P, P_pub) with t = 1.
Method:
1. The equation of E is set to E = E : y^2 = x^3 + 1, as is always
the case for type-1 curves. The actual encoding of E is ignored.
2. The encoding of q is parsed as (a, b, c, s), and its value set to
q = 2^a + s * 2^b + c.
3. The encoding of p is parsed as the integer r, from which p is
given by p = 12 * r * q . 1.
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4. P is reconstructed from its encoding (x, y) using the point
decompression technique of Algorithm 7.1.2 (DecodePoint).
5. P_pub is similarly reconstructed from its encoding using Algorithm
7.1.2 (DecodePoint).
7.3. Master secret blocks
Algorithm 6.3.1 (BFencodeMaster): encodes a BF master secret in an
exportable format.
Input:
a master secret integer s between 2 and q - 1.
Output:
a master secret block sigma, represented as a byte string.
Method:
1. Sigma is constructed as the unsigned big-endian byte-string
encoding of s of length 8 * Ceiling(lg(p)).
Algorithm 7.3.2 (BFdecodeMaster): decodes a BF master secret from a
block in exportable format.
Input:
a master secret block sigma, represented as a byte string.
Output:
a master secret integer s in between 2 and q - 1 .
Method:
1. s = Value(sigma), where sigma is interpreted in the unsigned big
endian convention.
7.4. Private key blocks
Algorithm 6.4.1 (BFencodePrivate): encodes a BF private key point in
an exportable format.
Input:
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a private key point S_id in E(F_p).
Output:
a private key block kappa, represented as a byte string.
Method:
1. kappa is obtained by applying Algorithm 7.1.1 (EncodePoint) to
S_id.
Algorithm 7.4.2 (BFdecodePrivate): decodes a BF private key point
from an exportable format.
Input:
a private key block kappa, represented as a byte string.
Output:
a private key point S_id in E(F_p).
Method:
1. Kappa is parsed and decoded into a point S_id in E(F_p) using
Algorithm 7.1.2 (DecodePoint).
7.5. Ciphertext blocks
Algorithm 7.5.1 (BFencodeCiphertext): encodes a BF ciphertext tuple
in an exportable format.
Input:
a ciphertext tuple (U, V, W) in E(F_p) x {0, . . . , 255}^20 x {0,
. . . , 255}*.
Output:
a ciphertext block gamma, represented as a byte string.
Method:
1. U = (x, y) is first encoded as a fixed-length string using
Algorithm 7.1.1 (EncodePoint).
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2. Gamma is obtained as the encoding of U, concatenated with the
fixed-length string V, and the variable length string W, both already
in byte-string format.
Algorithm 7.5.2 (BFdecodeCiphertext): decodes a BF ciphertext tuple
from an exportable format.
Input:
a ciphertext block gamma, represented as a byte string.
Output:
a ciphertext tuple (U, V, W) in E(F_p) x {0, . . . , 255}^20 x {0,
. . . , 255}*.
Method:
1. Gamma is parsed as a 3-tuple comprising a fixed-length encoding
of U, followed by a 20-byte string V, followed by an arbitrary-length
string W.
2. U in E(F_p) is then recovered by applying Algorithm 7.1.2
(DecodePoint) on its encoding.
8. The Boneh-Boyen BB1 cryptosystem
This chapter describes the algorithms constituting the first of the
two Boneh-Boyen identity-based cryptosystems proposed in [2]. The
description follows the practical implementation given in [2].
8.1. Setup
Algorithm 8.1.1 (BBsetup). Randomly selects a set of master secrets
and the associated public parameters.
Input:
a curve type t (currently required to be fixed to t = 1),
a security parameter n (currently required to take values n >=
1024).
Output:
a set of common public parameters,
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a corresponding master secret.
Method:
1. Depending on the selected type t:
(a) If t = 1, then Algorithm 8.1.2 (BBsetup1) is executed.
2. The resulting master secret and public parameters are separately
encoded as per the application protocol requirements.
