John Forbes Nash, Jr., by Wikipedia

[admin]

John Nash's Letters to NSA
https://www.nsa.gov/news-features/press ... tters1.pdf
Declassified: 2011

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE 39, MASS.
DEPARTMENT OF MATHEMATICS

Dear Major Grosjean,

I have written RAND concerning the machine description. This was handwritten and was sent to NSA late last spring, I believe, or sent to someone there. Essentially the same machine description was once sent to a Navy communication center in Washington, I think.

I have discussed the machine and general exponential conjecture with R.C. Blanchfield and A.M. Gleason who have worked for NSA. Recently a conversation with Prof. Hoffman here indicated that he has recently been working on a machine with similar objectives. Since he will be consulting for NSA I shall discuss my ideas with him. He has developed minimal redundancy coding methods.

I hope my handwriting, etc. do not give the impression I am just a crank or circle-squarer. My position here is Assist. Prof. of math. My best known work is in game theory (reprint sent separately). I mention these things only in the interest of securing a most careful consideration of the machine and ideas by your most competent associates.

If the machine description does not turn up, I will prepare another. Also I shall be happy to provide any additional information or answer any queries to the best of my ability.

With many thanks for your prompt reply, I am

Sincerely Yours,

John Nash

________________________________________________________________

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE 39, MASS.

DEPARTMENT OF MATHEMATICS

letter concerns ENCIPHERING

Dear Sirs:

An enciphering-deciphering machine (in general outline) of my invention has been sent to your organization by way of the RAND corporation. In this letter I make some remarks on a general principle relevant to enciphering in general and to my machine in particular. This principle seems quite important to me, and I have some reason to believe you may not be fully aware of it.

Consider an enciphering process with a finite “key”, operating on binary messages. Specifically, we can assume the process described by a function

Yi=F(α,α2,...αr;Xi,Xi−1,Xi−2,...Xi−n) where the α’s, X’s, and Y’s are mod2 and where if Xi is changed, with the other X’s and α’s left fixed then Yi is changed.

The α’s denote the “key” containing r bits of information. n is the maximum span of the “memory” of the process. If n were ∞ the arguments given below would not be basically altered.

To consider the resistance of an enciphering process to being broken we should assume that at some times the enemy knows everything but the key being used and to break it need only discover the key from this information.

We see immediately that in principle the enemy needs very little information to begin to break down the process. Essentially, as soon as r bits of enciphered message have been transmitted the key is about determined. This is no security, for a practical key should not be too long. But this does not consider how easy or difficult it is for the enemy to make the computation determining the key. If this computation, although possible in principle, were sufficiently long at best then the process could still be secure in a practical sense.

The most direct computation procedure would be for the enemy to try all 2r possible keys, one by one. Obviously this is easily made impractical for the enemy by simply choosing r large enough.

In many cruder types of enciphering, particularly those which are not auto-coding, such as substitution ciphers [letter for letter, letter pair for letter pair, triple for triple ..] shorter means for computing the key are feasible, essentially because the key can be determined piece meal, one substitution at a time.

So a logical way to classify enciphering processes is by the way in which the computation length for the computation of the key increases with increasing length of the key. This is at best exponential and at worst probably a relatively small power of r, αr2 or αr3, as in substitution ciphers.

Now my general conjecture is as follows: For almost all sufficiently complex types of enciphering, especially where the instructions given by different portions of the key interact complexly with each other in the determination of their ultimate effects on the enciphering, the mean key computation length increases exponentially with the length of the key, or in other words, with the information content of the key.

The significance of this general conjecture, assuming its truth, is easy to see. It means that it is quite feasible to design ciphers that are effectively unbreakable. As ciphers become more sophisticated the game of cipher breaking by skilled teams, etc., should become a thing of the past.

The nature of this conjecture is such that I cannot prove it, even for a special type of cipher. Nor do I expect it to be proven. But this does not destroy its significance. The probability of the truth of the conjecture can be guessed at on the basis of experience with enciphering and deciphering.

If qualified opinions incline to believe in the exponential conjecture, then I think we (the U.S.) can not afford not to make use of it. Also we should try to keep track of the progress of foreign nations towards “unbreakable” types of ciphers.

Since the U.S. presumably does not want other nations to use ciphers we cannot expect to break, this general principle should probably be studied but kept secret.

I believe the enciphering-deciphering machine I invented and had transmitted to the N.S.A. via RAND has this “unbreakable” property. In addition it has several other advantages in that the same physical machine would function both for ciphering and deciphering and that it is auto-synchronizing and recovers after isolated errors in transmission. These properties are not typical of enciphering systems which are autocoding. Also it is suitable for an all electronic, ultra rapid, embodiment.

***

I do not expect any informative answer to this letter, yet it would be nice to have some sort of answer. I would be happy to explain more fully anything which is not clear in my letter, or to amplify on it.

I have been treating my ideas as information deserving some secrecy precautions, yet I feel it is important to communicate them to the right people. I hope the material in this letter can obtain prompt consideration by very highly competent men, versed in the field.

Sincerely,

John Nash
Asst. Prof. Math.

________________________________________________________________

Serial: 531
25 Jan 1955

Mr. John Nash
Department of Mathematics
Massachusetts Institute of Technology
Cambridge 39, Massachusetts

Dear Mr. Nash:

Your recent letter, received 18 January 1955 is noted. Technicians at this Agency recall a very interesting discussion with you which took place approximately four years ago, and will welcome the opportunity to examine your ideas on the subject of cryptography.

A check within this Agency has, unfortunately, disclosed no information on your machine. A description of the principles involved will be appreciated.

Sincerely,

E.M. Gibson
Lt. Col., AGC
Assistant Adj. Gen.

cc: AG
C/S
COMSEC (3)
412

M/R: In Jan 1955, Mr. Nash offered general remarks on cryptography and requested evaluation of descriptive material which he had forwarded through Rand Corp. NSA Ser 236, 12 Jan 55 informed Mr. Nash that the material had not arrived. Mr. Nash in letter rec’d 18 Jan 55 states the material was sent to NSA and to a Navy Communication Center in Wash. late last spring. A check of Agency records and discussions with various individuals (R/D mathematicians and persons who might have had contact with Rand Corp.) within the Agency has uncovered nothing concerning the system. This correspondence requests a description of the machine.

