How do I prove Berman's theorem?

Question

Berman's theorem states

If a unary language ( a language with all the strings of the type $1^i$, $ i > 0 $ ) is NP-Complete then P = NP.

I tried reducing SAT to a given unary language $L$ assuming it is NP-Complete. But I can't think of a way such that after applying the reduction so that SAT gets solved in polynomial time. How do I proceed further?

This is an exercise from Sanjeev Arora and Boaz Barak , but not homework.

Answer 1

Please refer this solution. Or see the paper by S. R. Mahaney, Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis (Journal of Computer and System Sciences, 25:130-143, 1982).