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148 Pages·2007·17.07 MB·English

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Quadratic Diophantine Equations Fibonacci’s De Practica e
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T. Andreescu , University of Dallas, Dallas, TX, USA; D . Andrica , Geometrie ic
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Notices
of the American Mathematical Society
December 2007 d b
a 1
Communications
c
1
1494 WHAT IS...a Cluster
Algebra?
Andrei Zelevinsky
1500 Emma Lehmer (1906–2007)
John Brillhart
σ = −1,σ˘ = 0
1502 Remembering Philip 1477 1458
Rabinowitz
Philip J. Davis and Aviezri S.
Fraenkel
1507 How to Write Your First
Paper
Features
Steven G. Krantz
1512 Sobolev Institute of
Mathematics Celebrates Its 1458 Starting with the Group SL (R)
2
Fiftieth Anniversary
Victor Alexandrov Vladimir V. Kisil
The group of two by two real matrices of
Commentary determinant one is a well studied elementary math-
ematical object. The author considers some of its less
1454 Letters to the Editor well known invariants and the geometries they imply.
1468 The Triumph of Numbers
and Karl Pearson—Book
1477 Irving Kaplansky (1917–2006)
Reviews
Reviewed by Brian Blank
Hyman Bass and T. Y. Lam
1497 How Mathematicians
Colleagues, students, and family recall the life and work
Think—A Book Review
of the late American mathematician.
Reviewed by Reuben Hersh
Notices Departments
About the Cover .........................................1516
of the American Mathematical Society
Mathematics People .....................................1517
EDITOR: Andy Magid
Kreps Awarded 2007 CME/MSRI Prize, Singapore National Science and
ASSOCIATE EDITORS: Technology Awards Given, NDSEG Fellowships Awarded,
Daniel Biss, Susanne C. Brenner, Bill Casselman
(Graphics Editor), Robert J. Daverman, Susan B. H. Neumann Award Given, Pi Mu Epsilon Student Paper Presentation
Friedlander, Robion Kirby, Steven G. Krantz, Awards.
Lisette de Pillis, Peter Sarnak, Mark Saul, John
Swallow, Lisa Traynor Mathematics Opportunities ...............................1519
SENIOR WRITER and DEPUTY EDITOR: NSF Computing Equipment and Instrumentation Programs, NRC-
Allyn Jackson
Ford Foundation Diversity Fellowships, NDSEG Fellowships, National
MANAGING EDITOR: Sandra Frost
Academies Graduate Fellowship Program, Call for Nominations for
CONTRIBUTING WRITER: Elaine Kehoe Waterman Award, MfA Fellowship Program, CMI Liftoff Program for
PRODUCTION ASSISTANT: Muriel Toupin Summer 2008, Call for Applications for Kovalevskaja Award, Virtual
PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Exhibit at the Smithsonian, ETS Visiting Scholars Program.
Murphy, Lori Nero, Karen Ouellette, Donna Salter,
Deborah Smith, Peter Sykes Inside the AMS ..........................................1522
ADVERTISING SALES: Anne Newcomb Trjitzinsky Memorial Awards Presented, Epsilon Scholarships Awarded
for 2007, Deaths of AMS Members.
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09/04
Letters to the Editor
Koblitz Article Misleading the paper due to my strong disagree- reality, the precise statement of a
I found Koblitz’s essay “The uneasy ment with its contents, but rather due theorem is crucial to its practical
relationship between mathematics to the nature of this paper which, in meaning. Indeed, while we all know
and cryptography” (Notices, Vol. 54, my opinion, is not a novel technical that the impossibility of angle trisec-
No. 8) misleading in several ways. contribution of the type sought by tion depends on the precise defini-
Most importantly, I believe that the journal. My opinion was that the tion of allowed operations, none of us
Koblitz’s views regarding the subject paper may only be published as a relies on this theorem to protect our
are based on several fundamental “position paper”. Since the authors credit card information. Here indeed
misconceptions. For example, he refused to revise the title of their cryptographers have sometimes mis-
seems to view the unfortunate (and paper accordingly, the editor-in-chief stepped and inadequately modeled
rare) cases in which flaws were found was forced to write a special preface the scenarios in which systems could
in published claimed “proofs” (of se- that explains that their paper is a be attacked, leading to systems that
curity) as indication that proofs are position paper. regardless of their formal analysis
useless (w.r.t. security). In my opin- were insecure in practice. But the
ion, these incidences merely reinforce —Oded Goldreich problem is not inherently with proofs
the importance of careful verification Weizmann Institute of Science of security but rather with cryptogra-
of proofs, which constitute our only [email protected] phy itself, a notoriously difficult sub-
way of distinguishing facts from con- ject which over its long history has
jectures. Furthermore, Koblitz often (Received August 27, 2007) seen many great minds miss subtle
confuses proofs with what is being points and design systems that were
proved, and consequently does not eventually broken.
