For the person asking for the function, it's a black box... the whole point is that we don't know the function. The only thing we can do is interact with it by inputing quantum states and receiving quantum states as an output as well. If you don't know what a black box function is, please don't answer it or ask for more information.

3. You are given a quantum blackbox for a function f : {0,1}" + {0,1}" that is known to be 4-to-1 and for which there exist S1, S2 E {0,1}", on + $1 + $2 # on such that for every x E {0,1}", f(x) = f( x 81) = f(x S2) = f(x 81 82). (a) Given unlimited oracle access, what information about sand s2 can be extracted? (b) Design a quantum algorithm that extracts that information at an expected query cost of O(n). 3. You are given a quantum blackbox for a function f : {0,1}" + {0,1}" that is known to be 4-to-1 and for which there exist S1, S2 E {0,1}", on + $1 + $2 # on such that for every x E {0,1}", f(x) = f( x 81) = f(x S2) = f(x 81 82). (a) Given unlimited oracle access, what information about sand s2 can be extracted? (b) Design a quantum algorithm that extracts that information at an expected query cost of O(n)