assume there are three group of high dimension vectors: {a_1, a_2, ..., a_N}, {b_1, b_2, ... , b_N}, {c_1, c_2, ..., c_N}. each of my vector can be represented as: x = a_i + b_j + c_k, where 1 <=i, j, k <= N. then the vector is encoded as (i, j, k) wich is then can be decoded as x = a_i + b_j + c_k. my question is, if there are two vector: x = (i_1, j_1, k_1), y = (i_2, j_2, k_2), is there a method to compute the euclidian distance of these two vector without decode x and y.