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Much of modern cryptography is based around working with boolean or arithmetic circuits. For example in Multi-Party Computation the 'famous' results allow for the secure computation of any function that can be represented as a boolean or arithmetic circuit.

I am wondering what the limits of these circuits are, what are some examples of functions are we not able to compute securely --- that is what functions are not representable as a boolean or arithmetic circuit? And in the real world does this reduce the effectiveness of modern cryptography or are these circuits enough to implement everything we require?

dtb
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The question should be which functions can we build a small circuit to solve. Any function mapping a fixed number of bits to a fixed output can be represented as a boolean circuit. But it may be very large. This brings is to a difficult topic with few lower bounds: https://en.m.wikipedia.org/wiki/Circuit_complexity

Meir Maor
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