Symbol

Description

P = (q, p, q', p')

Parameter specification for an instantiation of PP-IBE

r

The order of the r-torsion group G.

q

Prime integer specifying the number of elements of G₁ and G_T. The security of the scheme in PP-IBE depends on q.

In our construction, q = r, because G₁ is chosen to be the torsion group of order r

$\mathcal{O}$

The point at infinity of an elliptic curve

p

Prime integer. Base of the prime field F_p from which points in E(F_p) are drawn

p'

Prime number, the base of $Z_{p^{\prime }}$ the prime field for digital credentials. p' is selected using q' such that q' | (p' − 1)

$Z_{p^{\prime }}$

The prime field within which calculations for digital credentials and their signatures are performed.

$Z_{q^{\prime }}$

Prime field, source of the attributes and exponents for digital credentials

#E(F_p)

The order of the base elliptic curve. E is specified in this document such that q | o.

$E\left(F_p\right)\left[q\right]$

The q-torsion group of curve E. In our construction $G_1=E\left(F_p\right)\left[q\right]$ .

k

Embedding degree k is the smallest positive integer k such that #G₁ | #E_k + 1. In our implementation the Tate pairing requires k as an argument.

F_p

Prime field upon which the base elliptic curve E(F_p) is drawn. F_p is the set of integers [0, p-1].

E(F_p)

The base elliptic curve. In this paper,

$E\left(F_p\right):y^2=x^3+1$ , $x,y\in F_p$

G₁

The set of points, a subset of E(F_p), that forms the domain for the bilinear pairing function. Our construction presents a symmetric pairing in which G₁ is a q-torsion subgroup of E(F_p). Group G₁ is the source of identity points for the protocol; as such, its size is a privacy and security parameter.

$E\left(F_{p^2}\right)\left[q\right]$

The extension of G₁ into $F_{p^2}$ . For a point $p_1\in E\left(F_p\right)\left[q\right]$ , $\phi \left(p_1\right)\in E\left(F_{p^2}\right)\left[q\right]$

$p_2=\phi \left(p_1\right)$

Distortion map which transforms a point from the torsion group ${\mathbb{G}}_1$ in the base curve E(F_p) to a point in the torsion group Gx in the elliptic curve on the extension field $E\left(F_{p^2}\right)$ . In this paper we use $\phi \left(x,y\right)=\left(\zeta x,y\right)$

$E\left(F_{p^2}\right)$

The elliptic curve on the extension field. The set of pairs drawn from elements of the polynomial ring which satisfy the elliptic curve characteristic equation. In this paper, #Math_274# x,y in $F_{p^2}$ , s.t. $y^2=x^3+1$ .

$F_{p^2}$

Field extension of F_p, elements of which have the form $c_{1}x+c_0$ with coefficients $c_1,c_0$ ϵ F_p. In this paper $G_T\subset F_{p^2}$ and $E\left(F_{p^2}\right)\subset F_{p^2}\times F_{p^2}$ .

$\widehat{e}:G_1\times G_1\to G_T$

Custom pairing function $\widehat{e}$ allowing two points from G₁ (possibly dependent) to be paired in a non-degenerate manner to a point in G_T. Our construction implements $\widehat{e}$ in terms of e and φ

$f:G_1\times G_1\to G_T$

A well-known pairing function, such as the Weil or the Tate pairing.

G

Source Group for the pairing. In our construction the source group G is the r-torsion, a subset of the points in Ep

G_t

Target Group. In our construction, the target group is $F_{p^2}$ , the extension field

$Z_{p^{\prime }}$

The prime field of integers modulo p'. used for calculation of digital credentials.

$Z_{p^{\prime }}^{*}$

The integers in $Z_{p^{\prime }}$ relatively prime to p'

$Z_{q^{\prime }}$

The prime field of integers mod q'. The values used for attributes and exponents within the digital credential are in $Z_{q^{\prime }}$

r

The order of the r-torsion group G.

q

Prime integer specifying the number of elements of G₁ and G_T. The security of the scheme in PP-IBE depends on q.

In our construction, q = r, because G₁ is chosen to be the torsion group of order r

$\mathcal{O}$

The point at infinity of an elliptic curve

p

Prime integer. Base of the prime field F_p from which points in E(F_p) are drawn

p'

Prime number, the base of $Z_{p^{\prime }}$ the prime field for digital credentials. p' is selected using q' such that q' | (p' − 1)

$Z_{p^{\prime }}$

The prime field within which calculations for digital credentials and their signatures are performed.

$Z_{q^{\prime }}$

Prime field, source of the attributes and exponents for digital credentials

#E(F_p)

The order of the base elliptic curve. E is specified in this document such that q | o.

$E\left(F_p\right)\left[q\right]$

The q-torsion group of curve E. In our construction #Math_297#.

k

Embedding degree k is the smallest positive integer k such that #G₁ | #E_k + 1. In our implementation the Tate pairing requires k as an argument.

F_p

Prime field upon which the base elliptic curve E(F_p) is drawn. F_p is the set of integers [0, p − 1].

E(F_p)

The base elliptic curve. In this paper,

$E\left(F_p\right):y^2=x^3+1$ , $x,y\in F_p$

G₁

The set of points, a subset of E(F_p), that forms the domain for the bilinear pairing function. Our construction presents a symmetric pairing in which G₁ is a q-torsion subgroup of E(F_p). Group G₁ is the source of identity points for the protocol; as such, its size is a privacy and security parameter.

$E\left(F_{p^2}\right)\left[q\right]$

The extension of G₁ into $F_{p^2}$ . For a point $p_1\in E\left(F_p\right)\left[q\right]$ , $\phi \left(p_1\right)\in E\left(F_{p^2}\right)\left[q\right]$

$p_2=\phi \left(p_1\right)$

Distortion map which transforms a point from the torsion group ${\mathbb{G}}_1$ in the base curve E(F_p) to a point in the torsion group Gx in the elliptic curve on the extension field $E\left(F_{p^2}\right)$ . In this paper we use $\phi \left(x,y\right)=\left(\zeta x,y\right)$

G

Source Group for the pairing. In our construction the source group G is the r-torsion, a subset of the points in 𝐸𝑝

$Z_{p^{\prime }}$

The prime field of integers modulo p'. used for calculation of digital credentials.

$Z_{p^{\prime }}^{*}$

The integers in $Z_{p^{\prime }}$ relatively prime to p'

$Z_{q^{\prime }}$

The prime field of integers mod q'. The values used for attributes and exponents within the digital credential are in $Z_{q^{\prime }}$