In 1950 Mr. Nash submitted material, in interview, which was evaluated by NSA as not suitable.

M.A. Lyons, 4128, 60372, in
[handwritten: C/Sec 2-2]

________________________________________________________________

Serial: 236
12 Jan 1955

Mr. John Nash
Department of Mathematics
Massachusetts Institute of Technology
Cambridge 39, Massachusetts

Dear Mr. Nash:

Reference is made to your recent letter concerning enciphering processes. The information regarding the general principles has been noted with interest. It will be considered fully, and particularly in connection with your enciphering-deciphering machine.

The description of your machine has not yet been received from the Rand Corporation. As soon as details are received, the machine will be studied to determine whether it is of interest to the Government.

The presentation for appraisal of your ideas for safeguarding communications security is very much appreciated.

Sincerely,

D.M. Grosjean
Major WAC
Actg. Asst. Adjutant General

cc: AG
C/S
COMSEC (3)
412

M/R: Mr. Nash offers remarks on a general principle relevant to enciphering in general and to his machine in particular. The machine, which he is sending via the Rand Corporation, has not yet been received.

This letter informs Mr. Nash that his remarks are being noted and that the machine will be studied as soon as details are received. This reply coordinated with Mr. M. M. Mathews, NSA-31. This is an interim reply.

M. A. Lyons, 4128, 30372, in

C/SEC 202

________________________________________________________________

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE 39, MASS.
DEPARTMENT OF MATHEMATICS

E.M. Gibson, Lt.Col., AGC, Asst. Adj. Gen.

Dear Sir:

Here is a description of my enciphering-deciphering machine.


Transmitting arrangement


Receiving arrangement

In the receiving arrangement the same components are used except for the addition of the retarder, which is a one-unit delay. The messages are to be sequences of binary digits (numbers mod 2). The machines work on a cycling basis, performing certain operations during each cycle.

During each cycle the adder, A, takes in two digits and adds them and sends on the sum obtained from the previous addition. The delay in this addition necessitates the retarder, R, in the receiving circuit.

The permuter will be described in more detail below. It takes in a digit from D during each cycle and also puts out a number. What it does, which is the choice between two permutations is determined by what digit (1 or 0) is in D at the time. The permuter always has a number of digits remembered within it. Each cycle it shuffles them around, changing some 1’s to zeros, sends one digit on, and takes in a digit from D.

In operation the input of the receiver is the output of the transmitter. So the input to R is the same as the input to D in the transmitter. Hence the output of P in the receiver is the same as the out-put of P in the transmitter, except for a one-unit lag.

So the adder A in the receiver gets: (1) the out-put of A in the transmitter, and (2) the previous input from P(trans.) to A(trans). Now since binary addition is the same as binary subtraction (i.e. + & - mod2 are the same) the output of A(receiv.) will be the previous input to A(trans.) from the input to the transmitter, i.e., it will be the clear or unciphered message.

The permuter, P, and decider, D, work as follows, illustrated by example:



The circles represent places where a digit can be stored. During each cycle either the red permutation of digits or the blue takes place. This is decided by the digit in D at the beginning of the cycle. The D digit moves to the first circle or storage place in P during the cycle after it has determined the choice of the permutation.

Both permutations should cycle through all the places in P, so that a digit would be carried through all of them and out under its action alone.

In addition to moving digits around the permutations can change 1’s to 0’s and v.v. For example



represents a shift of the digit in the left circle to the right with this change



The “key” for the enciphering machine is the choice of the permutations. If there are n storage points in P, not counting the first one, which receives the digit from D, then there are [n!2n+1]2 possible keys.

***

I guess I can rely on your people to check on the possession of this machine of the various properties I claimed for it in a previous letter. I hope the correspondence I have sent in receives careful attention from the most qualified people, because I think the basic points involved are very important.

Sincerely,
John Nash
Assist. Prof. Math.

P.S. Various devices could be added to the machine, but I think it would generally be better to enlarge the permuter than to add anything. Of course error-correcting coding could occasionally be a useful adjunct.

________________________________________________________________

Serial: 1358
3 Mar 1955

Mr. John Nash
Department of Mathematics
Massachusetts Institute of Technology
Cambridge 39, Massachusetts

Dear Mr. Nash:

Reference is made to your letter received in this Agency on 17 February 1955.

The system which you describe has been very carefully examined for possible application to military and other government use. It has been found that the cryptographic principles involved in your system, although ingenious, do not meet the necessary security requirements for official application.

Unfortunately it is impossible to discuss any details in this letter. Perhaps in the future another opportunity will arise for discussion of your ideas on the subject of cryptography.

Although your system cannot be adopted, its presentation for appraisal and your generosity in offering it for official use are very much appreciated.

It is regretted that a more favorable reply cannot be given.

Sincerely,
E.M. Gibson
Lt. Col., AGC
Assistant Adj. Gen.

cc: AG
C/S
COMSEC (3)
412

(M/R attached)

M/R: In Jan 55 Mr. Nash offered general remarks on cryptography and requested valuation of descriptive material which he had forwarded through Rand Corp. The Material was not received from Rand Corp. Dr. Campaigne received a letter from Mr. Nash enclosing a copy of the letter (5 Apr 54) from Rand which transmitted this material to NSA. This material was found in R/D files. In the meantime Mr. Nash sent a handwritten description of his enciphering-deciphering machine.

Mr. Nash proposes a permuting cipher-text auto-key principle which has many of the desirable features of a good auto-key system; but it affords only limited security, and requires a comparatively large amount of equipment. The principle would not be used alone in its present form and suitable modification or extension is considered unlikely, unless it could be used in conjunction with other good auto-key principles.

This correspondence informs Mr. Nash that his system does not meet necessary security requirements; and expresses pleasure at the thought of an opportunity to discuss Mr. Nash’s ideas on cryptography again. Such a discussion took place in 1950 when Mr. Nash submitted material, in interview, which was evaluated by NSA as unsuitable.

An interesting pamphlet on Non-Cooperative Games [1], written by Mr. Nash was also sent to this Agency by the author for our information.

Dr. Campaigne has been informed that the reply has been written and is not interested in further coordination.

