Malaysian Journal of Mathematical Sciences, August 2017, Vol. 11(S)
Special Issue: The 5th International Cryptology and Information Security Conference (New Ideas in Cryptology)
Cyclotomic Cosets, Codes and Secret Sharing
Wong, D. C. K.
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
In this paper, all 2-cyclotomic cosets modulo \(p^n\) are constructed when 2 is a primitive root modulo \(p^n\). When the order of 2 is \(\frac{p-1}{2}\) modulo \(p\) and the order of 2 is \(\frac{p(p-1)}{2}\) modulo \(p^2\), we construct all 2-cyclotomic cosets modulo \(p^2\). Also, when 2 has order \(\frac{p^2(p-1)}{2}\) modulo \(p^3\), we derive all 2-cyclotomic cosets modulo \(p^3\). Furthermore, four
results on all \(s\)-cyclotomic cosets modulo \(pq\) are obtained by considering three different possible orders of \(s\) modulo \(p\) and \(q\), for distinct odd primes \(p\), \(q\). Finally, we use the 2-cyclotomic cosets modulo 9, 25 and 49 to construct binary codes of length 9, 15
and 49, respectively, and hence the access sets for the secret sharing scheme based on some of these families of binary codes are discussed.
Keywords: Cyclotomic cosets, minimum distance, secret sharing, cyclic codes, idempotents
