Decisional Supersingular Sample Clauses

Decisional Supersingular. Isogeny (DSSI) problem. Let Ei be another supersingular curve defined over Fp2 . Decide whether Ei is lei -isogenous to E. → ⟨ ⟩ Computational Supersingular Isogeny (CSSI) problem Let the map φi : E Ei be an isogeny whose kernel is [mi]Pi + [ni]Qi , where mi and ni are chosen at random from Z/lei Z and not both divisible by li. Given Ei and the values φi(Pj), φi(Qj), for j /= i, find a generator Ri of ⟨[mi]Pi + [ni]Qi⟩. → / ⟨ ⟩ Supersingular Computational Xxxxxx-Xxxxxxx (SSCDH) problem Let φi : E → Ei be an isogeny whose kernel is equal to ⟨[mi]Pi + [ni]Qi⟩, and let φj : E Ej (j = i) be an isogeny whose kernel is [mj]Pj + [nj]Qj , where mi, ni (respectively mj, nj) are chosen at random from Z/lei Z (respectively j Z/lej Z ) and not both divisible by xx (respectively lj). Given the curves Ei, Ej and the points φi(Pj), φi(Qj), φj(Pi), φj(Qi), find the j-invariant of E/⟨[mi]Pi + [ni]Qi, [mj]Pj + [nj]Qj⟩. Supersingular Decision Xxxxxx-Xxxxxxx (SSDDH) problem Given a tuple sampled with probability 1/2 from one of the following two distributions: • (Ei, φi(Pj), φi(Qj), Ej, φj(Pi), φj(Qi), Eij), wherein the quantities Ei, φi(Pj), φi(Qj), Ej, φj(Pi), and φj(Qi) are as in the SSCDH problem and Eij E/⟨[mi]Pi + [ni]Qi, [mj]Pj + [nj]Qj⟩, • (Ei, φi(Pj), φi(Qj), Ej, φj(Pi), φj(Qi), EC), wherein the quantities Ei, φi(Pj), φi(Qj), Ej, φj(Pi), and φj(Qi) are as in the SSCDH problem and EC E/⟨[m′i]Pi + [ni′ ]Qi, [m′j ]Pj + [n′j ]Qj⟩, where m′, n′ (respectively m′ , n′ ) are chosen at random from Z/lei Z (re- i i j j i j spectively Z/lej Z) and not both divisible by li (respectively lj), determine from which distribution the tuple is sampled.