Linear complexity of modulo-m power residue sequences

Abstract

The linear complexity of m-phase power residue sequences is investigated for the case when m is composite. For each factor of m, the linear complexity and the characteristic polynomial of the shortest linear feedback shift register that generates this version of the sequence can be deduced and these results can then be combined using the Chinese remainder theorem to derive the m-phase values. These values are shown to depend on the categories of the length of the sequence computed modulo of each factor of m, rather than on the category of the length modulo-m itself. For a given length, the highest values of linear complexity results from constructing the sequences using those values of the primitive element which lead to non-zero categories for each factor of m.