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Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely)


Something to do with ``Brandt matrices"?

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    $\begingroup$ I just skimmed the papers linked in both answers, and want to point out that they are using slightly different constructions. The paper IgorRivin links to uses super singular elliptic curves and obtains a family of Ramaujan graphs of constant degree $\ell+1$; the paper SebiCioaba links to uses ordinary curves and obtains Ramanujan graphs of slowly growing degree. $\endgroup$ – David E Speyer Apr 8 '15 at 20:52
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The papers "Do All Elliptic Curves of the Same Order Have the Same Difficulty of Discrete Log?" by David Jao, Stephen D. Miller, Ramarathnam Venkatesan http://arxiv.org/abs/math/0411378 and "Expander graphs based on GRH with an application to elliptic curve cryptography" by the same authors http://arxiv.org/abs/0811.0647 might contain what you are looking for.

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See the very nice paper by Charles, Goren, Lauter, which describes these things in detail.

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