Download Equidistribution in Number Theory: An Introduction by Andrew Granville, Zeév Rudnick PDF

By Andrew Granville, Zeév Rudnick

Written for graduate scholars and researchers alike, this set of lectures presents a dependent advent to the idea that of equidistribution in quantity thought. this idea is of growing to be value in lots of parts, together with cryptography, zeros of L-functions, Heegner issues, top quantity thought, the speculation of quadratic varieties, and the mathematics elements of quantum chaos.;

The quantity brings jointly major researchers from a number fields, whose available displays show attention-grabbing hyperlinks among likely disparate components.

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Let δ > 0, let e, ϕ(m) be arbitrary and let (m be prime or) m = pℓ be the product of two primes. Define ∆ = (p−1, ℓ−1). If t/∆ > m3/4+δ , then the sequence un is uniformly distributed modulo m. Here the main tool in the proof is a very well-known exponential sum bound due to A. Weil. We recall that, if p is prime, f ∈ Z[x] with degree deg f = d, and such that f is non-constant modulo p, then p e p f (x) < d p1/2 . x=1 This bound is very good if the degree d is not large. In our case we have k n un is given by ue0 and this leads to polynomials having monomials xe of very high degree.

In this elliptic case the proof eventually reduces to estimates for exponential sums over finite fields in which an earlier reliance on Weil’s bound is replaced by bounds due to Bombieri (Bombieri, 1966). We remark that one obtains the same result if, in the sum Wa (U, V), we replace the x-coordinate in the exponential by the y-coordinate or indeed by any nonconstant rational function in x and y. In the same fashion as earlier, the above theorem implies bounds for the exponential sum t t e p ax(uG) + bx(vG) + cx(uvG) , u=1 v=1 and hence uniformity of distribution results for Diffie–Hellman triples on an elliptic curve, that is the triples x(uG), x(vG), x(uvG) .

M − 2}, 0 if θ xy ∈ {2, 4, 6, . . , m − 1}. 4 one may deduce a reasonable lower bound for the complexity of this function. 5. Let m be an odd integer, δ > 0 be real, and assume the period t of θ modulo m satisfies t ≥ m10/11+δ . Then, the communication complexity of the Diffie–Hellman bit operation satisfies the bound ψ( f ) ≥ 11 δ − o(1) n. 32 EXPONENTIAL SUMS, AND CRYPTOGRAPHY 49 This lower bound provides, a fortiori, a lower bound for the communication complexity of the Diffie–Hellman secret key θ xy itself.

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