In response to: “Cryptocurrencies and Blockchains” (Vol. 2, No. 4).


To the editors:

Jean-Paul Delahaye, professor emeritus of computer science at the University of Lille, regularly publishes articles on mathematics and computer science in popular journals. He is thus very comfortable presenting in an accessible manner, for the non-specialist, subjects as difficult as hash functions, proofs of work, blockchains, and other complex subjects associated with modern cryptography. He is also intimately acquainted with the fields of mathematics and computer science, as opposed to the current trend of researchers who are increasingly specialized in a very narrow area of their discipline.

I do not know Delahaye personally, but I have corresponded with him in relation to his publications. I have also expressed my astonishment at his facility with so many subjects and the frequency with which he publishes, while I struggle to write when it is a matter of popularizing my own research topics. He replied that it entertained him! Delahaye’s knowledge and analytical skills are exemplified by this article. However, if it is meant to say nearly everything about bitcoin, there are still a few aspects that should be deepened.

Although it uses bitcoin as an example, Delahaye’s analysis is not limited to this single cryptocurrency. For this reason, I will discuss it first. Initially, bitcoin was, for me, a mysterious currency that was sometimes the subject of articles in the national press about its volatility when converted into dollars or euros, or its possible use as a means of financial evasion for the Chinese.1

Curious as to whether this impression was merely my own private view, I took a mini-survey among a few friends at universities in different countries. Unsurprisingly, in response to the question, “Have you ever used bitcoin currency, and if so, why?” the answers were all negative. That is, with the exception of a young computer science lecturer, a member of my immediate family: “I have a portfolio on quadrigacx.com, because I think in the long run, the value of bitcoin will increase. It has already tripled since I bought it.” At that moment, the whole enterprise suddenly became more concrete for me, and the idea of acquiring some bitcoins briefly crossed my mind. How to proceed? By buying them, or by mining them?

From an economic point of view, before one can reflect upon on the bitcoin phenomenon, or any one of the seven hundred cryptographic currencies created since then, it seems necessary to return to the definition of what a currency is.

Although it is difficult to pinpoint the date at which the first currencies appeared, it is believed that they arose along with the development of trade and the emergence of commerce.2 One might imagine that, if two economic agents wish to exchange product A for product B with a delay in time, then a product C, to which these agents attribute a value equivalent to that of A and of B, could serve as a temporal intermediary for the transaction. In this case, product C must have an intrinsic value, and this value must persist over time in a stable manner. Metals, which were relatively rare in antiquity, offer these necessary qualities.

Imagine, for example, a transaction taking place in Mesopotamia more than four and a half thousand years before the kingdom of Akkad. Nidaba (the goddess of the harvest) would like to buy a sheep (A) from Sumuqan (the god of animals) who possesses a herd. She gives him a nugget of native copper (C) in exchange for the sheep. A few months later, after the harvest, Sumuqan buys a certain quantity of barley (B) from Nidaba by giving her back this nugget (C). In this imaginary case, as in the actual case of the coins that appeared later in antiquity, the material of exchange remains almost unchanged for years. No cost is induced by maintaining this means of payment over time. The situation changes when, for various reasons—metal shortages in a flourishing economy, fear of appropriations like that of Charles I of England in 1640, etc.—one moves from using a means of payment with an intrinsic value to one that represents that value, such as certificates of deposit, assignats, or notes.

In thirteenth and fourteenth century China, the Yuan dynasty attempted to impose the exclusive use of paper money. Marco Polo devoted a whole chapter of his Travels to this remarkable innovation, and his statements have been confirmed by research. The notes were printed on the liber, or inner bark, of the mulberry. Notes that were worn but still legible could be redeemed at face value upon payment of a three percent tax.3 Here the cost of maintaining a symbol was not zero, even if it was still minimal.

