One-way permutation on a family of surfaces. (Q1909436)

For the superelliptic curve \(C/F_p\) defined by the equation \(y^2= x^5+a\), where \(p\) is an odd prime such that \(p^2+1\) is square-free and \(\gcd (5,p-1)= \gcd (5,p^2-1)= 1\), the Jacobian of the curve \(C/ F_p\) is a cyclic group where the discrete logarithm problem is apparently harder than on the elliptic curve. The authors conjecture that the bijective mapping \(i\mapsto g+g+ \cdots+ g\) (\(i\) times) from the set \(\Omega= \{0,1,2 ,\dots, p^2\}\) to \(J(C/F_p)\), generated by \(g\), is one-way. They construct an efficiently computable bijective mapping from \(J(C/ F_p)\) onto \(\Omega\) and obtain a one-way permutation on \(\Omega\) as a composition of these two bijective mappings.