8.1.1. Type-1 curve implementation
BBsetup1 takes a security parameter n as input. For type-1 curves,
the scale of n corresponds to the modulus bit-size believed of
comparable security in the classical Diffie-Hellman or RSA public-key
cryptosystems. For this implementation, allowed values of n are
limited to 1024, 2048, and 3072, which correspond to the equivalent
security level ranging from 80-, 112- and 128-bit symmetric keys
respectively.
Algorithm 7.1.2 (BBsetup1): randomly establishes a master secret and
public parameters for type-1 curves.
Input:
a security parameter n, either 1024, 2048 or 3072.
Output:
a set of common public parameters (t, k, E, p, q, P, P_1, P_2, P_3,
v),
a corresponding triple of master secrets (alpha, beta, gamma).
Method:
1. Determine the subordinate security parameters n_p and n_q as
follows:
(a) n_p = n / 2, which will determine the size of the field F_p.
(b) if n = 1024, n_q = 160; if n = 2048, n_q = 224; if n = 3072,
n_q = 256, which will determine the size of the subgroup order q.
2. Construct the elliptic curve and its subgroup of interest, as
follows:
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(a) Select an arbitrary n_q-bit prime q, i.e., such that
Ceiling(lg(p)) = n_q. For better performance, q is chosen as a
Solinas prime, i.e., a prime of the form q = 2^a +/- 2^b +/- 1 where
0 < b < a.
(b) Select a random integer r such that p = 12 * r * q . 1 is an
n_p-bit prime, i.e., such that Ceiling(lg(p)) = n_p.
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 : y2 = x3 + 1 (mod p).
(b) Let P = [12 * r]P.
(c) If P = 1, 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_p2 obtained using the modified Tate pairing of
Algorithm 3.5.1 (Pairing).
5. (t, k, E, p, q, P, P_1, P_2, P_3, v) are the common public
parameters, where t = 1, k = 2, and E : y^2 = x^3 + 1.
6. (alpha, beta, gamma) constitute the master secret.
8.2. Public key derivation
BBderivePubl takes an identity string id and a set of public
parameters, and returns an integer h_id.
Algorithm 7.2.1 (BBderivePubl): derives the public key corresponding
to an identity string.
Input:
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an identity string id,
a set of common public parameters (t, k, E, p, q, P, P_1, P_2, P_3,
v).
Output:
an integer h_id modulo q.
Method:
1. Let h_id HashToRangeq(id), using Algorithm 3.1.1 (HashToRange).
8.3. Private key extraction
BBextractPriv takes an identity string id, and a set of public
parameters and corresponding master secrets, and returns a private
key consisting of two points D_0 and D_1.
Algorithm 8.3.1 (BBextractPriv): extracts the private key
corresponding to an identity string.
Input:
an identity string id,
a set of common public parameters (t, k, E, p, q, P, P_1, P_2, P_3,
v).
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.
2. Calculate the point D_0 as follows:
(a) Let hid = HashToRange(q, id), using Algorithm 3.1.1
(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:
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(a) Let D_1 = [r]P.
4. The pair of points (D_0,D_1) constitutes the private key for id.
8.4. Encryption
BBencrypt takes three inputs: a set of public parameters, an identity
id, and a plaintext m. The plaintext is intended to be a short random
session key, although messages of arbitrary size are in principle
allowed.
Algorithm 7.4.1 (BBencrypt): encrypts a short message or session key
for an identity string.
Input:
a plaintext string m of size |m| bytes,
a recipient identity string id,
a set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
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. Let the public parameter set be comprised of a prime p, a curve
E/F_p, the order q of a large prime subgroup of E(F_p), four points
P, P_1, P_2, P_3, of order q in E(F_p), and an extension field
element v of order q in F_p2 .
2. Select a random integer s in the range 1 to q . 1.
3. 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 .
4. Calculate the point C_0 as follows:
(a) Let C_0 = [s]P.
5. Calculate the point C_1 as follows:
(a) Let _hid = HashToRangeq(id), using Algorithm 3.1.1
(HashToRange).
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(b) Let y = s * h_id in F_q.
(c) Let C_1 = [y]P_1 + [s]P_3.
6. Obtain canonical string representations of certain elements:
(a) psi = Canonical(p, k, 1, w) using Algorithm 3.3.1 (Canonical),
the result of which is a canonical byte-string representation of w.
