By Raphael Pass, Abhi Shelat

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3 45 Exponentiation modulo N Given a, x, N, we now demonstrate how to efficiently compute a x mod N. Recall that by efficient, we require the computation to take polynomial time in the size of the representation of a, x, N. Since inputs are given in binary notation, this requires our procedure to run in time poly(log( a), log( x ), log( N )). 1. 5 computes a x mod N in time O(log( x ) log2 ( N )). i Proof. Rewrite a x mod N as ∏i xi a2 mod N. Since multiplying and squaring modulo N take time log2 ( N ), each iteration of the loop requires O(log2 ( N )) time.

SAT is conjectured not to be solvable in polynomial-time—this is the famous conjecture that P = NP. See Appendix B for definitions of P and NP. 2 Randomized Computation A natural extension of deterministic computation is to allow an algorithm to have access to a source of random coin tosses. Allowing this extra freedom is certainly plausible (as it is conceivable to generate such random coins in practice), and it is believed to enable more efficient algorithms for computing certain tasks. Moreover, it will be necessary for the security of the schemes that we present later.

In the rest of this text, any adversarial algorithm A will implicitly be a non-uniform PPT. 2 One-Way Functions At a high level, there are two basic desiderata for any encryption scheme: — it must be feasible to generate c given m and k, but — it must be hard to recover m and k given only c. This suggests that we require functions that are easy to compute but hard to invert—one-way functions. Indeed, these functions turn out to be the most basic building block in cryptography. There are several ways that the notion of one-wayness can be defined formally.