If you boil down VIFF, or any MPC program you can boil it down to 5 primitives.

These primitives are
x + y = z       Addition
x * y = z       Multiplication
-x = z          Negation
z = Share(x)    Secret sharing
z = Reveal(x)   Revealing of a secret sharing

My question is now: What are the second order primitives?

What are the basic features that, make a programmers life easy but not essential. I would propose the following 5:
z = f(x,y)     Function calls
for(0 to n)    A for call with a fixed number of iterations.
z = Rand(x)    Generating random values in Zp
x = y          Equation
x < y          Comparison
while( x )     A while loop on x, where x is a revealed value

The first 2 primitives are easy since they are already implemented using the underlying programming language. Should creating random values be on this list? Or should it maybe be among the first order primitives, a basic building block of MPC. Equation and comparison are obvious second order primitives and has been my research topic for many years. The last on the list is the while loop, I have included it because I have seen many times that you want to compute some function, open a variable, check some information and then redo the function if the revealed value was false. This can be found when creating random values less than some threshold, creating RSA primes, and many other algorithms. Also it is not trivial to program if you want it to be efficient.


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