(b) Let l = Ceiling(8 * lg(p)), the number of bytes needed to
represent integers in F_p, and represent each of these F_p elements
as a big-endian zero-padded byte-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.
7. Encrypt the message m into the string y as follows:
(a) Compute an encryption key h_0 as a dual-pass hash of w via its
representation psi:
i. Let zeta = SHA1(psi), using the SHA1 hashing algorithm; the
result is a 20-byte string.
ii. Let xi = SHA1(zeta || psi), using the SHA1 hashing
algorithm; the result is a 20-byte string.
iii. Let h = xi || zeta, the 40-byte concatenation of the
previous two SHA1 outputs.
(b) Let y = HashStream(|m|, h) XOR m, which is the bit-wise
exclusive-OR of m with the first |m| bytes of the pseudo-random
stream produced by Algorithm 3.2.1 (HashStream) with seed h.
8. 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.
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ii. Let eta = SHA1(sigma), using the SHA1 hashing algorithm to
get a 20-byte string.
iii. Let mu = SHA1(eta || sigma), using the SHA1 hashing
algorithm to get a 20-byte string.
iv. Let h = mu || eta, the 40-byte concatenation of the
previous two SHA1 outputs.
(b) Build the tag u as the encryption of the integer s with the
one-time pad h:
i. Let rho = HashToRangeq(h) to get an integer in Z_q.
ii. Let u = s + rho (mod q).
9. The complete ciphertext is given by the quadruple (u, C_0, C_1,
y).
8.5. Decryption
BBdecrypt takes three inputs: a set of public parameters, a private
key (D_0, D_1), and a ciphertext parsed as (u, C_0, C_1, y). It
outputs a message m, or signals an error if the ciphertext is invalid
for the given key.
Algorithm 7.5.1 (BBdecrypt): decrypts a short message or session key
using 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}*.
a set of public parameters.
Output:
a decrypted plaintext m, or an invalid ciphertext flag.
Method:
1. Let the public parameter set be comprised of a prime p, a curve
E/F_p, the order q of a large prime subgroup of E(F_p), four points
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P, P_1, P_2, P_3, of order q in E(F_p), and an extension field
element v of order q in F_p^2 .
2. 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).
3. Obtain canonical string representations of certain elements:
(a) psi = Canonical(p, k, 1, w), using Algorithm 4.3.1
(Canonical); the result is a canonical byte-string representation of
w.
(b) Let l = Ceiling(8 * lg(p)), the number of bytes needed to
represent integers in F_p, and represent each of these F_p elements
as a big-endian zero-padded byte-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.
4. 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 = SHA1(psi), using the SHA1 hashing algorithm to
get a 20-byte string.
ii. Let xi = SHA1(zeta || psi), using the SHA1 hashing
algorithm to get a 20-byte string.
iii. Let h = xi || zeta, the 40-byte concatenation of the
previous two SHA1 outputs.
(b) Let m = HashStream(|y|, h)_XOR y, which is the bit-wise
exclusive-OR of y with the first |y| bytes of the pseudo-random
stream produced by Algorithm 3.2.1 (HashStream) with seed h.
5. 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):
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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 = SHA1(sigma) using the SHA1 hashing algorithm to
get a 20-byte string.
iii. Let mu = SHA1(eta || sigma), using the SHA1 hashing
algorithm to get a 20-byte string.
iv. Let h = mu || eta, the 40-byte concatenation of the
previous two SHA1 outputs.
(b) Unblind the encryption randomization integer s from the tag u
using h:
i. Let rho = HashToRangeq(h) to get an integer in Z_q.
ii. Let s = u - rho (mod q).
6. Verify the ciphertext consistency according to the decrypted
values:
(a) Test whether the equality w = v^s holds in F_p2 .
(b) Test whether the equality C_0 = [s]P holds in E(F_p).
7. Adjudication and final output:
(a) If either of the tests performed in step 6 fails, the
ciphertext is rejected, and no decryption is output.
(b) Otherwise, i.e., when both tests performed in step 6 succeed,
the decrypted message is output.
9. Wrapper methods for the BB1 system
This section describes a number of wrapper methods providing the
identity-based cryptosystem functionalities using concrete encodings.