Koblitz Misrepresents
distinguish between the inadequacy In fact, the only way to systemati-
Cryptography
of the claim (e.g., an unsatisfactory cally improve practical security is to
definition of security) and the in- In the famous joke, a mathematician insist on precise modeling, and study
correctness of its proof. Finally, he would not infer the color of a sheep’s these models using mathematical
often uses unsound reasoning (e.g., right side from its left side. But Neal proofs, on the way refining the mod-
inferring that last-minute conference Koblitz, in his article on “The uneasy els and identifying and correcting
submissions indicate a rush to pub- relationship between mathematics subtle weaknesses in protocols. In-
lish minor results). and cryptography” makes quite a few deed, Koblitz’s anecdote on the MQV
The foregoing flaws dominate broad generalizations from a handful and HMQV protocols demonstrates
the series of papers by Koblitz and of anecdotes. precisely how careful definitions and
Menezes (see references in Koblitz’s Koblitz’s disparagement of secu- insistence on proofs can direct an
essay). For a discussion of the main rity proofs is particularly misleading. incremental process towards more
flaws, the interested reader is re- Proofs of security of cryptographic secure protocols.
ferred to my essay http://eprint. protocols are standard mathemati-
iacr.org/2006/461. Let me just cal proofs and in that sense are no —Boaz Barak
stress that, in contrary to Koblitz’s more “over-hyped” (to use Koblitz’s Princeton University
belief, the fact that this essay does term) than proofs in calculus. Ko- [email protected]
not criticize the papers of Koblitz blitz gives examples of mistakes in
and Menezes for inadequate refer- security proofs, but as we know such (Received August 30, 2007)
ences to prior work does not mean examples can be found in any area of
that such cases are not numerous. mathematics. He also criticizes these
On the contrary. Koblitz’s essay suf- proofs for relying on unproven con- Publication of Koblitz’s Article
fers from the same problems, and jectures. This is indeed most often Questioned
in addition it provides a distorted the case, as is not surprising in such I was shocked and dismayed that the
account of my own essay (e.g., the a young and vibrant field. Eventually AMS Notices published Neal Koblitz’s
(legitimate) controversy regarding we might prove these conjectures article [“The uneasy relationship be-
the “Random Oracle Model” is far (although some seem as hard as the tween mathematics and cryptog-
from being the focus of my essay and hardest open problems in mathemat- raphy”, September 2007] without,
was certainly not the source of my ics) but regardless, it’s much better to apparently, any editorial oversight.
concerns regarding the Koblitz and use a protocol proven secure under As one who works in the field of
Menezes papers). a well-defined and widely believed “provable security”, I vehemently
I also wish to correct Koblitz’s conjecture than a protocol with no disagree with Koblitz’s main argu-
account of the events related to the analysis at all. ment—more on this below—but this
publication of his paper with Mene- Koblitz points out the obvious is not my primary complaint. Instead,
zes in the Journal of Cryptography. I truth that in cryptography, as in what I found abhorrent is that the
did not object to the publication of any mathematical field that models article crosses the line from academic
1454 Notices of the AMs VoluMe 54, NuMber 11
Letters to the Editor
argument to personal screed, from of publishing “contributions” of this driven design” methodology is a tes-
constructive criticism to belliger- nature. tament to the fundamental role of the
ent name-calling. I cannot imagine theory of cryptography in bringing
the Notices publishing a similarly —Jonathan Katz more secure systems to practice.
disparaging article about any other University of Maryland There is no better way to as-
academic discipline. [email protected] sess the value of the HMQV pro-
By another fault of the editors,
tocol than reading the paper itself
readers were not given the oppor- (Received August 30, 2007
posted under http://eprint.iacr.