Nowadays, for euro coins, maintenance is more significant, since the cost of minting a one or two cent coin is three cents. These coins are easily misplaced and some countries have ceased to manufacture them.4 As for notes, the situation is different. To get an idea of this, one must consider manufacturing costs, which are difficult to estimate. Since their introduction in 2002, euro notes have been printed throughout Europe. The Bank of France continues to produce all of the five, ten, and twenty euro notes printed in France, but it does not report on manufacturing costs. On April 4, 2017, a new fifty euro banknote will be put into circulation. Ten billion will be produced at a cost of six hundred million euros, i.e., six cents per note.5 One can deduce the lower limit on the cost of maintaining these notes on the basis that they will replace existing notes printed in 2002 (and making the false assumption that these have not yet been replaced)—i.e., 0.4 cents, or 0.08 % of face value per year. Most of these notes have likely already been replaced at least once, since most of them bear the signature of Mario Draghi, who began his tenure as President of the European Central Bank in 2011. Thus it appears that the cost of maintaining this means of payment is at least 0.16 % of face value per year. If the new notes last fifteen years, this is approximately ten times cheaper than during the Yuan dynasty, whose notes used a less abundant natural resource.

In the case of bitcoins, their maintenance is not, of course, linked to manufacturing costs. Rather, it is connected to the considerable electricity requirements for the mining calculations which make it possible to create new bitcoins and for the maintenance of the blockchain by 5,000 or 6,000 miners. The purchase and replacement of dedicated computer equipment is another factor.

Delahaye indicates that, if one turns to an article by Pierre Buntix, the electrical power consumed by the network of miners could become equivalent to that of Denmark by 2020. Buntix was, in fact, commenting on a more accurate article by Sebastiaan Deetman that proposes two scenarios:6 the pessimistic one to which Delahaye refers, and another, much more optimistic, which provides a range of 470 megawatts to 14 gigawatts—i.e., thirteen nuclear reactors or seven thousand windmills, each generating two megawatts, measuring between eight and one hundred and thirty meters in height, and weighing up to three hundred tons. Assuming the optimistic scenario, mining a single bitcoin in 2020 will require 5,500 kilowatt hours and produce four tons of CO2! In France, the cost of this amount of electricity (including subscription) for an individual who would like to start mining is about nine hundred and forty euros, slightly higher than the current bitcoin price. And yet, as a result of nuclear power generation, electricity prices in France are among the lowest in Europe. The cost considered here is just that of the electricity. It does not take into account the purchase of specific equipment: the computer and specialized ASIC circuits that have been available since 2013. A possible objection might be that the value of a bitcoin may have quintupled, or even increased tenfold in four years, making their mining a profitable activity. But it may also be subject to crashes.7

It is evident, however, that the activity of a miner is not as trivial as one might imagine when reading Delahaye’s article, which states, in the French version:

On le voit Nakamoto a imaginé un système complexe mais très subtil. Ce système assure qu’il existe et existera toujours des volontaires, les mineurs, pour s’occuper de la blockchain du bitcoin et pour se surveiller les uns les autres

(“One sees that Nakamoto has conceived a system that is both complex and very subtle. It ensures its own continued existence and that there will always exist volunteers, the miners, to take care of the bitcoin blockchain and to monitor each other.”)

In any case, as part of my quest to obtain bitcoins, I consulted the site bitcoin.fr. This site indicates that the preferred method of acquisition is to use the marketplaces. Indeed, the website warns against the temptation to become a miner:

Complex and expensive, the mining of bitcoins is not at all a profitable activity for an individual. … In general, mining of crypto-currencies is advisable in Europe only for persons who have their own generator of green electricity (solar, wind, geothermal, hydraulic) for domestic purposes, and are not able to resell surplus production on the public network.

This warning is in line with Deetman’s cost analysis. But, if this activity is not profitable for an individual, it may be of interest, for other reasons, to multinationals or nation states.

There are, at most, sixteen million bitcoins in circulation.8 The price of a bitcoin is about US$900, leading to a current capitalization of around US$14 billion. Let us suppose that this currency is recognized as reliable, as seems to be the case, according to Delahaye. As the author of the original bitcoin paper is still not known and the mining software is free, there is nothing to prevent a nation state from duplicating this currency a significant number of times (without even seeking to improve the system as the other cryptocurrencies have done). We would then have: bitcoin1, bitcoin2, …, bitcoinn—each with its own blockchain.

The United States, for example, could then resort to these bitcoini to secure, or to discreetly move, their debt from country to country. A debt which amounted to US$19 trillion at the end of January 2016.9 Around 1,350 copies of bitcoin would suffice. If we combine the figures with the electrical power required to maintain them, it would be necessary to have power plants producing, according to Deetman’s lower hypothesis, 634 gigawatts, almost twice current global power production, which is now at 327 gigawatts.

