Opinion 41: The British Government Should Declassify Turing's Counterexample to the Riemann Hypothesis

By Doron Zeilberger

Written: April 1, 2000

Post April Fool's Day Disclaimer: While this "opinion" was initially meant as my annual April Fool's "joke" to my E-correspondants, and of course, the part about Turing is fictional, the issues it raises, i.e. concealing fundamental discoveries in the name of "security" are pertinent for all times of the year.

I only recently got over my anger at the British government for holding back, for so long, the fact that the so-called Diffie-Hellman and RSA public-key encryption algorithms were anticipated by Malcolm Williamson and Cliff Cocks respectively, several years before they were rediscovered by civilians, and that the whole concept was conceived, several years before Merkle-Diffie-Hellman, by the late James Ellis.

But, in mitigation, one may argue that Ellis, Cocks, and Williamson were sworn to secrecy, and being ``top secret'' added a romantic aspect to their life that made up for the loss of fame and potential fortune. Besides, their algorithms were rediscovered by so-called academics, so the damage of this non-release to the scientific community was only having to wait for a few years.

Much more serious, in my opinion, is the `denial' of the British Defence Agency (BDA) of the very plausible rumors, that were confirmed by inside sources who spoke on the (obvious) condition of anonymity, that Alan Turing, the brilliant cracker of the ENIGMA, found a counter-example to the Riemann Hypothesis, during his service at Bletchley Park. Turing's interest in the Riemann Hypothesis is well-known, and he must have been very frustrated to be only able to publish a numerical algorithm for finding the first N zeros (Proc. London Math. Soc. v. 3 (1953) 99-117).

The reason that the British authorities have not yet declassified this breakthrough is open to speculation. Some rumors claim that Turing devised a cryptosystem whose complexity is O(n**a), where a is the absolute value of the smallest zero of the zeta function that does not lie on the critical line. Even though it is a polynomial-time algorithm, the astronomical value of a makes it "practically exponential".

Another possible reason is that British officials, brainwashed by the conventional wisdom of main-stream mathematicians, doubt the validity of this counterexample, and refrain from revealing it in fairness to Turing himself, whose permission can no longer be solicited, this way making up for their shameful interference with his personal life.

But, for whatever reason, be it a paranoia left over from the cold war, or "noble" intentions, a discovery of such magnitude belongs to the whole world-wide mathematical community. Flat denials, of course, are not to be trusted! The BDA should either publish the Turing purported counter-example, so that it can be checked by experts like Odlyzko and te-Riele, or agree to an appointment of an independent counsel to investigate and possibly dispel the rumors.

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