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 GT. 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 |
| The point at infinity of an elliptic curve |
p | Prime integer. Base of the prime field Fp from which points in E(Fp) are drawn |
p' | Prime number, the base of the prime field for digital credentials. p' is selected using q' such that q' | (p' − 1) |
| The prime field within which calculations for digital credentials and their signatures are performed. |
| Prime field, source of the attributes and exponents for digital credentials |
#E(Fp) | The order of the base elliptic curve. E is specified in this document such that q | o. |
| The q-torsion group of curve E. In our construction . |
k | Embedding degree k is the smallest positive integer k such that #G1 | #Ek + 1. In our implementation the Tate pairing requires k as an argument. |
Fp | Prime field upon which the base elliptic curve E(Fp) is drawn. Fp is the set of integers [0, p-1]. |
E(Fp) | The base elliptic curve. In this paper, , |
G1 | The set of points, a subset of E(Fp), 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(Fp). Group G1 is the source of identity points for the protocol; as such, its size is a privacy and security parameter. |
| The extension of G1 into . For a point , |
| Distortion map which transforms a point from the torsion group in the base curve E(Fp) to a point in the torsion group Gx in the elliptic curve on the extension field . In this paper we use |
| 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 , s.t. . |
| Field extension of Fp, elements of which have the form with coefficients ϵ Fp. In this paper and . |
| Custom pairing function allowing two points from G1 (possibly dependent) to be paired in a non-degenerate manner to a point in GT. Our construction implements in terms of e and φ |
| 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 |
Gt | Target Group. In our construction, the target group is , the extension field |
| The prime field of integers modulo p'. used for calculation of digital credentials. |
| The integers in relatively prime to p' |
| The prime field of integers mod q'. The values used for attributes and exponents within the digital credential are in |
r | The order of the r-torsion group G. |
q | Prime integer specifying the number of elements of G1 and GT. 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 |
| The point at infinity of an elliptic curve |
p | Prime integer. Base of the prime field Fp from which points in E(Fp) are drawn |
p' | Prime number, the base of the prime field for digital credentials. p' is selected using q' such that q' | (p' − 1) |
| The prime field within which calculations for digital credentials and their signatures are performed. |
| Prime field, source of the attributes and exponents for digital credentials |
#E(Fp) | The order of the base elliptic curve. E is specified in this document such that q | o. |
| 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 | #Ek + 1. In our implementation the Tate pairing requires k as an argument. |
Fp | Prime field upon which the base elliptic curve E(Fp) is drawn. Fp is the set of integers [0, p − 1]. |
E(Fp) | The base elliptic curve. In this paper, , |
G1 | The set of points, a subset of E(Fp), 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(Fp). Group G1 is the source of identity points for the protocol; as such, its size is a privacy and security parameter. |
| The extension of G1 into . For a point , |
| Distortion map which transforms a point from the torsion group in the base curve E(Fp) to a point in the torsion group Gx in the elliptic curve on the extension field . In this paper we use |
G | Source Group for the pairing. In our construction the source group G is the r-torsion, a subset of the points in 𝐸𝑝 |
| The prime field of integers modulo p'. used for calculation of digital credentials. |
| The integers in relatively prime to p' |
| The prime field of integers mod q'. The values used for attributes and exponents within the digital credential are in |