A secret sharing scheme such as Shamir's secret sharing allow to perform addition and multiplication for secret values so far as there is at least 3 participants. Addition of two secret values is done locally at each party by adding the corresponding local shares, so it is possible to do addition even with only 2 parties. In the case of multiplication, a degree reduction step is obligatory, because multiplication increases the degree of the shares and this makes it impossible for 2 parties to perform multiplications without the aid of a third one.
It is possible to construct a multiplicative sharing scheme that works for two parties, but it wouldn't be additive.
Is any one aware of a secret sharing scheme for two parties (without the need for a third party) that is BOTH additive and multiplicative, or is it impossible ?
So far I know that it is impossible to construct scalar product protocol with unconditional security for two parties. But I don't suppose that it necessarily prevents the construction of an algebraic (additive+multiplicative) secret sharing scheme.
Update: I am aware of homomorphic encryption and the existence of algebraic homomorphic encryption schemes. However homomorphic encryption is not unconditionally secure unlike secret sharing which is.