Fig. 3
From: Efficient privacy-preserving variable-length substring match for genome sequence
Example of a search when \(V=(2,0,3,1)\), \(p_0=2\), and \(\ell =4\). The goal is to compute \([\![ V^{(4)}[2] \, ]\!]=[\![2]\!]\). Here we assume \(\mathcal {B}\) generates \(r^0=1, r^1=2, r^2=1\). In Step 1 of Search phase, \(P_0\) and \(P_1\) jointly compute \(\mathsf {Reconst}([\![R^0[2]\,]\!]_0, [\![R^0[2]\, ]\!]_1)\) to obtain \(R^0[2]=0\). (\(R^0[2]\) is randomized by \(r^0\), so any element of V is leaked.) In a similar way, \(P_0\) and \(P_1\) compute \(R^1[0] = 3\) and \(R^2[3] = 1\). In Step 2, \(P_0\) and \(P_1\) output \([\![R^3[1]\, ]\!]_0\) and \([\![R^3[1]\, ]\!]_1\) respectively. Since \(R^0[2]=V[2]+r^0\), \(R^1[V[2]+r^0] = V[V[2]+r^0-r^0]+r^1\), \(R^2[V[V[2]]+r^1] = V[V[V[2]] +r^1-r^1 ]+r^2\), and \(R^3[V[V[V[2]]]+r^2] = V[V[V[V[2]]] +r^2-r^2]\), ss-ROTÂ successfully computes \([\![ V^{(4)}[2] \, ]\!]\)