Breaking the Enigma Cipher

Warsaw 1980.

Received on 13.5.1977

**
Introduction
**
.
*Cryptology
*, i.e., the science of ciphers, has applied since the very beginning
some mathematical methods, mainly the elements of probability theory an
statistics. Mechanical and electromechanical ciphering devices, introduced to
practice in the twenties of our century, broadened considerably the field of
applications of mathematics in cryptology. This is particularly true for the
theory of permutations, known since over a hundred years
^{
(1)
}
, called
formerly
*the theory of substitutions
*. Its application by Polish
cryptologists enabled, in turn of years 1932-33, to break the German Enigma
cipher, whichh subsequently exerted a considerable influence on the course of
1939-1945 war operation upon the European and African as well as the Far East
war theatres (see [1]-[4]). The present paper is intended to show, necessarily
in great brevity and simplification, some aspects of the Enigma cipher
breaking, those in particular which used the theory of permutations. This
paper, being not a sistematic outline of the process of breaking the Enigma
cipher, presents however its important part.

It should be mentioned that the present paper is the first publication on the mathematical background on the Enigma cipher breaking. There exist, however, several reports related to this topic by the same author: one - written in 1942 - can be found in the General Wladyslaw Sikorski Historical Institute in London. and the other - written in 1967 - is deposited in the Military Historical Institute in Warsaw.

## Phot. 1 |

PART I - THE MACHINE

**
1.
**
**
Description of the machine
**
.
Enigma, a work of the German engineer Scherbius, is an electromechanical
device. Phot. 1 gives an image of the machine and enables the description of
its operation to be abbreviated.

Enigma has a 26-letter keyboard behind which a board with 26 letters illuminated from below with bulbs is located. The main ciphering device, partially visible in the photograph, consists of three ciphering drums put on a common axle and a fourth - a stationary one - the so called reverting drum witch with the use of a lever can be shifted toward and outward of the ciphering drums. The three ciphering drums carry upon their circumferences the letters of the alphabet (Phot.2), the upper of witch are visible by small windows of a lid. Alongside there are seen, protruding a bit, handles enabling manipulations with the drums. Each of the three ciphering drums has on its one side 26 concentric fixed contacts and of the other side - 26 spring contacts. The fixed contacts are connected with the spring contacts in an irregular way with the use of insulated wires housed within the ebonite boxes of drums. The reverting drums has spring contacts only and they are connected in pairs, also in an irregular way. The connections of the four drums form the main ciphering part and are the main secret of Emigma. The block diagram of the electric circuit with drums and other parts nad their notation is shown i Fig. 1. On the right of the drums a dry battery of low voltage (4 V) is placed, in front of the machine, before the keyboard, there is a device like a telephon switch-board. Six pairs of plugs with connecting cords enable an interchange of 12 among 26 letters of the alphabet.

Phot. 2 |

Pressing a key cause the right drum to turn by the 1/26th of the round angle. Simultaneously, the electric circuit is closed for the current flowing from the pressed key through the plugboard, all three ciphering drums, the reverting drum and back through the ciphering drums and again through the plugboard. One of the bulbs is shining then and there can be seen a letter which is different from that on the pressed key. If, in the preceding position of the drum, the key marked with the letter illuminated just now had been pressed, then the bulb marked with the same letter as the previous key would shine.The enigma machine serves thus both for changing the plain text into the ciphered one and for the reverse transformation without need of any additional manipulations. Each subsequent pressing of a key causes the rotation of the right drum by the 1/26th of the round angle to be continued and another bulb to be shone. The middle and the left drums also rotate but much less frequently and their rotations will be negleeted in our considerations.