M.A. Lyons, 4128/60372/rwb

_______________

Notes:

1. Non-cooperative games are the best-known areas of game theory, including such games as the prisoner’s dilemma and where Nash did almost all of his game theory work; results on cooperative games appeared later, are more complex & harder, and accordingly are much more obscure.

[admin]

Secondary
https://www.gwern.net/docs/cs/1955-nash
Accessed: November 5, 2017

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

National Cryptologic Museum Opens New Exhibit on Dr. John Nash, NSA:

When people hear the name John Nash, many recall the movie A Beautiful Mind, in which actor Russell Crowe portrays the mathematical genius whose game-theory research as a graduate student at Princeton University earned him the Nobel Memorial Prize in Economic Sciences in 1994.

The National Cryptologic Museum’s newest exhibit, An Inquisitive Mind: John Nash Letters, features copies of correspondence between Dr. Nash and the National Security Agency (NSA) from the 1950s when he was developing his ideas on an encryption-decryption machine.

At the height of his career in mathematics, Dr. Nash wrote a series of letters to NSA, proposing ideas for such a machine. While the agency acknowledged his ideas, they were never adopted. The letters were preserved with NSA’s analysis in a collection of unsolicited correspondence received in 1955.

Cryptographer Ron Rivest implemented Nash’s cryptosystem in Python and assigned an examination to his class:

Problem 1-2. Cryptosystem proposal by Nash

In the 1950’s the mathematician John Nash (now famous for his work on game theory, and also the subject the movie A Beautiful Mind), privately proposed to the National Security Agency (NSA) an idea for a cryptosystem. This proposal was just declassified (two weeks ago!); a copy of it is available on the class website as Handout # 3, together with some relevant correspondence with the NSA. Note that this proposal was not accepted by the NSA, who said that it didn’t meet their security requirements.

What reasons, if any, can you figure out for this rejection? (Note that this assignment is a bit of an open problem! Do the best you can, and write up what you can figure out. We don’t know the answer to this question ourselves…)

Bloggers on the significance of the letter; Aaron Roth:

In it, John Nash makes the distinction between polynomial time and exponential time, conjectures that there are problems that -cannot- be solved faster than in exponential time, and uses this conjecture as the basis on which the security of a cryptosystem (of his own design) relies. He also anticipates that proving complexity lower bounds is a difficult mathematical problem…These letters predate even Gödel’s [1956] letter to Von Neumann, which goes into much less detail about complexity, and yet has also been taken to anticipate complexity theory and the P vs. NP problem.

Noam Nisan:

He then goes on to put forward an amazingly prescient analysis anticipating computational complexity theory as well as modern cryptography. In the letter, Nash takes a step beyond Shannon’s information-theoretic formalization of cryptography (without mentioning it) and proposes that security of encryption be based on computational hardness - this is exactly the transformation to modern cryptography made two decades later by the rest of the world (at least publicly…). He then goes on to explicitly focus on the distinction between polynomial time and exponential time computation, a crucial distinction which is the basis of computational complexity theory, but made only about a decade later by the rest of the world…He is very well aware of the importance of this conjecture and that it implies an end to the game played between code-designers and code-breakers throughout history. Indeed, this is exactly the point of view of modern cryptography…Surprisingly, for a mathematician, he does not even expect it to be solved. Even more surprisingly he seems quite comfortable designing his encryption system based on this unproven conjecture. This is quite eerily what modern cryptography does to this day: conjecture that some problem is computationally hard; not expect anyone to prove it; and yet base their cryptography on this unproven assumption.

Markus Kroetzsch:

Some caution is needed here…it seems clear to me that Nash did foresee important ideas of modern cryptography. This is great and deserves recognition. However, it seems also very clear that he did not foresee (in fact: could not even imagine) modern complexity theory. Why else would he say that he does not think that the exponential hardness of the problem could ever be proven? It is true that the computational hardness of key tasks in modern cryptography is an unsolved problem, but we have powerful tools to prove hardness results in many other cases. So if Nash had anticipated complexity theory as such, then his remark would mean that he would also have foreseen these difficulties…Overall, it seems clear that a prediction of complexity theory or its current incapacity with respect to cryptographic problems cannot be found in this text, which does by no means diminish the originality of the remarks on cryptography. One could also grant him a certain mathematical intuition that some computational problems could be inherently hard to solve, although I don’t see any hint that he believes that such hardness could ever become a precise mathematical property.

Geoffrey Watson agrees:

This is very interesting as a historical document, but not sure that it supports the interpretations being put on it…a reasonable historical conclusion is that these ideas were just going around in the usual way. The conjecture is a pretty muddled bit of thinking. It presumably means that Nash thinks that there are such exponentially hard ciphers, but to convert almost all sufficiently complex types of enciphering, especially where the instructions given by different portions of the key interact complexly with each other in the determination of their ultimate effects on the enciphering into a prescient anticipation of complexity theory is a big ask.

Shiro Kawai implemented Nash’s system in Scheme.

Philonus Atio comments that (as one might expect for very early work):

I broke it in about 1 hour and I’m no expert in cryptanalysis. It is weak.

Animats suggests why the NSA may’ve rejected the system:

What Nash seems to be describing is a linear feedback shift register. This has potential as a cryptosystem, but isn’t a very good one. As the NSA pointed out, it affords only limited security. When Nash wrote this, Friedman had already developed the theory that allowed general cryptanalysis of rotor-type machines. But that was still highly classified. Friedman, of course, was responsible for breaking the Japanese Purple cipher, plus many others. Before Friedman, cryptanalysis was about guessing. After Friedman, it was about number crunching.

Friedman was the head cryptanalyst at NSA at the time. Within NSA, it would have been known that a linear feedback shift register was a weak key generator. So this idea was, properly, rejected. At least NSA looked at it. Friedman’s hard line on that subject was No new encryption system is worth looking at unless it comes from someone who has already broken a very hard one.

The fact that a problem is NP-hard isn’t enough to make it a good key generator. The Merkle-Hellman knapsack cryptosystem, the first public-key cryptosystem published, is based on an NP-hard problem. But, like many NP-hard problems, it’s only NP-hard in the worst case. The average case is only P-hard. (Linear programming problems, and problems which can be converted to a linear programming problem, are like that.) So that public-key system was cracked. We still don’t have cryptosystems which are provably NP-hard for all cases. Factoring and elliptic curves are as good as it gets, and there’s still the possibility that a breakthrough could make factoring easy.

