Algebraic manipulation detection codes from algebraic curves

摘要

A method for protecting encoded data from algebraic manipulation includes receiving a data word sεK^d to be protected, randomly selecting two integers a ε{0, . . . , q−1} and b ε{0, . . . , √q−1}, finding a point (α, β) on a Hermitian curve over a field F_q that corresponds to the randomly selected integers (a, b) from a mapping (a, b)(α, β)=(u_a, u_a^√q+1z+v_b), where; $$ u_a:=\{\begin{array}{cc}0& \mathrm{if}a=0,\\ {\gamma }_1^{a-1}& \mathrm{otherwise}\end{array},V_b:=\{\begin{array}{cc}0& \mathrm{if}b=0,\\ {\gamma }_2^{b-1}& \mathrm{otherwise}\end{array},$$and z is an element of the field F_q of unit trace, and where γ₁ is a fixed primitive element of the field F_q and γ₂ is a primitive element of a field F_√q⊂F_q, and calculating a sum f_s(α, β)=α^i_d+1β^j_d+1+Σ_k=1^dα^i_kβk^j_k for a set of d+1 integers pairs I ={(i_k,j_k)}_k=1^d+1, where the encoded word is a triple (s, (α, β),f_s(α, β)).