tunity to read a companion article
org/2005/176. In particular, the in-
containing a countervailing point of Koblitz’s Arguments
view. Without dissecting Koblitz’s Disingenuous troduction and concluding remarks
section in the paper, unchanged since
arguments point-by-point, let me Addressing Neal Koblitz’s disingenu-
the original publication, already con-
assure readers that proofs in mod- ous arguments against theoretical
tain answers to many of the points
ern cryptography are as meaningful cryptography in his recent article in
raised by Koblitz against our meth-
as proofs in any other field. Can a the Notices requires far more elabora-
scheme that has been proven secure tion than allowed by the space allo- odology. Also note the preface where
still succumb to a real-world attack? cated for this letter (see http://www. I comment on a correction pointed
Yes, but this does not invalidate the ee.technion.ac.il/~hugo/ams- out by Alfred Menezes that, contrary
proof. (A proof is given with respect letter). Let me thus focus only on to Koblitz’s misleading account, did
to a particular definition; any single some of Koblitz’s unfounded claims not change in any essential way the
definition is not appropriate for all against my work on the HMQV proto- results and value of the work, neither
possible environments in which a col that he uses as a way to discredit with respect to its provability nor
scheme may be deployed.) Are most the entire field of complexity-based the substantial practical benefits of
results in cryptography conditional? cryptography (what he refers to as HMQV.
Yes, but this has been shown to be “provable security”) and to deny the Let me end by stressing a very
inherent until the P​ vs. N​P​ question significant achievements of this field, important point in understanding the
is settled, and should not hold back in particular its important contribu-
role of theory when designing and
research. Do mistakes happen? Oc- tions to the practice of cryptogra-
analyzing real-world cryptographic
casionally, though rarely. But this phy.
systems: By its very nature, there
surely does not diminish the impor- Contrary to what Koblitz claims,
is no (and cannot be) empirical evi-
tance of proofs in the first place. the HMQV work represents a prime
dence for the security of a design.
Frankly, I cannot understand why example of the success of theoreti-
Indeed, no concrete measurements
any mathematician would discour- cal cryptography, not only in laying
or simulations can show that attacks
age the use of definitions, proofs, rigorous mathematical foundations
against a cryptographic scheme are
and formal reasoning in any field. for cryptography at large, but also in
(Indeed, these elements have helped its ability to guide us in the design not feasible. The only way to do so
cryptography progress from an art of truly practical solutions to real- is to develop a formal mathemati-
to a science.) Koblitz’s article clari- world problems. Indeed, the HMQV cal model and language in which
fies his motivation: sheer elitism. key-agreement protocol that resulted to reason about such schemes. The
According to Koblitz, cryptographers from this work not only improved area of theoretical cryptography and
publish papers of “little originality” significantly on its predecessor, the its applications has been remarkably
and containing “tiny improvements”; MQV protocol, in terms of analysis successful in developing such mod-
when we do publish something of and security guarantees, but the pro- els. They are certainly not perfect
potential interest, it is likely to be tocol itself became more practical, and will be further improved over
wrong. According to Koblitz, cryp- improving performance and lowering time, but the foundations laid so
tographers are simply incapable of the dependency on external mecha-
far are outstanding. Whoever finds
writing correct proofs, hence his nisms such as trust in certification
them insufficient should be encour-
admonition that anyone other than authorities and key derivation func-
aged to improve upon them or come
“trained mathematicians” simply tions.
up with alternatives. Emotional and
give up on the goal. This is snobbery This double improvement, in both
unfounded attacks against a whole
at its purest. security and performance, is no coin-
research area and its individuals, as
Publication of Koblitz’s article cidence. It is the very understanding
carried by Koblitz, are of no use.
has the potential to cause serious that one obtains through the process

damage: not to the field of cryptog- of formally proving (or disproving) a
raphy—which will continue to do cryptographic protocol that allows —Hugo Krawczyk
fine with or without Koblitz’s sup- us to eliminate safety margins that IBM T. J. Watson Research Center
port—but to the future involvement are often added to cryptographic [email protected]
of mathematicians in this field. In schemes when there is not enough
the future, the editors should more confidence in the strength of the (Received September 3, 2007)
carefully weight the pros and cons design. The success of this “proof-
DeceMber 2007 Notices of the AMs 1455
Letters to the Editor
ASSISTANT, ASSOCIATE OR FULL PROFESSOR
(2 POSITIONS)
CENTER FOR ALGORITHMS AND Reply to Katz, Goldreich, and HMQV protocol had been deployed
INTERACTIVE SCIENTIFIC SOFTWARE (CAISS) Krawczyk in its original form as published,
not only would the advertised “prov-
Jonathan Katz misstates what I wrote
FLSA Status: Exempt
able security” guarantee have been
in my article and attributes to me
Compensation: Assistant
false, but in certain settings HMQV
Professor: $38,801 - $67,092, things I never said, all to justify accus-
could have been breached by a mali-
Associate Professor: $50,321 - ing me of “sheer elitism” and “snob-
$80,020, Professor: $62,268 - bery at its purest”. I never objected cious adversary. That’s not a minor
$95,997 (Salaries effective matter. (See http://eprint.iacr.