A Brief History of NP-Completeness, 1954-2012, David S. Johnson 2012:

The other famous mathematician whose letters foreshadowed the theory of NP-completeness was John Nash, Nobel Prize winner for Economics and subject of both the book and the movie A Beautiful Mind. In 1955, Nash sent several handwritten letters about encryption to the United States National Security Agency, which were not declassified and made publicly available until 2012 [1]. In them, he observes that for typical key-based encryption processes, if the plain texts and encrypted versions of some small number of messages are given, then the key is determined. This is not technically correct, since in addition there must be sufficient entropy in the plain texts, but Nash’s arguments apply as well to the problem of finding some key consistent with the encryptions. His central observation was that even if the key is determined, it still may not be easy to find.

If the key is a binary string of length r, exhaustive search will work (as it did for Gödel), but takes time exponential in r. For weak cryptosystems, such as substitution ciphers, there are faster techniques, taking time O(r2) or O(r3), but Nash conjectured that for almost all sufficiently complex types of enciphering, running time exponential in the key length is unavoidable.

This conjecture would imply that P≠NP, since the decryption problem he mentions is polynomial-time equivalent to a problem in NP: Given the data on plain and encrypted texts and a prefix x of a key, is there a key consistent with the encryptions which has x as a prefix? It is a stronger conjecture, however, since it would also rule out the possibility that all problems in NP can, for instance, be solved in time nO(logn) n^{\mathcal{O}(log n)} , which, although non-polynomial, is also not what one typically means by exponential. Nash is also making a subsidiary claim that is in essence about the NP-hardness of a whole collection of decryption problems. This latter claim appears to be false. Nash proposed an encryption scheme of the type he specified, but the NSA observed in private notes that it provided only limited security, and since the publication of the letters modern researchers have found it easy to break [2]. Also, like Gödel, Nash did not make the leap from low-order polynomial time to polynomial time in general. He did however, correctly foresee the mathematical difficulty of the P versus NP problem. He admitted that he could not prove his conjecture, nor did he expect it to be proved, even if it were true.

[admin]

National Cryptologic Museum Opens New Exhibit on Dr. John Nash
by nsa.gov
January 27, 2012
Date Posted: May 3, 2016 | Last Modified: May 3, 2016

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

FORT MEADE, MD - When people hear the name "John Nash," many recall the movie A Beautiful Mind, in which actor Russell Crowe portrays the mathematical genius whose game-theory research as a graduate student at Princeton University earned him the Nobel Memorial Prize in Economic Sciences in 1994.

Renowned mathematician Dr. John Nash wrote a series of letters to NSA in the 1950s proposing a new encryption-decryption machine. Copies of his letters are on display at the National Cryptologic Museum. (NSA photo.)Renowned mathematician Dr. John Nash wrote a series of letters to NSA in the 1950s proposing a new encryption-decryption machine. Copies of his letters are on display at the National Cryptologic Museum. (NSA photo.) The National Cryptologic Museum's newest exhibit, "An Inquisitive Mind: John Nash Letters," features copies of correspondence between Dr. Nash and the National Security Agency (NSA) from the 1950s when he was developing his ideas on an encryption-decryption machine.

At the height of his career in mathematics, Dr. Nash wrote a series of letters to NSA, proposing ideas for such a machine. While the agency acknowledged his ideas, they were never adopted. The letters were preserved with NSA's analysis in a collection of unsolicited correspondence received in 1955.

A check within this Agency has, unfortunately, disclosed no information on your machine. A description of the principles involved will be appreciated.

***

This correspondence requests a description of the machine. In 1950 Mr. Nash submitted material, in interview, which was evaluated by NSA as not suitable.

***

In Jan 55 Mr. Nash offered general remarks on cryptography and requested valuation of descriptive material which he had forwarded through Rand Corp. The Material was not received from Rand Corp. Dr. Campaigne received a letter from Mr. Nash enclosing a copy of the letter (5 Apr 54) from Rand which transmitted this material to NSA. This material was found in R/D files. In the meantime Mr. Nash sent a handwritten description of his enciphering-deciphering machine.

-- John Nash's Letters to NSA

The unclassified letters and the agency's analysis, portions of which were classified, remained protected in NSA's records center until 2011, when the entire collection was reviewed and declassified. The entire collection is being formally accessioned to the National Archives and Records Administration and will be available for public viewing later this year.

Copies of Nash's letters to NSA are on display at the National Cryptologic Museum with complete copies available for review in the museum's library and on the museum's web page. The Nash letters were also recently featured on the National Geographic Channel's (NGC), "Inside the NSA" program. For more information on the program, please visit the NGC web site: http://channel.nationalgeographic.com/channel/.

The National Cryptologic Museum is located at the intersection of Maryland Route 32 and the Baltimore-Washington Parkway (I-295), adjacent to the headquarters of NSA. Hours of operation are 9:00 a.m. to 4:00 p.m. Monday through Friday (except federal holidays), and 10:00 a.m. to 2:00 p.m. on the 1st and 3rd Saturdays of each month.

For more information on this press release, call the NSA Public Affairs Office at 301-688-6524. For information on museum tours, educational programs, and hours and days of operation, click on the National Cryptologic Museum tab at NSA.gov. Admission and parking at the museum are free.

[admin]

John Nash’s Letter to the NSA
by Noam Nisan
February 17, 2012

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

The National Security Agency (NSA) has recently declassified an amazing letter that John Nash sent to it in 1955. It seems that around the year 1950 Nash tried to interest some US security organs (the NSA itself was only formally formed only in 1952) in an encryption machine of his design, but they did not seem to be interested. It is not clear whether some of his material was lost, whether they ignored him as a theoretical professor, or — who knows — used some of his stuff but did not tell him. In this hand-written letter sent by John Nash to the NSA in 1955, he tries to give a higher-level point of view supporting his design:

In this letter I make some remarks on a general principle relevant to enciphering in general and to my machine in particular.