**
2.
The way of ciphering
**
. The Enigma ciphering machine
can be used in many ways. In Germany military and paramilitary units till
September 15, 1938, the following regulations were obeyed: the criptographer
set first the drums to be prescribred, valid for a given day, basic position
and performed reccomended changes of letters on the plugboard by putting the
plugs into suitable sockets. Next, he chose the individual key for a message,
consisting of three letters which were ciphered twice, thus obtaining six
letters placed at the beginning of the message. Thus the individual keys for
the given day had the following properties:

(1) all individual message keys were ciphered in the same basic position unknown to the cryptologist;

(2) each individual key was ciphered twice, so that the first letter meant the same as the fourth, the second - the same as the fifth and so on.

If
a sufficent number of messages (approximately 80) of the same day are
available, then, in general, all alphabet letters are present in their six
initial places. In each place of the message they form a one-to-one
transformation of the set of letters on itself and hence are permutations.
These permutations, denoted subsequentely by letters from
*A
* to
*F
*,
are unknown to the cryptologist. But the transitions from the first letter of
each message to the fourth one, from the second to the fifth and from the third
to the sixth form also permutations which, contrary to the previous ones, are
entirely known to the cryptologist since they are the products
*AD, BE, CF
*of
the above mentioned permutations. They can be represented as the products of
disjoint cycles, and then take a very characteristic form, different, in
general, for each day, e.g.

*AD
= (dvpfkxgzyo) (eijmunqlht) (bc) (rw) (a) (s)
*
*
*

(1)
*BE
= (blfqveoum) (hjpswizrn) (axt) (cgy) (d) (k)
*
*
*

*CF
= (abviktjgfcqny) (duzrehlxwpsmo)
*
.

Such a set of permutation resulting from the beginnings of messages forms a key for the Enigma secret.By the use of such sets for a few days only, it was possible to reconstruct the whole machine and afterwards each set enabled to reconstruct the keys changed daily in many years and thus to read messages ciphered with Enigma. We will pay more attention to this set due to its importance.

We
know from the description of the machine that if the pressure of key, say
*x
*,
causes a bulb
*y
* to shine, then, in turn, pressing the key
* y
*causes
a bulb
*x
* to shine, whichh is obviously connected with the operation of
the reverting drum. This is the reason why the all unknown permutations from
*A
*
to
*F
* consist only of transpositions. If the criptographer in ciphering
twice the individual key had pressed in the first place an unknon key
*x
*
to receive the letter
*a
*and had pressed the same key
*x
* in the
fourth place to receive the letter
*b
* , then by pressing in the first
place the key
*a
* he should receive the letter
*x
*, and by pressing
in the fourth place the key
*x
* - the letter
*b
*. Hence we have the
successive operation:
*a
* onto
*x
* and then
*x
*onto
* b,
*which
is called
*multiplications of permutations
*. So we see that by writing
consecutively the letters
*ab
* we obtain a fragment of the permutation
*AD
*
which is a product of unknown permutations
*A
* and
*D
*.

Let us take yet a small example. Let

*
dmq
vbn, von puy, puc fmq
*

denote
the beginnings, i.e. the twice-ciphered initial keys of three among
approximately 80 messages obtained of a given day. From the first and fourth
letters we see that
* d
*is substituted for
*v
*,
*v
* for
*p
*,
and
*p
* for
*f
*. In this way we get a fragment of permutation
*AD
*,
namely
*dvpf
*. In the same way from the second and fifth letters we see
that o
*
*is substituted for
*u
*,
*u
*for
*m
*, and
*m
*
for
*b
*. We get a fragment of permutation
*BE,
*namely
*oumb.
*And,
finally, from the third and sixth letters we see that
* c
*is substituted
for
*q
*,
*q
* for
*n
*, and
*n
* for
* y.
* We get a
fragment of permutation
*CF,
*namely
*cqny.
*The beginnings of
further messages enable to collect the complete permutations
*AD, BE, CF
*.
This set, due to its form and primary importance, will be called the
*characteristic
set
*or, directly,
*the characteristic of a given day
*.

