By Raphael Pass, Abhi Shelat

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T. that given k and m ∈ {0, 1}n produces a ciphertext c. 3. t. that given a ciphertext c and key k produces a message m ∈ {0, 1}n ∪ ⊥. 4. g. 14 = 1111. This security parameter is used to instantiate the “security” of the scheme; larger parameters correspond to more secure schemes. The security parameter also establishes the running time of Gen, and therefore the maximum size of k, and thus the running times of Enc and Dec as well. Stating that these three algorithms are “polynomial-time” is always with respect to the size of their respective inputs.

By inspection, if n ≤ 2, the procedure returns after at most 2 recursive calls. Assume the hypothesis holds for a < 2n . Now consider an instance with a < 2n+1 . We identify two cases. 1. If b < 2n , then the next recursive call on (b, a mod b) meets the inductive hypothesis and makes at most 2n recursive calls. Thus, the total number of recursive calls is less than 2n + 1 < 2(n + 1). 2. If b > 2n , than the first argument of the next recursive call on (b, a mod b) is upper-bounded by 2n+1 since a > b.

They describe the amount of work required for this task as follows: We spent half a year on 80 processors on polynomial selection. This was about 3% of the main task, the sieving, which was done on many hundreds of machines and took almost two years. 2 GHz AMD Opteron processor with 2 GB RAM per core, sieving would have taken about fifteen hundred years. 2 There are many primes The problem of characterizing the set of prime numbers has been considered since antiquity. Euclid, in Book IX, Proposition 20, noted that there are an infinite number of primes.