He tries to make sure that he will be taken seriously:

I hope my handwriting, etc. do not give the impression I am just a crank or circle-squarer. My position here is Assist. Prof. of Math. My best known work is in game theory (reprint sent separately).

He then goes on to put forward an amazingly prescient analysis anticipating computational complexity theory as well as modern cryptography. In the letter, Nash takes a step beyond Shannon’s information-theoretic formalization of cryptography (without mentioning it) and proposes that security of encryption be based on computational hardness — this is exactly the transformation to modern cryptography made two decades later by the rest of the world (at least publicly…). He then goes on to explicitly focus on the distinction between polynomial time and exponential time computation, a crucial distinction which is the basis of computational complexity theory, but made only about a decade later by the rest of the world:

So a logical way to classify enciphering processes is by t he way in which the computation length for the computation of the key increases with increasing length of the key. This is at best exponential and at worst probably at most a relatively small power of r, ar^2 or ar^3, as in substitution ciphers.

He conjectures the security of a family of encryption schemes. While not totally specific here, in today’s words he is probably conjecturing that almost all cipher functions (from some — not totally clear — class) are one-way:

Now my general conjecture is as follows: for almost all sufficiently complex types of enciphering, especially where the instructions given by different portions of the key interact complexly with each other in the determination of their ultimate effects on the enciphering, the mean key computation length increases exponentially with the length of the key, or in other words, the information content of the key.

He is very well aware of the importance of this “conjecture” and that it implies an end to the game played between code-designers and code-breakers throughout history. Indeed, this is exactly the point of view of modern cryptography.

The significance of this general conjecture, assuming its truth, is easy to see. It means that it is quite feasible to design ciphers that are effectively unbreakable. As ciphers become more sophisticated the game of cipher breaking by skilled teams, etc., should become a thing of the past.

He is very well aware that this is a conjecture and that he cannot prove it. Surprisingly, for a mathematician, he does not even expect it to be solved. Even more surprisingly he seems quite comfortable designing his encryption system based on this unproven conjecture. This is quite eerily what modern cryptography does to this day: conjecture that some problem is computationally hard; not expect anyone to prove it; and yet base their cryptography on this unproven assumption.

The nature of this conjecture is such that I cannot prove it, even for a special type of ciphers. Nor do I expect it to be proven.

All in all, the letter anticipates computational complexity theory by a decade and modern cryptography by two decades. Not bad for someone whose “best known work is in game theory”. It is hard not to compare this letter to Goedel’s famous 1956 letter to von Neumann also anticipating complexity theory (but not cryptography). That both Nash and Goedel passed through Princeton may imply that these ideas were somehow “in the air” there.

ht: this declassified letter seems to have been picked up by Ron Rivest who posted it on his course’s web-site, and was then blogged about (and G+ed) by Aaron Roth.

Edit: Ron Rivest has implemented Nash’s cryptosystem in Python. I wonder whether modern cryptanalysis would be able to break it.

[admin]

John Nash and a Beautiful Mind on Strike
by Johns Hopkins Medicine
Jun 01, 2007, 7:41 am PDT
© 2007 Johns Hopkins University.

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

John F. Nash, Jr., won the Nobel Prize for economics in 1994 for his early work in developing game theory, but he is best known for being diagnosed with paranoid schizophrenia and for being the subject of the film "A Beautiful Mind," which won an Academy Award for Best Picture in 2001.

Thousands of attendees at the 2007 annual meeting of the American Psychiatric Association in San Diego crowded into a huge hall to see John Nash in person and to listen to him present a lecture.

There was a standing ovation as he walked up to the podium. He proceeded to read a lengthy speech that was difficult to follow at times. He had a somewhat awkward manner and tended to speak in a monotone. However, there was a little humor in his talk.

The focus of Dr. Nash's lecture was what he called "minds on strike," which he described as minds not doing their duty of thinking in an acceptable fashion. In his wide-ranging remarks, he talked about the nervous system, evolution, economics, and a grand design for terrestrial species.

As species evolved on Earth into more complex organisms, he said, there arose a need for a differentiation of roles. He tied this together with appropriate social behaviors, noting that society is the collective of individuals interacting with nature.

Nash pointed out that even non-standard behaviors may involve roles that are important for society, and he offered the clergy as an example. He suggested that medical doctors do not produce food, but ultimately are important for society, as well.

Nash went on to discuss insanity from an economic standpoint, noting that because insane people might not be able to perform their expected social roles, their minds are unable to function properly for society and are, therefore, "on strike." He said that his mind, for instance, had gone on strike repeatedly; however, his message was one of hope because he left open the possibility of the mind functioning properly again.

He did speak about his own delusions and suggested that these beliefs were as real as anything else he had experienced. The delusional thoughts came to him as naturally as other information. He had both paranoid and grandiose delusional thoughts.

At his most delusional, he believed that there was a conspiracy and that he was being monitored in a police-state world like that described in George Orwell's novel "1984." He believed that family members who had aided in having him placed in psychiatric hospitals were simply duped by the authorities. He also sensed that he was a very important person, and he noted that when he was awarded the Nobel Prize, history finally caught up with his grandiose delusion.

Especially interesting were his thoughts about mental health counselors. During the years that his psychotic symptoms were very pronounced, he believed that if he could communicate with a real angel, then everything would be clarified for him. He now thinks that someday a very complex computer program will be able to simulate a counselor and be useful for some therapy sessions.

At the end of his remarks, he argued that nature favors diversity. The inevitable result of diversity in human behavior is that the human being is vulnerable to insanity. It is a price that society pays for progress in other ways.

An important lesson of John Nash's life is that genius does not protect one against mental illness.

[admin]

John Forbes Nash
by J J O'Connor and E F Robertson
history.mcs.st-andrews.ac.uk
Accessed November 5, 2017

NOTICE: THIS WORK MAY BE PROTECTED BY COPYRIGHT

YOU ARE REQUIRED TO READ THE COPYRIGHT NOTICE AT THIS LINK BEFORE YOU READ THE FOLLOWING WORK, THAT IS AVAILABLE SOLELY FOR PRIVATE STUDY, SCHOLARSHIP OR RESEARCH PURSUANT TO 17 U.S.C. SECTION 107 AND 108. IN THE EVENT THAT THE LIBRARY DETERMINES THAT UNLAWFUL COPYING OF THIS WORK HAS OCCURRED, THE LIBRARY HAS THE RIGHT TO BLOCK THE I.P. ADDRESS AT WHICH THE UNLAWFUL COPYING APPEARED TO HAVE OCCURRED. THANK YOU FOR RESPECTING THE RIGHTS OF COPYRIGHT OWNERS.