to cryptographers making a carefully
9/19/2007)
org/2005/205 for detailed explana-
reasoned, rigorous argument in sup-
College Web Site:www.ccny.cuny.edu
tions of the security flaws that have
port of a claim. Indeed, in my papers
Notice Number: FY13784
been found in HMQV.) Indeed, if
with Menezes on “provable security”
Closing Date:Open until filled.
Krawczyk believes that fallacies in
we give detailed explanations of the
POSITION DESCRIPTION AND DUTIES
proofs are so unimportant, then why
need for precision in definitions and
The City College of the City University of New York as
bother to give proofs at all?
part of its continuing efforts to achieve the highest security analysis, and we describe

quality in teaching and research is making available some of the best examples of early
two new tenure-track positions in one or other of —Neal Koblitz
and more recent research along these
the Department of Mathematics or Computer University of Washington
lines. In my article what I took issue
Science, or the School of Education. These [email protected]
positions will be closely associated with CAISS, the with was all the hype, misleading ter-

Center for Algorithms and Interactive Scientific minology, and easily misunderstood
Software. CAISS has embarked on a number of (Received September 14, 2007)
and misinterpreted “theorems” that
projects involving symbolic computation (see our
one finds in much of the “provable se-
website:www.caissny.organd the associated site
www.grouptheory.org). The objective is to build curity” literature. It is hard to escape
on past efforts culminating in an ongoing, the impression that mathematical
innovative, high-risk, creative, research group with
jargon and the theorem-proof para-
wide-ranging interests.
digm are often used to kick dust in
It is expected that successful applicants will have
almost limitless creative license with the the eyes of outsiders.
expectation that funding from a number of sources, It is Oded Goldreich, not me, who
both government and commercial, will flow gives a misleading version of the
from these efforts. These positions promise to
events surrounding his last-minute
be extremely exciting opportunities for the
right candidates. effort to prevent publication of my
Collaboration with the Computer Science and article with Menezes in the Jour-
Mathematics Departments and the School of nal of Cryptology. Our paper had
Education, and working on some research with one gone through the refereeing process
or two undergraduates will be expected.
almost two years before, and had
Teaching responsibilities will be reduced for the
been judged to be of sufficient tech-
first two years (and possibly the third year) to one
course per semester. nical novelty to merit acceptance.
QUALIFICATION REQUIREMENTS Goldreich’s essay “On post-modern
A PhD in Mathematics, Computer Science or a cryptography” finds fault with our
closely related field is required. Rank and salary to article not on technical, but rather
be determined depending on the experience of
on philosophical grounds. Calling
the applicant.
Menezes and me “post-modern [and]
Energy, a high degree of programming skills,
reactionary”, he is incensed by some
interest in open source and zero learning curve
software, a deep knowledge of the Linux and of our conclusions—notably, that
Microsoft operating systems, evidence of an ability “our confidence in the random oracle
to build and design software, new learning tools
assumption is unshaken” and that
and/or academic accomplishments and a
cryptography “is as much an art as
willingness to go out on a limb and work very hard
are a must. a science”. Whatever Goldreich’s rea-
TO APPLY sons might have been for attempting
Applicants should send a curriculum vitae and at to block our article on the eve of its
least 3 letters of recommendation to: Mary Tramel, publication, in the scientific world
Mathematics Dept, CAISS NAC 8/133, City
such conduct by an editorial board
College of New York, 160 Convent Ave, New York,
NY 10031, E-mail: [email protected]. member is irregular and improper.
Hugo Krawczyk’s letter itself is
The City University of New York is an Equal
Employment Opportunity/Affirmative an illustration of what I find so ex-
Action/Immigration Reform asperating in the “provable security”
and Control Act/ Americans with
Disabilities Act Employer field. In order to advertise his work
as “a prime example of the success
of theoretical cryptography,” Kraw-
czyk minimizes the fact that his
published proof was fallacious. If the
1456 Notices of the AMs VoluMe 54, NuMber 11