Strongly Multiplicative Ramp Schemes from High Degree Rational Points on Curves

In this work we introduce a novel paradigm for the construction of ramp schemes with strong multiplication that allows the secret to be chosen in an extension field, whereas the shares lie in a base field. When applied to the setting of Shamir’s scheme, for example, this leads to a ramp scheme with strong multiplication from which protocols can be constructed for atomic secure multiplication with communication equal to a linear number of field elements in the size of the network. This is also achieved by the results from Cramer, Damgaard and de Haan from EUROCRYPT 2007. However, our new ramp scheme has an improved privacy bound that is essentially optimal and leads to a significant mathematical simplification of the earlier results on atomic secure multiplication. As a result, by considering high degree rational points on algebraic curves, this can now be generalized to algebraic geometric ramp schemes with strong multiplication over a constant size field, which in turn leads to low communication atomic secure multiplication where the base field can now be taken constant, as opposed to earlier work.