The London-Paris Number Theory Seminar held its 22nd meeting last June, in London. It had come a long way in the 11 years since I founded it – alongside a handful of colleagues from the two capitals – as a way to do something useful with a French research grant.

The idea in 2006 had been to cover the expenses of bringing a small group of London number theorists to Paris for a day that autumn; London would return the favour the following spring. The timing was right. Number theory is a sport at which the English have excelled at least since G. H. Hardy brought the self-taught S. Ramanujan from South India to Trinity College, Cambridge more than 100 years ago. But while Oxford, Cambridge and a few other outposts such as Bristol had for many years been notable centres for number theory, London was not. The capital was certainly no match for Paris, which, since the Second World War, has remained unchallenged as the mathematical capital of Europe, if not of the world.

Nevertheless, London had begun to close the gap in recent years, and is, by now, a stop on the international number theory circuit. That partly explains why, although the grants that launched it expired long ago, the seminar has become an institution, meeting in both Paris and London every year, and more than doubling in size since its first years, when travelling participants had to book their Eurostar tickets well in advance, to make use of the deep discounts on offer for doing so.

Having sweated over the seminar’s budget during those first few rounds, I was glad not to have to think about it this time. The organisers had managed to turn a one-day seminar into a fully fledged two-day conference, but I refrained from speculating about how this had happened until, halfway through the conference dinner, the London-based organiser sitting next to me asked me whether I wasn’t curious to know the secret.

Indeed I was. And, thus, I learned that the seminar’s magic money tree grew in Bristol and was called the Heilbronn Institute for Mathematical Research. Named after Hans Heilbronn, a German-Jewish refugee who had put Bristol on the number theory map in the 1930s, the institute had been planted in the city in 2005 by the UK spy centre GCHQ – which therefore owned it “100 per cent”, my colleague claimed (although it is run as a partnership with the University of Bristol).

Tom Leinster, a category theorist at the University of Edinburgh, has called the Heilbronn Institute GCHQ’s “mathematical brand”. If you are not a mathematician, you may find it surprising that agencies such as GCHQ and the US’ National Security Agency are so attentive to number theory, especially if you got your picture of mathematics from Hardy’s *A Mathematician’s Apology* . The Cambridge don, a fervent pacifist, is famous for boasting in his book that he had “never done anything useful”. But you should not forget that the NSA’s motto – “collect it all, exploit it all” and so on – does not merely describe its activities of surveillance and data harvesting. Exploiting data that have been encrypted entails finding a way to undo the encryption. Modern methods of encryption, including the protocols that protect your personal data when you make purchases online, make extensive use of number theory. Therefore, GCHQ and the NSA are intimately interested in number theory – and number theorists.

Number theory became an asset in modern cryptography with the invention in 1977 of the Rivest-Shamir-Adleman (RSA) public-key cryptosystem. The idea of public-key encryption is that the encoding of a message using a public key – entering your credit-card details on a commercial website, for example, where the public key is built into the software – is a rapid computation, but decoding it takes too long to be of use to spies or hackers – unless they have the private key. In RSA, the public key is the product, N, of two very large prime numbers, p and q – these days, each one is typically 500 digits long – together with an encryption key. The private key would allow the refactorisation of N back into p and q – a computation that is otherwise believed to be impossible to carry out more quickly than by trial and error.

The public key cryptosystems used in the most familiar applications are generated by commercially available algorithms based on arithmetic that is often more sophisticated than RSA, but the principle is always the same: encryption is fast enough to be commercially viable, but decryption is impossibly slow without the secret key. Impossibly slow, that is, unless you have a *back door* .

The documents that Edward Snowden leaked in 2013 contained evidence that the NSA had compromised commercial encryption by pressing the US National Institute for Standards and Technology to certify encryption algorithms with such back doors: secret built-in flaws that would allow the NSA or GCHQ or anyone else in on the secret – including hackers – to circumvent the slow process of figuring out the private key. Tom Hales, Andrew Mellon professor of mathematics at the University of Pittsburgh, was so deeply disturbed by Snowden’s revelations that, in February 2014, he published a self-contained explanation of the mathematics behind the back doors in *Notices of the American Mathematical Society (AMS Notices)*, the society’s house journal. He concludes: “It is no secret that the NSA employs some of the world’s keenest cryptographic minds…in my opinion, an algorithm that has been designed by [the] NSA with a clear mathematical structure giving them exclusive back door access is no accident, particularly in light of the Snowden documents.”

More than a few number theorists volunteered their skills to help the NSA after the September 11 attacks. By the time of the Snowden disclosures 12 years later, the agency’s shine had clearly worn off as far as mathematicians were concerned. For many years, the agency had sponsored a grants programme administered by the AMS. In the wake of the Snowden story, I helped to organise a debate on the relationship between mathematicians and the NSA, published in *AMS Notices*. It was easy enough for deputy editor Allyn Jackson and I to find colleagues willing to speak out against cooperation with the NSA. University of Chicago professor Alexander Beilinson, who left Russia in the 1980s, went so far as to call for “making the NSA and its ilk socially unacceptable – just as, in the days of my youth, working for the KGB was socially unacceptable for many in the Soviet Union”. Some of the most eminent specialists were deeply critical of the practices that Snowden had uncovered, and found the “collect it all” mentality counterproductive, even on purely technical grounds. On the other hand, it was practically impossible to find a mathematician not on the NSA payroll who was willing to put in a good word for the agency, although we certainly tried.

