Abstract This paper presents an efficient algorithm for computing 11 th -power residue symbols in the cyclo-tomic field ℚ ( ζ 11 ) , $ \mathbb{Q}\left( {{\zeta }_{11}} \right), $ where 11 is a primitive 11 th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7 th -power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.