Symbol | Description |

| Parameter specification for an instantiation of PP-IBE |

| The order of the r-torsion group |

| Prime integer specifying the number of elements of q. In our construction, |

$\mathcal{O}$ | The point at infinity of an elliptic curve |

| Prime integer. Base of the prime field E(F) are drawn _{p} |

| Prime number, the base of ${Z}_{{p}^{\prime}}$ the prime field for digital credentials. |

${Z}_{{p}^{\prime}}$ | 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 |

| The order of the base elliptic curve. |

$E\left({F}_{p}\right)\left[q\right]$ | The q-torsion group of curve |

| Embedding degree |

| Prime field upon which the base elliptic curve F_{p} is the set of integers [0, p-1]. |

| The base elliptic curve. In this paper, $E\left({F}_{p}\right):{y}^{2}={x}^{3}+1$ , $x,y\in {F}_{p}$ |

| The set of points, a subset of G_{1} is a q-torsion subgroup of E(F). Group _{p}G_{1} 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 |

${p}_{2}=\phi \left({p}_{1}\right)$ | Distortion map which transforms a point from the torsion group ${\mathbb{G}}_{1}$ in the base curve |

$E\left({F}_{{p}^{2}}\right)$ | 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# |

${F}_{{p}^{2}}$ | Field extension of F. In this paper ${G}_{T}\subset {F}_{{p}^{2}}$ and $E\left({F}_{{p}^{2}}\right)\subset {F}_{{p}^{2}}\times {F}_{{p}^{2}}$ . _{p} |

$\widehat{e}:{G}_{1}\times {G}_{1}\to {G}_{T}$ | Custom pairing function $\widehat{e}$ allowing two points from 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. |

| Source Group for the pairing. In our construction the source group |

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

${Z}_{{p}^{\prime}}$ | The prime field of integers modulo |

${Z}_{{p}^{\prime}}^{*}$ | The integers in ${Z}_{{p}^{\prime}}$ relatively prime to |

${Z}_{{q}^{\prime}}$ | The prime field of integers mod |

| The order of the |

| Prime integer specifying the number of elements of q. In our construction, |

$\mathcal{O}$ | The point at infinity of an elliptic curve |

| Prime integer. Base of the prime field E(F) are drawn _{p} |

| Prime number, the base of ${Z}_{{p}^{\prime}}$ the prime field for digital credentials. |

${Z}_{{p}^{\prime}}$ | 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 |

| The order of the base elliptic curve. |

$E\left({F}_{p}\right)\left[q\right]$ | The |

| Embedding degree |

| Prime field upon which the base elliptic curve F_{p} is the set of integers [0, p − 1]. |

| The base elliptic curve. In this paper, $E\left({F}_{p}\right):{y}^{2}={x}^{3}+1$ , $x,y\in {F}_{p}$ |

| The set of points, a subset of G_{1} is a q-torsion subgroup of E(F). Group _{p}G_{1} 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 |

${p}_{2}=\phi \left({p}_{1}\right)$ | Distortion map which transforms a point from the torsion group ${\mathbb{G}}_{1}$ in the base curve |

| Source Group for the pairing. In our construction the source group |

${Z}_{{p}^{\prime}}$ | The prime field of integers modulo |

${Z}_{{p}^{\prime}}^{*}$ | The integers in ${Z}_{{p}^{\prime}}$ relatively prime to |

${Z}_{{q}^{\prime}}$ | The prime field of integers mod |