New PDF release: A Course in Cryptography

By Raphael Pass, Abhi Shelat

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2 below. 2: Graph of π (n) for the first thousand primes By inspecting this curve, at age 15, Gauss conjectured that π ( x ) ≈ x/ log x. Since then, many people have answered the question with increasing precision; notable are Chebyshev’s theorem (upon which our argument below is based), and the famous Prime Number Theorem which establishes that π ( N ) approaches N ln N as N grows to infinity. 3 (Chebyshev) For x > 1, π ( x ) > x 2 log x Proof. Consider the integer X= 2x x = (2x )! )2 x+x x x + ( x − 1) ( x − 1) ··· x+1 1 Observe that X > 2x (since each term is greater than 2) and that the largest prime dividing X is at most 2x (since the largest numerator in the product is 2x).

One may argue about the choice of polynomial-time as a cutoff for efficiency, and indeed if the polynomial involved is large, computation may not be efficient in practice. There are, however, strong arguments to use the polynomial-time definition of efficiency: 21 22 chapter 2. computational hardness 1. ) because converting from one representation to another only affects the running time by a polynomial factor. 2. This definition is also closed under composition which may simplify reasoning in certain proofs.

2 (Running-time) An algorithm A is said to run in time T (n) if for all x ∈ {0, 1}∗ , A( x ) halts within T (| x |) steps. A runs in polynomial time if there exists a constant c such that A runs in time T (n) = nc . 3 (Deterministic Computation) An algorithm A is said to compute a function f : {0, 1}∗ → {0, 1}∗ if for all x ∈ {0, 1}∗ , A, on input x, outputs f ( x ). We say that an algorithm is efficient if it runs in polynomial time. One may argue about the choice of polynomial-time as a cutoff for efficiency, and indeed if the polynomial involved is large, computation may not be efficient in practice.

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A Course in Cryptography by Raphael Pass, Abhi Shelat

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