A secret sharing scheme such as [Shamir's secret sharing](http://en.wikipedia.org/wiki/Shamir%27s_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. 



Is there a two-party multiplicative and additive secret sharing scheme ?
Unconditionnaly secure 2-party computation does not exist (unfortunately). This is derived from the impossibility of Oblivious Transfer. Also note that unconditionnaly secure OT is also impossible if the 2 parties are quantum.