It is therefore understandable that the adoption of one or more cryptocurrencies to replace existing currencies, under the present conditions of operation in the global computer park, will not be easily achievable. More important than the uncertain future of the bitcoin is the use of blockchains which, according to Delahaye, can be used for other applications such as secure voting, provably fair online games, property contracts, etc.:

Comme pour le courrier électronique ou Internet qui ne sont aux mains d’aucune autorité et conduisent donc à une meilleure appropriation de l’information pour tous, et des pratiques démocratiques nouvelles de communication entre citoyens, il semble que dans le domaine de la monnaie et plus généralement de toutes les interactions exigeant de la confiance (engagements, contrats, propriétés, jeux, etc.) tout pourrait fonctionner sans autorité centrale dominante de contrôle.

These applications will come up against the barriers of available energy, a barrier that will become higher and higher, and eventually impassable, with the depletion of fossil fuel resources.

This aspect is, of course, taken into account by Delahaye, who sees the variants of the Nakamoto protocol as tweaking the principle of the blockchain according to protocols which avoid the serious problems posed by energy-expensive mining. These variants are based on limiting the number of miners, leading to private or semi-private blockchains: “Partout dans le monde, elles [les banques] sont aujourd’hui en train de développer et de tester cette technologie issue de l’invention géniale du mystérieux Nakamoto.” (“Throughout the world, [banks] are currently engaged in developing and testing this technology, the result of the extraordinary invention of the mysterious Nakamoto.”)

In flagrant contradiction to the lofty aim of democratic practice is the appropriation of the network by a large number of people with no other link than membership in a network of miners, and the restricted membership of this network. In any case, the difficulty of becoming a miner, as a result of the extremely high investment and operating costs, does, in fact, lead to the emergence of a community of unknown bitcoin miners, which could turn out to be members of government agencies (which is probably the case for many groups of hackers), multinationals, or mafia networks. There is a practical risk that the network will become far removed from any democratic ideal.

This situation is reminiscent of the emergence of free, or pirate, radio and CB radio during the 1970s (and from 1981 onwards in France following the end of the state monopoly). The most profitable of these rapidly became commercial radio stations controlled by financial groups.10

Reading Delahaye’s article leads to other questions which are not merely of an economic nature. Two of them may seem anecdotal in view of my remarks thus far. I will consider them before concluding this letter with a few computing and mathematical aspects that a reader who might not wish to enter into technicalities may safely ignore.

First of all, many currencies possess subunits. The euro has the cent, or centime. The French franc once had the centime—and also the sou, hence the expression “avoir des sous” (to have money). The dollar has the cent—five cent coins are called nickels, ten cent coins are dimes, and twenty-five cent coins are quarters—and so on. As in classic currencies, there is a subunit of the bitcoin that Delahaye does not mention. This is the satoshi, named in communal homage after Satoshi Nakamoto, our mysterious inventor, and a topic to which I shall return shortly.11

The satoshi has an extremely low value. It takes one hundred million satoshi to make a single bitcoin! A satoshi is therefore currently worth about 0.0009 euro cents (other units are also in use, such as the millibitcoin, mBTC or microbitcoin, μBTC—also called a bit—which is worth a hundred satoshi). What is the point of such a small subunit, which seems unusual for a currency? Do its creators think that the value of the bitcoin will soar to the degree that a satoshi will be equivalent to one or more euro cents in a few years time? On the contrary, inflation could lead to devaluation of the currency. There are, however, real world situations in which notes are issued with a high face value to buy current assets. An extreme case is that of Zimbabwe, where in 2009 the central bank issued a $100,000,000,000,000 (one hundred trillion Zimbabwe dollar) note, worth US$30.12 The relationship between the face value of this note and the Zimbabwe dollar is one million times greater than that linking the bitcoin to the satoshi. This example shows that what may seem strange at first sight is sometimes not as strange as certain real world situations.

The second question is related to the creation of bitcoin by an unknown person hiding behind the Satoshi Nakamoto pseudonym. Despite much research, no one knows who he is, or even whether he is still alive. It also not known whether the bitcoins he possesses still exist. Why behave this way? Has there ever been an example of a widely disseminated scientific publication whose true author is unknown? From an economic point of view, Nakamoto’s approach was nothing less than brilliant. He managed to create financial value from the scientific idea of a blockchain without patenting it and without significant investment (since mining the first bitcoins was easy) by applying it to the creation of the bitcoin. His anonymity assures his safety and that of his fortune, which has been valued at eight hundred million euros, without needing to pay for any form of physical protection. This need for security could explain why he has preserved his anonymity, while avoiding the celebrity that he would undoubtedly acquire, and that some have tried to appropriate by pretending to be him. His disappearance, or other unknown motivations, may also explain the continuation of this anonymity.

