Is there a two-party multiplicative and additive secret sharing scheme ? 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. 
 A: Yes, there is such scheme. It was recently suggested by Craig Gentry.
Reference: "Fully Homomorphic Encryption Using Ideal Lattices" by Craig Gentry.
http://domino.research.ibm.com/comm/research_projects.nsf/pages/security.homoenc.html/$FILE/stocdhe.pdf
A: I believe the answer to your question is yes.  See Cramer, Damgard, Maurer, "General Secure Multi-party Computation from any Linear Secret-Sharing Scheme," Eurocrypt 2000.
A: 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.
A: Two operation homomorphic secret sharing scheme has been proposed by Benaloh in the paper , "Secret sharing homomorphisms: Keeping shares of the secret secret" . This scheme allows homomorphic properties at two levels of dividing the secrets ( and their sub secrets) .
Note: This is not same as Homomorphic secret sharing with both add and mult at same level of. This means add at one secret level and mult at sub secret level .