The following functions are presently given based on the Boneh-
Franklin algorithms.
9.1. Private key generator (PKG) setup
Algorithm 9.1.1 (BBwrapperPKGSetup): randomly selects a PKG master
secret and a set of public parameters.
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Input:
a curve type t,
a security parameter n.
Output:
a common public parameter block pi,
a corresponding master secret block sigma.
Method:
1. Perform Algorithm 8.1.1 (BBsetup) on parameters t and n, producing
a set of public parameters and master secret.
2. Apply Algorithm 10.2.1 (BBencodeParams) on the public parameters
obtained in step 1 to get the public parameter block pi.
3. Apply Algorithm 10.3.1 (BBencodeMaster) on the master secrets
obtained in step 1 to get the master secret block sigma.
9.2. Private key extraction by the PKG
Algorithm 9.2.1 (BBwrapperPrivateKeyExtract): extraction by the PKG
of a private key corresponding to an identity.
Input:
a master secret block sigma,
a corresponding public parameter block pi,
an identity string id.
Output:
a private key block kappa_id.
Method:
1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
block pi to obtain the public parameters, comprising a prime p, the
parameters of a curve E/F_p with some embedding degree k, the order q
of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
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order q in E(F_p), and an element v of order q in the extension field
F_p^k of degree k.
2. Apply Algorithm 10.3.2 (BBdecodeMaster) on the master secret block
sigma to obtain the master secret (alpha, beta, gamma).
3. Perform Algorithm 8.3.1 (BBextractPriv) on the identity id, using
the decoded public parameters and master secret, to produce a private
key (D_0, D_1).
4. Apply Algorithm 10.4.1 (BBencodePrivate) on the private key to
produce a private key block kappa_id.
9.3. Session key encryption
Algorithm 9.3.1 (BBwrapperSessionKeyEncrypt): encrypts a short
message or session key for an identity.
Input:
a public parameter block pi,
a recipient identity string id,
a plaintext string m (possibly comprising the concatenation of a
pair of random session keys for symmetric encryption and message
authentication purposes on a larger plaintext).
Output:
a ciphertext block omega.
Method:
1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
block pi to obtain the public parameters, comprising a prime p, the
parameters of a curve E/F_p with some embedding degree k, the order q
of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
order q in E(F_p), and an element v of order q in the extension field
F_p^k .
2. Perform Algorithm 8.4.1 (BBencrypt) on the plaintext m for
identity id using the decoded set of parameters, to obtain a
ciphertext quadruple (u, C_0, C_1, y).
3. Apply Algorithm 10.5.1 (BBencodeCiphertext) on the ciphertext (u,
C_0, C_1, y) to obtain a string representation of omega.
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Algorithm 9.3.2 (BBwrapperSessionKeyDecrypt): decrypts a short
message or session key using a private key.
Input:
a public parameter block pi,
a private key block kappa,
a ciphertext block omega.
Output:
a decrypted plaintext string m, or an error flag signaling an
invalid ciphertext.
Method:
1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
block pi to obtain the public parameters, comprising a prime p, the
parameters of a curve E/F_p with some embedding degree k, the order q
of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
order q in E(F_p), and an element v of order q in the extension field
F_p^k.
2. Apply Algorithm 10.4.2 (BBdecodePrivate) on kappa to obtain the
private key points (D_0, D_1).
3. Apply Algorithm 10.5.2 (BBdecodeCiphertext) on omega to obtain a
ciphertext quadruple (u, C_0, C_1, y).
4. Perform Algorithm 8.5.1 (BBdecrypt) on (u, C_0, C_1, y) using the
private key (D_0, D_1) and the decoded set of public parameters, to
obtain decrypted plaintext m, or an invalid ciphertext flag.
(a) If the decryption was successful, return the plaintext string
m.
(b) Otherwise, raise an error condition.
10. Concrete encoding guidelines for BB1
This section specifies a set of concrete encoding schemes for the
inputs and outputs of the previously described algorithms. ASN.1
encodings are specified in Section 11 of this document.
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10.1. Encoding of points on a curve
We refer to the description of Algorithm 7.1.1 (EncodePoint) and
Algorithm 7.1.2 (DecodePoint).