But, curiously, his example is not unique. Consider the case of Nicolas Bourbaki: a secret group of French-speaking mathematicians who began to write and publish mathematical texts collectively under this pseudonym towards the end of the 1930s.13 For years, the composition of this group, which changed constantly due to a 50-year age limit, remained strictly confidential. This did not prevent its publications from exerting a strong influence on the mathematical community, even into the 1980s.14 Obviously, their secretiveness was not motivated by financial gain. I would have liked to see Delahaye consider this parallel in his article, and perhaps other, similar examples.

The appearance of bitcoin has not only triggered attempts at speculation, it has also engendered an immense amount of reflection across a number of fields. Some mathematicians have found, for example, that

the mathematical part of the founding paper of Satoshi Nakamoto suffers from an approximation. The probability of success of a double expenditure is estimated by coarsely replacing a negative binomial law by a Poisson law, as explained by Meni Rosenfel in 2012.15

I will conclude by examining two further aspects of bitcoin. Delahaye’s article acknowledges that “la mise en place d’un protocole bitcoin doit aussi son existence à la puissance informatique dont chacun dispose aujourd’hui … et qui n’était pas disponible il y a seulement 15 ans ” (“the implementation of a bitcoin protocol also owes its existence to the computing power that each person has today … and which was not available a mere 15 years ago.”)

The first aspect concerns computer development in the coming decades. Will it change the analysis of the importance of blockchains? It is very difficult to know what these computers will be like, as there is much ongoing research into quantum computers, multi-core processors, exaflopic computers, and so on.16 I will consider, within the limited scope of this letter, only the case of the memristor, first conceived in 1971 by Leon Chua at the University of California at Berkeley. A complete theory has recently been published, of which I was a co-author.17 This electronic component is a type of memory that remembers the amount of electricity that flows through it.

Hewlett-Packard (HPE) labs are working to create a computer with a revolutionary architecture called “The Machine” in which, among other things, calculations would be centered around an enormous number of these components. In practice, while memristors, also called ReRAM, have been built, they have not yet reached the stage of industrial production. This may change soon following the creation of an alliance between HPE and SanDisk. The power of a current computing farm will then be available in a simple desktop computer. Within the framework of this new architecture, where data transfer does not occupy the majority of processor working time, what will become of the principle of proofs of work which, according to Delahaye, leads to an arms race among miners?

My last point concerns two essential mathematical components of the bitcoin system: the ECDA transaction signature algorithm, based on the algebraic properties of elliptic curves, and the hash functions, whose importance “in cryptography is such that great care is taken to develop and subject them to all sorts of tests.” In 2002, SHA-256, the hash function used by bitcoin, which operates on 32 bits, became a federal standard for information processing in the United States. Modern cryptography, generally speaking, uses algebraic properties and number theory, as in, for example, the RSA algorithm, which is based on the difficulty of factoring a number having many digits. Current methods have been the subject of intensive cryptanalysis in an effort to break their encoding. The main risk, aside from the (at this stage, hypothetical) threat posed by the advent of quantum computing, is that these methods are based on a very limited number of processes, which will, sooner or later, become vulnerable. This is not really taken into account by Delahaye.

A new form of cryptography is currently being developed, whose foundations are chaos theory and the theory of strange attractors. We can already build hash functions of a particular type based on mechanisms that are radically different to existing algebraic cryptographic methods.18 This cryptography-based chaos has not yet reached maturity, but in the coming years it may well become essential for cryptography, cryptocurrencies, and the Internet of Things (IoT).

René Lozi


Jean-Paul Delahaye replies:

I thank René Lozi for his recollections and thoughts about the origin of currencies and their deep nature. The suppositions Lozi imagines—concerning the voluntary multiplication of digital currencies, similar to bitcoin, by the United States administration in order to solve the problem of its debt—seem to me rather unrealistic. This could not work unless each cryptographic currency they introduced was successful on a comparable scale to that of bitcoin. Given that no cryptographic currency now in competition with bitcoin has even close to a tenth of its capitalization, such a situation seems unlikely.