Among British mathematicians, Edinburgh’s Leinster was probably the first, and certainly the most visible, to question the links between mathematicians and surveillance agencies. In April 2014, he launched the *London
Mathematical Society Newsletter* ’s brand new “members’ opinion” section, asking “Should Mathematicians Cooperate with GCHQ?” But it was his *New Scientist* opinion piece in the same month, under the print headline “Ethical calculus”, that briefly brought him – and mathematics’ centrality to surveillance – to the attention of the international press. Leinster’s piece was cited on news sites from Sweden to Guatemala to Bulgaria; most of the press coverage also repeated Beilinson’s KGB comment. In September of that year, I was even interviewed about the AMS debate for French TV – but, by then, media executives – and mathematicians themselves – were moving on to other concerns and it was never broadcast.

The question about whether we should cooperate with the security agencies has not gone away, however. The problem is that research funds have to come from somewhere: the survival of number theory depends on it. One veteran colleague likens mathematical research to a kidney: no matter where it gets its funding, the output is always pure and sweet, and any impurities are filtered out in the paperwork. Our cultural institutions have long since grown accustomed to this excretory function, and that includes our great universities. Henry VIII was a morally ambiguous character, to say the least, and a pioneer in eavesdropping, as well as in cryptography; but neither Hardy nor his friend and future anti-nuclear campaigner Bertrand Russell refused their fellowships at Trinity on that account.

It would be nice if the state could provide its own kidneys and impose an impermeable barrier between the budgets for research that is socially progressive – or at least innocuous – and the military and surveillance functions about which the less we know, the better. But states don’t work that way; for the most part, they never have.

The only alternative to public funding, from whatever source, is private philanthropy. America’s great private universities are monuments to the past and present generosity of some of the country’s wealthiest citizens. That is not, however, what is most appealing about them. I find it demeaning to have to express gratitude for my research funding to despots such as the Emir of Kuwait, whose foundation used to sponsor a generous lecture series at Cambridge, or leading hedge fund managers and data miners, whose status as “ultra high net worth individuals” also affords them the opportunity to function publicly as philanthropists.

As the blogger Mathbabe – also known as Cathy O’Neil, author of *Weapons of Math Destruction* – put it in January 2014, “We lose something when we consistently take money from rich people, which has nothing to [do]with any specific rich person who might have great ideas and great intentions.” But she was not thinking of the feeling of being demeaned so much as the loss of control of how decisions are made: “The entire system depends on the generosity of someone who could change his mind at any moment.”

The more basic problem is that the very existence of ultra high net worth individuals entails the concentration of power beyond democratic oversight. Among billionaire patrons, Jim Simons stands out for his commitment to the goals and priorities of mathematicians themselves – which is natural, given that he was a distinguished geometer before his management of the wildly successful hedge fund Renaissance Technologies made him fabulously wealthy. But the same high-frequency trading algorithms that fuelled Simons’ philanthropy gave us Breitbart News, courtesy of Robert Mercer, Simons’ former colleague at Renaissance. Mercer was much in the news last year after it was revealed that, through his connection to the data-mining firm Cambridge Analytica, he used psychologically targeted advertising on social media to intervene in the Brexit and Trump polls, possibly tipping the balance in both cases. Even outspoken liberal billionaires such as Facebook’s Mark Zuckerberg and Google’s Sergei Brin, who have been subsidising pure mathematics indirectly through their co-sponsorship of the extravagant Breakthrough Prizes, have built their fortunes on mathematical techniques that are no less threatening to privacy than GCHQ surveillance.

In the past decade, mathematics has taken its place alongside the other sciences as the source of dramatic threats to the foundations of organised human life: the explosive growth of markets in financial derivatives; erosion of privacy through corporate as well as government surveillance; targeted psychological manipulation through data mining; the supposedly objective algorithms exposed in O’Neil’s book that are used to make judgements about us, such as our loan eligibility, but that actually embed various destructive biases. Mathematicians have been reluctant to recognise that if our work interests generous donors, it is often precisely because it is “useful” according to a definition that Hardy proposed near the beginning of the First World War: “its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life”.

We will have to overcome this reluctance and draw uncomfortable conclusions. Wherever you turn as a mathematician, you’re going to be someone’s kidney: practically every potential source of research funds is tainted in some way. But professional maturity seems to require that we just get used to it, and get on with our careers. Indeed, the quest to free mathematics from its dependence on unsavoury funding sources has its cautionary tale in the person of the late Alexander Grothendieck. Considered by many to be the most influential and original mathematician of the postwar period, he spent the “golden age” of his career at the Institut des Hautes Études Scientifiques, a small-scale version of Princeton’s Institute for Advanced Study, built outside Paris in large part as a refuge for his brilliance. When he learned in 1970 that the Institut had accepted a small grant from the French Ministry of the Army, Grothendieck, never one to compromise, resigned his position, in effect ending his research career; he spent the last decades of his life as a hermit in the Pyrenees. Most of his former French colleagues felt, and still feel, that his defence of principle was not worth the price that both he and mathematics paid.

Upon closer examination, however, two different sorts of maturity come into focus. One is bittersweet and tinged with resignation – and, in my experience, this is the kind that predominates in university departments. It generally comes down to the observation that “everyone else does it”. Everyone, that is, accepts funds from ethically dubious donors; to refuse to do so would not change the world: it would merely deprive us of scarce resources.

The second kind of maturity admits the accuracy of this observation but draws a different conclusion. The immense privilege of devoting our lives to the research projects that we have chosen freely imposes on us the obligation to speak out when our work is used for destructive ends, or when the sources of our funding do not share our values.

**Michael Harris is professor of mathematics at Columbia University and Paris Diderot University.**

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Print headline: *Doing sums for spooks*

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