10.2. Public parameters blocks
Algorithm 10.2.1 (BBencodeParams): encodes a BB1 public parameter set
in an exportable format.
Input:
a set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
Output:
a public parameter block pi, represented as a byte string.
Method:
1. Separate encodings for k, E, p, q, P, P_1, P_2, P_3 are obtained
as follows:
(a) If t = 1, execute Algorithm 10.2.3 (BBencodeParams1).
2. The separate encodings as well as a type indicator flag for t are
then serialized in any suitable manner as dictated by the
application.
Algorithm 10.2.2 (BBdecodeParams): imports a BB1 public parameter
block from a serialized format.
Input:
a public parameter block pi, represented as a byte string.
Output:
a set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
Method:
1. Identify from the appropriate flag the type t of curve upon which
the parameter block is based.
2. Then:
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(a) If t = 1, execute Algorithm 10.2.4 (BBdecodeParams1).
10.2.1. Type-1 implementation
Algorithm 10.2.3 (BBencodeParams1): encodes a BB1 type-1 public
parameter set in an exportable format.
Input:
a set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v)
with t = 1.
Output:
separate encodings for each of the k, E, p, q, P, P_1, P_2, P_3
components (v is redundant and omitted).
Method:
1. E : y^2 = x^3 + a * x + b and k = 2 are represented as a constant
string, such as the empty string, since the coefficients a and b and
the embedding degree k are invariant for type-1 curves.
2. p = 12 * r * q . 1 is represented as the smaller integer r,
encoded, e.g., using a big-endian byte-string representation.
3. q = 2^a + s* 2^b + c, where a, b are small and both c and s are
either 1 or -1 is compactly represented as the 4-tuple (a, b, c, s).
4. P = (x_P , y_P ) in F_p x F_p is represented using the point
compression technique of Algorithm 7.1.1 (EncodePoint).
5. Each of P_1, P_2, and P_3 is similarly encoded using Algorithm
7.1.1 (EncodePoint).
Algorithm 10.2.4 (BBdecodeParams1): decodes the components of a BB1
type-1 public parameter block.
Input:
separate encodings for each one of k, E, p, q, P, P_1, P_2, P_3.
Output:
a set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v)
with t = 1.
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Method:
1. The equation of E is set to E E : y^2 = x^3 + 1, as is always
the case for type-1 curves.
2. The embedding degree is set to k = 2 for type-1 curves.
3. The encoding of q is parsed as (a, b, c, s), and its value set to
q = 2^a + s * 2^b + c.
4. The encoding of p is parsed as the integer r, from which p is
given by p = 12 * r * q . 1.
5. P is reconstructed from its encoding (x, y) using the point
decompression technique of Algorithm 7.1.2 (DecodePoint).
6. Each of P_1, P_2, and P_3 is reconstructed in a similar manner
from its encoding using Algorithm 7.1.2 (DecodePoint).
7. The extension field element v is reconstructed as v = Pairing(E,
p, q, P_1, P_2) using Algorithm 4.5.1 (Pairing).
10.3. Master secret blocks
Algorithm 10.3.1 (BBencodeMaster): encodes a BB1 master secret in an
exportable format.
Input:
a master secret triple of integers (alpha, beta, gamma) in (Z+_q
)^3.
Output:
a master secret block sigma, represented as a byte string.
Method:
1. Encode each integer as an unsigned big-endian byte-string of fixed
length Ceiling(8 * lg(q)), or, when q is a Solinas prime q = 2^a +/-
2^b +/- 1, of length Ceiling((a + 1) / 8):
(a) sigma_alpha to represent alpha.
(b) sigma_beta to represent beta.
(c) sigma_gamma to represent gamma.
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2. Sigma = sigma_alpha || sigma_beta || sigma_gamma is the
concatenation of these strings.
Algorithm 10.3.2 (BBdecodeMaster): decodes a BB1 master secret from a
block in exportable format.
Input:
a master secret block sigma, represented as a byte string.
Output:
a master secret triple of integers (alpha, beta, gamma) in (Z+_q
)^3.
