The Thirteenth Power Residue Symbol

This paper presents an efficient deterministic algorithm for computing $13$\textsuperscript{th}-power residue symbols in the cyclotomic field $\mathbb{Q}(\zeta_{13})$, where $\zeta_{13}$ is a primitive $13$\textsuperscript{th} root of unity. The new algorithm finds applications in the implementation of certain cryptographic schemes and closes a gap in the \textsl{corpus} of algorithms for computing power residue symbols

New Assumptions and Efficient Cryptosystems from the ee-th Power Residue Symbol

The $e$-th power residue symbol $\left(\frac{\alpha}{\mathfrak{p}}\right)_e$ is a useful mathematical tool in cryptography, where $\alpha$ is an integer, $\mathfrak{p}$ is a prime ideal in the prime factorization of $p\mathbb{Z}[\zeta_e]$ with a large prime $p$ satisfying $e \mid p-1$, and $\zeta_e$ is an $e$-th primitive root of unity. One famous case of the $e$-th power symbol is the first semantic secure public key cryptosystem due to Goldwasser and Micali (at STOC 1982). In this paper, we revisit the $e$-th power residue symbol and its applications. In particular, we prove that computing the $e$-th power residue symbol is equivalent to solving the discrete logarithm problem. By this result, we give a natural extension of the Goldwasser-Micali cryptosystem, where $e$ is an integer only containing small prime factors. Compared to another extension of the Goldwasser-Micali cryptosystem due to Joye and Libert (at EUROCRYPT 2013), our proposal is more efficient in terms of bandwidth utilization and decryption cost. With a new complexity assumption naturally extended from the one used in the Goldwasser-Micali cryptosystem, our proposal is provable IND-CPA secure. Furthermore, we show that our results on the $e$-th power residue symbol can also be used to construct lossy trapdoor functions and circular and leakage resilient public key encryptions with more efficiency and better bandwidth utilization

Primary Elements in Cyclotomic Fields with Applications to Power Residue Symbols, and More

Higher-order power residues have enabled the construction of numerous public-key encryption schemes, authentication schemes, and digital signatures. Their explicit characterization is however challenging; an algorithm of Caranay and Scheidler computes $p$-th power residue symbols, with $p \le 13$ an odd prime, provided that primary elements in the corresponding cyclotomic field can be efficiently found. In this paper, we describe a new, generic algorithm to compute primary elements in cyclotomic fields; which we apply for $p=3,5,7,11,13$. A key insight is a careful selection of fundamental units as put forward by Dénes. This solves an essential step in the Caranay--Scheidler algorithm. We give a unified view of the problem. Finally, we provide the first efficient deterministic algorithm for the computation of the 9-th and 16-th power residue symbols