Born: 13 June 1928 in Bluefield, West Virginia, USA
Died: 23 May 2015 in New Jersey, USA

John F Nash's father, also called John Forbes Nash so we shall refer to him as John Nash Senior, was a native of Texas. John Nash Senior was born in 1892 and had an unhappy childhood from which he escaped when he studied electrical engineering at Texas Agricultural and Mechanical. After military service in France during World War I, John Nash Senior lectured on electrical engineering for a year at the University of Texas before joining the Appalachian Power Company in Bluefield, West Virginia. John F Nash's mother, Margaret Virginia Martin, was known as Virginia. She had a university education, studying languages at the Martha Washington College and then at West Virginia University. She was a school teacher for ten years before meeting John Nash Senior, and the two were married on 6 September 1924.

Johnny Nash, as he was called by his family, was born in Bluefield Sanitarium and baptised into the Episcopal Church. He was [2]:-

... a singular little boy, solitary and introverted ...

but he was brought up in a loving family surrounded by close relations who showed him much affection. After a couple of years Johnny had a sister when Martha was born. He seems to have shown a lot of interest in books when he was young but little interest in playing with other children. It was not because of lack of children that Johnny behaved in this way, for Martha and her cousins played the usual childhood games: cutting patterns out of books, playing hide-and-seek in the attic, playing football. However while the others played together Johnny played by himself with toy airplanes and matchbox cars.

His mother responded by enthusiastically encouraging Johnny's education, both by seeing that he got good schooling and also by teaching him herself. Johnny's father responded by treating him like an adult, giving him science books when other parents might give their children colouring books.

Johnny's teachers at school certainly did not recognise his genius, and it would appear that he gave them little reason to realise that he had extraordinary talents. They were more conscious of his lack of social skills and, because of this, labelled him as backward. Although it is easy to be wise after the event, it now would appear that he was extremely bored at school. By the time he was about twelve years old he was showing great interest in carrying out scientific experiments in his room at home. It is fairly clear that he learnt more at home than he did at school.

Martha seems to have been a remarkably normal child while Johnny seemed different from other children. She wrote later in life (see [2]):-

Johnny was always different. [My parents] knew he was different. And they knew he was bright. He always wanted to do things his way. Mother insisted I do things for him, that I include him in my friendships. ... but I wasn't too keen on showing off my somewhat odd brother.

His parents encouraged him to take part in social activities and he did not refuse, but sports, dances, visits to relatives and similar events he treated as tedious distractions from his books and experiments.

Nash first showed an interest in mathematics when he was about 14 years old. Quite how he came to read E T Bell's Men of Mathematics is unclear but certainly this book inspired him. He tried, and succeeded, in proving for himself results due to Fermat which Bell stated in his book. The excitement that Nash found here was in contrast to the mathematics that he studied at school which failed to interest him.

He entered Bluefield College in 1941 and there he took mathematics courses as well as science courses, in particular studying chemistry, which was a favourite topic. He began to show abilities in mathematics, particularly in problem solving, but still with hardly any friends and behaving in a somewhat eccentric manner, this only added to his fellow pupils view of him as peculiar. He did not consider a career in mathematics at this time, however, which is not surprising since it was an unusual profession. Rather he assumed that he would study electrical engineering and follow his father but he continued to conduct his own chemistry experiments and was involved in making explosives which led to the death of one of his fellow pupils. [2]:-

Boredom and simmering adolescent aggression led him to play pranks, occasionally ones with a nasty edge.

He caricatured classmates he disliked with weird cartoons, enjoyed torturing animals, and once tried to get his sister to sit in a chair he had wired up with batteries.

Nash won a scholarship in the George Westinghouse Competition and was accepted by the Carnegie Institute of Technology (now Carnegie-Mellon University) which he entered in June 1945 with the intention of taking a degree in chemical engineering. Soon, however, his growing interest in mathematics had him take courses on tensor calculus and relativity. There he came in contact with John Synge who had recently been appointed as Head of the Mathematics Department and taught the relativity course. Synge and the other mathematics professors quickly recognised Nash's remarkable mathematical talents and persuaded him to become a mathematics specialist. They realised that he had the talent to become a professional mathematician and strongly encouraged him.

Nash quickly aspired to great things in mathematics. He took the William Lowell Putnam Mathematics Competition twice but, although he did well, he did not make the top five. It was a failure in Nash's eyes and one which he took badly. The Putnam Mathematics Competition was not the only thing going badly for Nash. Although his mathematics professors heaped praise on him, his fellow students found him a very strange person. Physically he was strong and this saved him from being bullied, but his fellow students took delight in making fun of Nash who they saw as an awkward immature person displaying childish tantrums. One of his fellow students wrote:-

He was a country boy unsophisticated even by our standards. He behaved oddly, playing a single chord on a piano over and over, leaving a melting ice cream cone melting on top of his cast-off clothing, walking on his roommate's sleeping body to turn off the light.

Another wrote:-

He was extremely lonely.

And a third fellow student wrote:-

We tormented poor John. We were very unkind. We were obnoxious. We sensed he had a mental problem.

He showed homosexual tendencies, climbing into bed with the other boys who reacted by making fun of the fact that he was attracted to boys and humiliated him. They played cruel pranks on him and he reacted by asking his fellow students to challenge him with mathematics problems. He ended up doing the homework of many of the students.

Nash received a BA and an MA in mathematics in 1948. By this time he had been accepted into the mathematics programme at Harvard, Princeton, Chicago and Michigan. He felt that Harvard was the leading university and so he wanted to go there, but on the other hand their offer to him was less generous than that of Princeton. Nash felt that Princeton were keen that he went there while he felt that his lack of success in the Putnam Mathematics Competition meant that Harvard were less enthusiastic. He took a while to make his decision, while he was encouraged by Synge and his other professors to accept Princeton. When Lefschetz offered him the most prestigious Fellowship that Princeton had, Nash made his decision to study there.