Method:
1. Parse sigma as sigma_alpha || sigma_beta || sigma_gamma, where
each substring is a byte string of fixed length Ceiling(8 * lg(q)),
or, when q is a Solinas prime q = 2^a +/- 2^b +/- 1, of length
Ceiling((a + 1) / 8)).
2. Decode each substring as an integer in unsigned big-endian byte-
string representation:
(a) alpha = Value(sigma_alpha).
(b) beta = Value(sigma_beta).
(c) gamma = Value(sigma_gamma).
10.4. Private key blocks
Algorithm 10.4.1 (BBencodePrivate): encodes a BB1 private key in an
exportable format.
Input:
a private key pair of points (D_0, D_1) in E(F_p) x E(F_p).
Output:
a private key block kappa, represented as a byte string.
Method:
1. Encode each point separately:
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(a) kappa_0 is obtained by applying Algorithm 7.1.1 (EncodePoint)
to D_0.
(b) kappa_1 is obtained by applying Algorithm 7.1.1 (EncodePoint)
to D_0.
2. Kappa = kappa_0 || kappa_1.
Algorithm 10.4.2 (BBdecodePrivate): decodes a BB1 private key from an
exportable format.
Input:
a private key block kappa, represented as a byte string.
Output:
a private key pair of point (D_0, D_1) in E(F_p) x E(F_p).
Method:
1. Decode each point separately:
(a) The first prefix of kappa is parsed and decoded into a point
D_0 in E(F_p) using Algorithm 7.1.2 (DecodePoint).
(b) The remainder of kappa is parsed and decoded into a point D_1
in E(F_p) using Algorithm 7.1.2 (DecodePoint).
10.5. Ciphertext blocks
Algorithm 10.5.1 (BBencodeCiphertext). Encodes a BB1 ciphertext tuple
in an exportable format.
Input:
a ciphertext tuple (u, C_0, C_1, y) in Z_q x E(F_p) x E(F_p) x {0,
. . . , 255}*.
Output:
a ciphertext block omega, represented as a byte string.
Method:
1. Let chi_0 be the fixed-length encoding of C_0 = (x_0, y_0) using
Algorithm 7.1.1 (EncodePoint).
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2. Let chi_1 be the fixed-length encoding of C_1 = (x_1, y_1) using
Algorithm 7.1.1 (EncodePoint).
3. Let nu be the encoding of u as an unsigned big-endian byte-string
of fixed length Ceiling(8 * lg(q)), or, when q is a Solinas prime q =
2^a +/- 2^b +/- 1, of length Ceiling((a + 1)/8).
4. Omega = chi_0 || chi_1 || nu || y is the concatenation of these
three strings and y.
Algorithm 10.5.2 (BBdecodeCiphertext): decodes a BB1 ciphertext tuple
from an exportable format.
Input:
a ciphertext block omega, represented as a byte string.
Output:
a ciphertext tuple (u, C_0 ,C_1, y) in Z_q x E(F_p) x E(F_p) x {0,
. . . , 255}*.
Method:
1. Omega is parsed as a quadruple comprising a fixed-length encoding
of C_0, a fixed-length encoding of C_1, a fixed-length encoding of u,
and the arbitrary-length string y:
(a) C_0 in E(F_p) is first recovered by applying Algorithm 7.1.2
(DecodePoint) on the first parsed component of omega.
(b) C_1 in E(F_p) is next recovered by applying Algorithm 7.1.2
(DecodePoint) on the second parsed component of omega.
(c) u in Z_q is then recovered from its unsigned big-endian byte-
string representation in the third parsed component of omega, of
length Ceiling(8 * lg(q)), or, when q is a Solinas prime q = 2^a +/-
2b +/- 1, of length Ceiling((a + 1)/8).
(d) y is finally taken as the remainder of omega.
11. ASN.1 module
This section defines the ASN.1 module for the encodings discussed in
sections 7 and 10.
IBCS { joint-iso-itu(2) country(16) us(840) organization(1)
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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 of the IBCS standard only supports IBE over
-- type-1 curves. In the current version, the Curve type is
-- always set to NULL, although future versions will use it.
