3 ideas
Ellie Arroway's team detects a signal from the star Vega, 26 light-years away, that pulses out consecutive prime numbers. Underneath it is a retransmission of the 1936 Berlin Olympics television broadcast, Hitler's opening address, because that was the first strong TV signal to leave Earth. A further layer holds tens of thousands of pages of plans for a machine, which can only be decoded once a primer is found that explains its notation.
A signal counting out prime numbers is chosen because no known astrophysical process produces a sequence that requires arithmetic to generate. Its content, not its strength, separates it from natural noise. Mathematics works as the first shared language because any civilisation able to build a transmitter must already know it.
The extraterrestrials say that deep within the digits of transcendental numbers such as pi there are messages left by whoever made the universe. Ellie computes pi far out and finds a long run of ones and zeros that forms a circle when laid out on a grid. The idea is that if the universe was designed, the evidence would sit in mathematical constants that no one inside the universe could have altered, not in scripture or miracles.