In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures. Usually those who decide not to learn through lectures turn to books but this appears not to be so for Nash, who decided not to learn mathematics "second-hand" but rather to develop topics himself. In many ways this approach was successful for it did contribute to him developing into one of the most original of mathematicians who would attack a problem in a totally novel way.

In 1949, while studying for his doctorate, he wrote a paper which 45 years later was to win a Nobel prize for economics. During this period Nash established the mathematical principles of game theory. P Ordeshook wrote:-

The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory. ... Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people.

Milnor, who was a fellow student, describes Nash during his years at Princeton in [6]:-

He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. However, my most vivid memory of this time is of the many games which were played in the common room. I was introduced to Go and Kriegspiel, and also to an ingenious topological game which we called Nash in honor of the inventor.

In fact the game "Nash" was almost identical to Hex which had been invented independently by Piet Hein in Denmark.

Here are three comments from fellow students:-

Nash was out of the ordinary. If he was in a room with twenty people, and they were talking, if you asked an observer who struck you as odd it would have been Nash. It was not anything he consciously did. It was his bearing. His aloofness.

Nash was totally spooky. He wouldn't look at you. he'd take a lot of time answering a question. If he thought the question was foolish he wouldn't answer at all. He had no affect. It was a mixture of pride and something else. He was so isolated but there really was underneath it all a warmth and appreciation of people.

A lot of us would discount what Nash said. ... I wouldn't want to listen. You didn't feel comfortable with the person.

He had ideas and was very sure they were important. He went to see Einstein not long after he arrived in Princeton and told him about an idea he had regarding gravity. After explaining complicated mathematics to Einstein for about an hour, Einstein advised him to go and learn more physics. Apparently a physicist did publish a similar idea some years later.

In 1950 Nash received his doctorate from Princeton with a thesis entitled Non-cooperative Games. In the summer of that year he worked for the RAND Corporation where his work on game theory made him a leading expert on the Cold War conflict which dominated RAND's work. He worked there from time to time over the next few years as the Corporation tried to apply game theory to military and diplomatic strategy. Back at Princeton in the autumn of 1950 he began to work seriously on pure mathematical problems. It might seem that someone who had just introduced ideas which would, one day, be considered worthy of a Nobel Prize would have no problems finding an academic post. However, Nash's work was not seen at the time to be of outstanding importance and he saw that he needed to make his mark in other ways. We should also note that it was not really a move towards pure mathematics for he had always considered himself a pure mathematician. He had already obtained results on manifolds and algebraic varieties before writing his thesis on game theory. His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis. He said in a recent interview:-

I developed a very good idea in pure mathematics. I got what became Real Algebraic Manifolds. I could have published that earlier, but it wasn't rushed to publication. I took some time in writing it up. Somebody suggested that I was a prodigy. Another time it was suggested that I should be called "bug brains", because I had ideas, but they were sort of buggy or not perfectly sound. So that might have been an anticipation of mental problems. I mean, taking it at face value.

In 1952 Nash published Real Algebraic Manifolds in the Annals of Mathematics. The most important result in this paper is that two real algebraic manifolds are equivalent if and only if they are analytically homeomorphic. Although publication of this paper on manifolds established him as a leading mathematician, not everyone at Princeton was prepared to see him join the Faculty there. This was nothing to do with his mathematical ability which everyone accepted as outstanding, but rather some mathematicians such as Artin felt that they could not have Nash as a colleague due to his aggressive personality.

Halmos received the following letter in early 1953 from Warren Ambrose relating to Nash (see for example [2]):-

There's no significant news from here, as always. Martin is appointing John Nash to an Assistant Professorship (not the Nash at Illinois, the one out of Princeton by Steenrod) and I'm pretty annoyed at that. Nash is a childish bright guy who wants to be "basically original," which I suppose is fine for those who have some basic originality in them. He also makes a damned fool of himself in various ways contrary to this philosophy. He recently heard of the unsolved problem about imbedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently worthwhile to justify his efforts; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium. Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a system of partial differential equations and if he could only [get] to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper - and Nash had only the vaguest notions about the whole thing. So it is generally conceded he is getting nowhere and making an even bigger ass of himself than he has been previously supposed by those with less insight than myself. But we've got him and saved ourselves the possibility of having gotten a real mathematician. He's a bright guy but conceited as Hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y.

Ambrose, the author of this letter, and Nash had rubbed each other the wrong way for a while. They had played silly pranks on each other and Ambrose seems not to have been able to ignore Nash's digs in the way others had learned to do. It had been Ambrose who had said to Nash:-

If you're so good, why don't you solve the embedding theorem for manifolds.

From 1952 Nash had taught at the Massachusetts Institute of Technology but his teaching was unusual (and unpopular with students) and his examining methods were highly unorthodox. His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics. His paper C1 isometric imbeddings was published in 1954 and Chern, in a review, noted that it:-

... contains some surprising results on the C1-isometric imbedding into an Euclidean space of a Riemannian manifold with a positive definite C0-metric.

Nash continued to develop this work in the paper The imbedding problem for Riemannian manifolds published in 1956. This paper contains his famous deep implicit function theorem. After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958. Nash, however, was very disappointed when he discovered that E De Giorgi had proved similar results by completely different methods.

The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it. One imagines that the Committee would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields' Medal but mental illness destroyed his career long before those decisions were made.

During his time at MIT Nash began to have personal problems with his life which were in addition to the social difficulties he had always suffered. Colleagues said:-

Nash was always forming intense friendships with men that had a romantic quality. He was very adolescent, always with the boys. He was very experimental - mostly he just kissed.

He met Eleanor Stier and they had a son, John David Stier, who was born on 19 June 1953. Eleanor was a shy girl, lacking confidence, a little afraid of men, did not want to be involved. She found in Nash someone who was even less experienced than she was and found that attractive. [2]:-

Nash was looking for emotional partners who were more interested in giving than receiving, and Eleanor, was very much that sort.

Nash did not want to marry Eleanor although she tried hard to persuade him. In the summer of 1954, while working for RAND, Nash was arrested in a police operation to trap homosexuals. He was dismissed from RAND.