--
IMPORTS Curve
FROM X9-62-module
{ iso(1) member-body(2) us(840) ansi-x9-62(10045) module(5) 1
};
ibcs OBJECT IDENTIFIER ::= {
joint-iso-itu(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)
}
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--
-- Supporting types
--
--
-- Encoding of a point on an elliptic curve E/Fp.
--
FpPoint ::= SEQUENCE {
x INTEGER,
y INTEGER
}
--
-- Encoding of a Solinas prime.
--
-- Encodes a Solinas prime of the form
-- q = 2^a + s * 2^b +c with the integers a, b, c, and s.
--
SolinasPrime ::= SEQUENCE {
a INTEGER,
b INTEGER,
c INTEGER { positive(1), negative(-1) },
s INTEGER { positive(1), negative(-1) }
}
--
-- Algorithms
--
ibe-algorithms OBJECT IDENTIFIER ::= {
ibcs1 ibe-algorithms(2)
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}
---
--- Boneh-Franklin IBE
---
bf OBJECT IDENTIFIER ::= { ibe-algorithms bf(1) }
--
-- Encoding of a BF public parameters block.
-- The only version currently supported is version 1.
-- For type-1 curves, the curve is fixed, so Curve is set to NULL
-- For the BF prime p and subprime q, we have q * r = p + 1,
-- and we encode the values of r and q in the public parameters.
-- The points P and P_pub are encoded as pointP and pointPpub
respectively.
--
BFPublicParamaters ::= SEQUENCE {
version INTEGER { v1(1) },
curve Curve { NULL },
r INTEGER,
q SolinasPrime,
pointP FpPoint,
pointPpub FpPoint
}
--
-- A BF private key is a point on an elliptic curve,
-- which is an FpPoint.
--
BFPrivateKeyBlock ::= FpPoint
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--
-- A BF master secret is an integer.
--
BFMasterSecret ::= INTEGER
--
-- BF ciphertext block
--
BFCiphertextBlock ::= SEQUENCE {
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 1.
-- The embedding degree is currently fixed to 2.
-- For type-1 curves, curve is set to NULL.
-- For the BB1 prime p and subprime q, we have q * r = p + 1,
-- and we encode the values of r and q in the public parameters.
--
BB1PublicParameters ::= SEQUENCE {
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Version INTEGER { v1(1) },
embedding-degree INTEGER { degree-2(2) },
curve Curve { NULL },
r INTEGER,
q SolinasPrime,
pointP FpPoint,
pointP1 FpPoint,
pointP2 FpPoint,
pointP3 FpPoint
}
--
-- BB1 master secret block
--
BB1MasterSecret ::= SEQUENCE {
alpha INTEGER,
beta INTEGER,
gamma INTEGER
}
--
-- BB1 private Key block
--
BB1PrivateKeyBlock ::= SEQUENCE {
pointD0 FpPoint,
pointD1 FpPoint
}
--
-- BB1 ciphertext block
--
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BB1CiphertextBlock ::= SEQUENCE {
pointChi0 FpPoint,
pointChi1 FpPoint,
nu INTEGER,
y OCTET STRING
}
END
12. Security considerations
This entire document discusses security considerations.
13. IANA considerations
All of the OIDs used in this document were assigned by the National
Institute of Standards and Technology (NIST), so no further action by
the IANA is necessary for this document.
14. Acknowledgments
This document is based on the IBCS #1 v2 document of Voltage
Security, Inc. Any substantial use of material from this document
should acknowledge Voltage Security, Inc. as the source of the
information.
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15. References
15.1. Informative references
[1] I. Blake, G. Seroussi, N. Smart, Elliptic Curves in
Cryptography, Cambridge University Press, 1999.
[2] D. Boneh, X. Boyen, Efficient selective-ID secure identity
based encryption without random oracles, In Proc. of EUROCRYPT
04, LNCS 3027, pp. 223238, 2004.
[3] D. Boneh, M. Franklin, Identity-based encryption from the Weil
pairing, In Proc. of CRYPTO 01, LNCS 2139, pp. 213229, 2001.
Authors Addresses
Xavier Boyen
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304
Email: xavier@voltage.com
Luther Martin
Voltage Security
1070 Arastradero Rd Suite 100
Palo Alto, CA 94304
Email: martin@voltage.com
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