The letter invokes a

flagrant contradiction [with] the lofty aim of democratic practice [by] the appropriation of the network by a large number of people with no other link than membership in a network of miners, and the restricted membership of this network.

This contradiction does indeed exist, but it is between an initial ideology at the heart of bitcoin, as much as one can know it, and what bitcoin has become in the real world, once its original creators lost control. Bitcoin is not democratic and inequality between holders of bitcoins is much greater than that which exists within our societies between holders of capital.

Still, bitcoin and the blockchain technology exist; it all seems robust and is of central importance for the future of networks, and even the economy. It is this reality, whose future evolution seems so difficult to anticipate, that is our problem today. No one can afford to ignore it.

Regarding the “one-hundred millionth” division of the bitcoin—i.e., the satoshi—its raison d’être is quite simple. Even if the bitcoin were to become extremely valuable, Nakamoto wanted, and hoped, that it could be as useful for small transactions—to buy a coffee!—as it would be for large transactions. Note in this regard that it would be easy enough to change the software so that an even finer division of bitcoin would be usable.


René Lozi is a mathematician in the Laboratoire Jean-Alexandre Dieudonné at the Université de Nice Sophia-Antipolis.

Jean-Paul Delahaye is a mathematician and professor emeritus in computer science at the University of Lille.

Translated from the French by the editors.

  1. Les diverses restrictions imposées par Pékin pour limiter les sorties de capitaux ont fait apparaître le bitcoin comme une alternative pour convertir des Yuans en monnaie étrangère. Elle aurait aussi aiguisé l’appétit des boursicoteurs chinois pour des placements spéculatifs. Résultats, les trois principales plates-formes chinoises d’échange en bitcoins concentrent aujourd’hui plus de 95 % des volumes mondiaux.
    Marie de Vergès, “Montagnes russes pour le bitcoin,” Le Monde, January 7, 2017.  
  2. Wikipedia, “History of Money.” 
  3. Hans Ulrich Vogel, Marco Polo Was in China: New Evidence from Currencies, Salts and Revenues (Leiden, Netherlands: Brill Online Books and Journals, 2012). 
  4. Marie Kostrz, “Combien coûte la fabrication de la monnaie et qui paye ?L’Obs, February 15, 2010. 
  5. Voici le nouveau billet de 50 euros,” Le Parisien, July 5, 2016. 
  6. Sebastiaan Deetman, “Bitcoin Could Consume as Much Electricity as Denmark by 2020,” Motherboard, March 2016. 
  7. Julien Pruvost and Philippine Robert, “Bitcoin : après le boom, le krach,” Liberation, January 19, 2016. 
  8. At most, because we do not know whether Nakamoto’s bitcoins have been lost. 
  9. La dette des Etats-Unis a dépassé les 19 000 milliards de dollars…,” BusinessBourse, February 4, 2016. 
  10. Thierry Lefebvre and Sébastien Poulain, eds., Radios libres, 30 ans de FM : la parole libérée ? (Paris, France: L’Harmattan. 2016). 
  11. Bitcoin Satoshi => USD
  12. Jonathan Chelet, “Le Zimbabwe, ce pays où tous les habitants sont multi-milliardaires,” Capital, June 15, 2016. 
  13. Maurice Mashaal, Bourbaki: A Secret Society of Mathematicians (Paris, France: Pour la Science, 2006). 
  14. All of its members are now known. One of them, Pierre Cartier, published an essay in Inference. See Pierre Cartier, Pierre Cartier, “Alexander Grothendieck: A Country Known Only by Name,” Inference: International Review of Science 1 no. 1 (2014). 
  15. Cyril Grunspan, Ricardo Pérez-Marco, and Dominique Guégan, “Séminaire de cryptofinance : Paiements sécurisés et non-sécurisés sur une blockchain,” Université Paris 8, 12 janvier 2017. 
  16. Adeline Raynal, “La course à l’exaflop, le Saint-Graal des informaticiens,” BFM.TV, May 23, 2016. 
  17. Mohammed-Salah Abdelouahab, René Lozi & Leon O. Chua, “Memfractance: A Mathematical Paradigm for Circuit Elements with Memory,” International Journal of Bifurcation and Chaos 24, no. 9, (2014), 10.1142/S0218127414300237. 
  18. René Lozi, “Emergence of Randomness from Chaos,” International Journal of Bifurcation and Chaos 22, no. 2 (2012), doi:10.1142/S0218127412500216. nbsp;