A note on ramp secret sharing schemes from error-correcting codes

In this paper, new multiplicative and strongly multiplicative linear ramp secret sharing schemes (LRSSSs) based on codes are presented by making use of the construction of LRSSSs without multiplication or strong multiplication in EUROCRYPT 2007 and a special isomorphism. Next, the sufficient conditions that new schemes have multiplication and strong multiplication are also presented. Then, we construct new (strongly) multiplicative LRSSSs based on algebraic geometry codes and give the sufficient conditions that they have multiplication and strong multiplication. Moreover, better LRSSSs with multiplication and strong multiplication based on algebraic geometric codes are presented. They are better than the existing LRSSSs with multiplication and strong multiplication based on algebraic geometric codes in CRYPTO 2006 and EUROCRYPT 2008. One of the advantages of our (strongly) multiplicative scheme is that it does not consume code-length in exchange for secret-length; the construction in CRYPTO 2006 and EUROCRYPT 2008 does consume code-length. Specially, we construct better LRSSSs with multiplication or strong multiplication when the dimension of the secret-domain l=1l=1 or 2.