One of Nash's students at MIT, Alicia Larde, became friendly with him and by the summer of 1955 they were seeing each other regularly. He also had a special friendship with a male graduate student at this time: Jack Bricker. Eleanor found out about Alicia in the spring of 1956 when she came to Nash's house and found him in bed with Alicia. Nash said to a friend:-

My perfect little world is ruined, my perfect little world is ruined.

Alicia did not seem too upset at discovering that Nash had a child with Eleanor and deduced that since the affair had been going on for three years, Nash was probably not serious about her. In 1956 Nash's parents found out about his continuing affair with Eleanor and about his son John David Stier. The shock may have contributed to the death of Nash's father soon after, but even if it did not Nash may have blamed himself. In February of 1957 Nash married Alicia; by the autumn of 1958 she was pregnant but, a couple of months later near the end of 1958, Nash's mental state became very disturbed.

At a New Year's Party Nash appeared at midnight dressed only with a nappy and a sash with "1959" written on it. He spent most of the evening curled up, like the baby he was dressed as, on his wife's lap. Some described his behaviour as stranger than usual. On 4 January he was back at the university and started to teach his game theory course. His opening comments to the class were:-

The question occurs to me. Why are you here?

One student immediately dropped the course! Nash asked a graduate student to take over his course and vanished for a couple of weeks. When he returned he walked into the common room with a copy of the New York Times saying that it contained encrypted messages from outer space that were meant only for him. For a few days people thought he was playing an elaborate private joke.

Norbert Wiener was one of the first to recognize that Nash's extreme eccentricities and personality problems were actually symptoms of a medical disorder. After months of bizarre behaviour, Alicia had her husband involuntarily hospitalised at McLean Hospital, a private psychiatric hospital outside of Boston. Upon his release, Nash abruptly resigned from MIT, withdrew his pension, and went to Europe, where he intended to renounce his US citizenship. Alicia left her newborn son with her mother, and followed the ill Nash. She then had Nash deported - back to the United States.

After their return, the two settled in Princeton where Alicia took a job. Nash's illness continued, transforming him into a frightening figure. He spent most of his time hanging around on the Princeton campus, talking about himself in the third person as Johann von Nassau, writing nonsensical postcards and making phone calls to former colleagues. They stoically listened to his endless discussions of numerology and world political affairs. Her husband's worsening condition depressed Alicia more and more.

In January 1961 the despondent Alicia, John's mother, and his sister Martha made the difficult decision to commit him to Trenton State Hospital in New Jersey where he endured insulin-coma therapy, an aggressive and risky treatment, five days a week for a month and a half. A long sad episode followed which included periods of hospital treatment, temporary recovery, then further treatment. Alicia divorced Nash in 1962. Nash spent a while with Eleanor and John David. In 1970 Alicia tried to help him taking him in as a boarder, but he appeared to be lost to the world, removed from ordinary society, although he spent much of his time in the Mathematics Department at Princeton. The book [2] is highly recommended for its moving account of Nash's mental sufferings.

Slowly over many years Nash recovered. He delivered a paper at the tenth World Congress of Psychiatry in 1996 describing his illness; it is reported in [3]. He was described in 1958 as the:-

... most promising young mathematician in the world ...

but he soon began to feel that:-

... the staff at my university, the Massachusetts Institute of Technology, and later all of Boston were behaving strangely towards me. ... I started to see crypto-communists everywhere ... I started to think I was a man of great religious importance, and to hear voices all the time. I began to hear something like telephone calls in my head, from people opposed to my ideas. ...The delirium was like a dream from which I seemed never to awake.

Despite spending periods in hospital because of his mental condition, his mathematical work continued to have success after success. He said:-

I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia.

In the 1990s Nash made a recovery from the schizophrenia from which he had suffered since 1959. His ability to produce mathematics of the highest quality did not totally leave him. He said:-

I would not treat myself as recovered if I could not produce good things in my work.

Nash was awarded (jointly with Harsanyi and Selten) the 1994 Nobel Prize in Economic Science for his work on game theory. In 1999 he was awarded the Leroy P Steele Prize by the American Mathematical Society:-

... for a seminal contribution to research.

Nash and Louis Nirenberg were awarded the Abel prize in 2015 for:

... striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.

A few days after picking up the prize in Norway, Nash and his wife Alicia were killed in an accident to their taxi on the New Jersey turnpike.

[ Louis Nirenberg] We flew back together, he and his wife and I, they had changed our flight, and so there was nobody to meet him or to meet me. I managed to contact my daughter who came and picked me up. We chatted for an hour at the airport. And when my daughter came, they said, “Well, I’ll take a taxi.” And then the horrible tragedy.

[John Nash] We’re going back to the U.S. on the airplane tomorrow when my [illegible]. That’s fate.

-- A Mind on Strike: John Nash Revisited, by Jim Rakete

_______________

References:

Biography in Encyclopaedia Britannica.
http://www.britannica.com/biography/John-F-Nash-Jr
Books:
S Nasar, A beautiful mind (London-New York, 1998).

Articles:

H W Kuhn, Introduction : Dedication to John F. Nash, Jr., A celebration of John F Nash, Jr., Duke Math. J. 81 (1) (1995), i-v.
H W Kuhn, J C Harsanyi, R Selten, J W Weibull, E van Damme, J F Nash Jr. and P Hammerstein,The work of John F Nash, Jr. in game theory, A celebration of John F Nash, Jr., Duke Math. J. 81 (1) (1995),1-29.
Madness and mathematics, The Times London, 28 Aug, 1996.
J Milnor, A Nobel prize for John Nash, The Mathematical Intelligencer 17 (3) (1995), 11-17.
John F Nash : 1999 Steele prizes, Notices Amer. Math. Soc. 46 (4) (1999), 457-462.
http://www.ams.org/notices/199904/comm-steele-prz.pdf
The work of John Nash in game theory: Nobel Seminar, December 8, 1994, J. Econom. Theory 69 (1) (1996), 153-185.
G Umbhauer, John Nash, un visionnaire de l'économie, Gaz. Math. 65 (1995), 47-69.
E van Damme, On the contributions of John C Harsanyi, John F Nash and Reinhard Selten, Internat. J. Game Theory 24 (1) (1995), 3-11.