Table 6 Basic Group field assumptions of various cryptographic operations.

Bilinear-pairing

\(\overline{E}\): \(y^{2}\) = \(x^{3}\) + x mod \(\hat{p}\)

\(\hat{e}\): \(G_{1}\) x \(G_{2}\) → \(G_{T}\)

\(G_{1}\)(P)

64 Bytes

q = 20 bytes

|\(G_{1} | = 128\) bytes

Elliptic curve

\(\overline{E}\): \(y^{2} = x^{3} +\) ax + b mod p

No

G(P)

20 Bytes

q = 20 bytes

|G|= 40 Bytes

Chebyshev polynomial

\(T_{n} \left( x \right) \equiv \left( {2xT_{n - 1} \left( x \right) - T_{n - 2} \left( x \right)} \right)\left( {mod P} \right);\)

\(n{ \succcurlyeq }2\)

No

\(Z_{p}^{*}\)

–

–

\(\left| {Z_{p}^{*} } \right| = 20\) Bytes

Dickson polynomial

\(D_{n} \left( {x,\alpha } \right) = x D_{n - 1} \left( {x,\alpha } \right) - \alpha D_{n - 2} \left( {x,\alpha } \right);\)

\(n{ \succcurlyeq }2\)

No

\(Z_{p}^{*}\)

–

–

\(\left| {Z_{p}^{*} } \right| = 20\) Bytes