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From: Exploit me. on 29 Mar 2010 12:59
On Mar 29, 9:40 am, mstem...(a)walkabout.empros.com (Michael Stemper)
wrote:
> In article <v30sq5ph6b29vrrmjril9ri62pfjenr...(a)4ax.com>, David C. Ullrich <ullr...(a)math.okstate.edu> writes:
>
> >On Fri, 26 Mar 2010 14:15:53 EDT, Maury Barbato <mauriziobarb...(a)aruba.it> wrote:
> >>G. A. Edgar wrote:
> >>> You need to define "Cauchy sequence" and "converge to
> >>> zero".
>
> >>The definitions are the usual ones.
>
> >>A sequence (x_n) of elements of F is said to be a
> >>Cuchy sequence if for every e > 0 in F, there is
> >>a positive integer N such that | x_m - x_n | < e
> >>for ever n, m > N.
>
> >That's not quite the usual definition; the usual
> >definition, or at least a very common definition,
> >is that the above holds for every _real_ e > 0.
>
> Interesting. The text that I'm studying from uses the "e in F" version
ABSTRACT: The bombe was an electromechanical machine devised by Alan
Turing and Gordon Welchman for breaking the. German Enigma cipher in
World War II. The way in which it used a reductio ad absurdum logic to
reduce 263 possibilities to a few is a unique example of ingenious
circuit design, which is described in detail. Its relationship to the
Polish version of the 'bombe' is explored. The importance of the
diagonal board is shown as is the threat that this device faced from
the German's use of the `Enigma-Uhr'.

KEYWORDS: Enigma machine, Bletchley Park, Alan Turing, Gordon
Welchman, Bombe, diagonal board, Ehigma Uhr, cryptanalysis.

INTRODUCTION

Today we perform cryptographic algorithms using state-of-the-art
semiconductor devices, so that long keys and heavy computation can be
used. We are accustomed to the idea that cryptanalysis needs stronger
computing resources than those employed by the crypto user. But this
is a recent phenomenon. For nearly all the long history of cryptology,
paper and pencil was the only technology for both cryptography and
cryptanalysis. In this period, ingenuity was the main tool of the
cryptanalyst. Success depended on the relative simplicity of pencil
and paper algorithms.

The change from pencil and paper to electronics went in three stages.
First there was the introduction of simple mechanical devices during
the 19th Century, but cryptanalysis did not gain much from this
change. Then the period from about 1930 to 1950 saw a transition from
simple mechanisms to electronics, by way of electro-mechanical
techniques, pioneered in telephone switching.

There were not many widely used electro-mechanical designs, but they
form a fascinating subject of study because of their short life and
their combination of mechanical and electrical ingenuity. Our subject
in this paper is a remarkable electromechanical machine for
cryptanalysis, the "bombe".

Because the cryptographic machine of World War II have mostly been
kept secret by Governments, there are few for study and those of the
German forces stand out. There were two on-line cipher machines in
wide use, the Siemens and Halske T52, [1-4], and the Lorenz SZ40/42,
[5], both appearing in a number of variants as the war progressed. The
largest quantity of cipher material was produced by an off-line
machine called Enigma.

British cryptanalysts took an interest in all three machines, but the
T52 was never routinely broken. The Lorenz machines were used to send
long composite messages containing intelligence of the highest value
between parts of the high command. To break this cipher, the Colossus
series of machines was developed, the first electronic digital
computers, though of a special kind.

Enigma began as a commercial machine and was developed in many ways,
creating a family of cipher machines[6]. We are only concerned here
with one version that provided copious intelligence to Bletchley Park,
but at least three other versions also received their attention.

The breaking of the Enigma cipher, first by a Polish team and then, on
an industrial scale, by the British at Bletchley Park is well known.
After a period during which the German users improved their procedures
and put a stop to early breaking techniques, the main tool for
breaking Enigma was the 'bombe'.

This used electrical connections (not voltages or currents) to
represent logical relations and it selected hypotheses by the method
of reductio ad absurdum, which was strikingly original and probably
unique in machine design.

In this paper, the design of the British bombe and its method of
operation will be described. It is known that the Polish team designed
and named the first bombe, but its design and even its name remain
obscure. At the end of this paper we try to understand this early
machine and see how much it anticipated the British one.

It seems clear that the British bombe was a very different and
altogether more complex machine. Its conception is credited to Alan
Turing, while Gordon Welchman augmented the design in a critical way
by adding the `diagonal board', which is described below.

Working at high speed in a highly parallel search, the bombe used an
advanced form of the technology of the Enigma machine itself. This
typifies the way in which a technology used for cryptography also
serves the cryptanalyst, by pushing it to the limit. In no sense is
the bombe's principle a brute force attack on the Enigma's key space,
for that would have been beyond any technology.

Just how the difficult problem of breaking the Enigma cipher was
solved is the theme of this paper.

The source of the information is chiefly `The Hut Six Story' by Gordon
Welchman, [7]. For the Polish bombe we rely on a paper by Marian
Rejewski, translated from the Polish and published in Annals of the
History of Computing, . The help of Frode Weierud (CERN) and Ralph
Erskine is gratefully acknowledged.

ENIGMA'S MECHANISM

Pressing down a key on the alphabetic keyboard of the Enigma machine
lights an alphabetic symbol which is the corresponding ciphertext
letter. The relationship, at that moment, between the 26 keys and the
26 lamps is a permutation (or `substitution) of the alphabet. Pressing
and releasing the key also moves at least one 'rotor' in the
mechanism, so that the permutation for each letter is different.

>
> --
> Michael F. Stemper
> #include <Standard_Disclaimer>
> The name of the story is "A Sound of Thunder".
> It was written by Ray Bradbury. You're welcome.

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