The output series of logic operation of binary numbers can be seen as random series of decimal numbers

There is a function of a,b and c.

c=f(a,b)

If the f() is "exclusive or", can you get any periodicity in the series of c? Do you know any formula to solve it?

The output series of logic operation of binary numbers can be seen as random series of decimal numbers. But there may be a invisible periodicity within it.

Is it possible to get a and b from the series of c?

$ ./xor
(c:0)=(a:1)^(b:1)
(c:3)=(a:1)^(b:2)
(c:2)=(a:1)^(b:3)
(c:5)=(a:1)^(b:4)
(c:4)=(a:1)^(b:5)
(c:7)=(a:1)^(b:6)
(c:6)=(a:1)^(b:7)
(c:9)=(a:1)^(b:8)
(c:8)=(a:1)^(b:9)
(c:11)=(a:1)^(b:10)

(c:3)=(a:2)^(b:1)
(c:0)=(a:2)^(b:2)
(c:1)=(a:2)^(b:3)
(c:6)=(a:2)^(b:4)
(c:7)=(a:2)^(b:5)
(c:4)=(a:2)^(b:6)
(c:5)=(a:2)^(b:7)
(c:10)=(a:2)^(b:8)
(c:11)=(a:2)^(b:9)
(c:8)=(a:2)^(b:10)