**
3.
The set of equations.
**
As was explained, after the
pressing of any key the current flows through a chain of machine devices before
causing a particular bulb to shine. Each of these devices gives a permutation
of letters. Let us denote by
* S
*the permutation caused by the plugboard,
by
*L, M, N
*- those caused by the three ciphering drums, respectively, and
by
*R
* - that caused by the reverting drum. The the passage of the
current is described by the product of permutations
*SNMLRL
*
*
^{
-1
}
*

*
P = (a b c d e f g h i j k l m n o p q r s t u v w x y
z).
*

Fig.2, in which the ciphering part of the
machine, i.e. the drums, is replaced by a 2-dimensional sliders, permits to
follow the path of the current before and after shifting the drum
*N
*.

From the figure we read easily that the
unknown permutations from
*A
* to
*F
* can be represented in the form

*
A = SPNP
^{-1
}MLRL
^{-1
}M
^{-1
}PN
^{-1
}P
^{-1
}S
^{-1
}
*

*
B = SP
^{2
}NP
^{-2
}MLRL
^{-1
}M
^{-1
}P
^{2
}N
^{-1
}P
^{-2
}S
^{-1
}
*

*
.....................................................
*

*
E = SP
^{5
}NP
^{-5
}MLRL
^{-1
}M
^{-1
}P
^{5
}N
^{-1
}P
^{-5
}S
^{-1
}
*

*
F = SP
^{6
}NP
^{-6
}MLRL
^{-1
}M
^{-1
}P
^{6
}N
^{-1
}P
^{-6
}S
^{-1
}
*

While the known products
*AD, BE, CF
* are given
by the formulas

*
AD = SPNP
^{-1
}MLRL
^{-1
}M
^{-1
}PN
^{-1
}P
^{3
}NP
^{-4
}MLRL
^{-1
}M
^{-1
}P
^{4
}N
^{-1
}P
^{-4
}S
^{-1
}
*

*
BE = SP
^{2
}NP
^{-2
}MLRL
^{-1
}M
^{-1
}P
^{2
}N
^{-1
}P
^{3
}NP
^{-5
}MLRL
^{-1
}M
^{-1
}P
^{5
}N
^{-1
}P
^{-5
}S
^{-1
}
*

*
CF = SP
^{3
}NP
^{-3
}MLRL
^{-1
}M
^{-1
}P
^{3
}N
^{-1
}P
^{3
}NP
^{-6
}MLRL-
^{1
}M
^{-1
}P
^{6
}N
^{-1
}P
^{-6
}S
^{-1
}.
*

The first part of our
task is, in principle, to solve this set of equations in which the left-hand
sides are known and the permutation
*P
* and its powers on the right
-hand sides as well, whereas the
permutations
*S, L, M, N, R
* are unknown. Since in this form the set is
certainly unsolvable, we have to simplify it.

The first step in this direction is purely
formal and consists of the substitution of a letter
*Q
* for the repeated
product
*
MLRL
^{-1
}M
^{-1
}
*
, which can be interpreted as an equivalent
reverting drum. Thus the number of unknowns is actually reduced to three:

*
AD = SPNP
^{-1
}QPN
^{-1
}P
^{3
}NP
^{-4
}QP
^{4
}N
^{-1
}P
^{-4
}S
^{-1
}
*

*
BE = SP
^{2
}NP
^{-2
}QP
^{2
}N
^{-1
}P
^{3
}NP
^{-5
}QP
^{5
}N
^{-1
}P
^{-5
}S
^{-1
}
*

*
CF = SP
^{3
}NP
^{-3
}QP
^{3
}N
^{-1
}P
^{3
}NP
^{-6
}QP
^{6
}N
^{-1
}P
^{-6
}S
^{-1
}.
*

**
4. Theorem on the product of transpositions
**
. The next step is more important. We aim
to get disjoint unknown permutations from
*A
* to
*F
*from known
products
*AD, BE, CF
*. As we explained earlier, the unknown permutations
consists only of transpositions, and the expressions
*AD, BE, CF
* are
their products. We can apply the following

THEOREM.
*
If two permutations of the same
degree consist only of disjoint transpositions, then their product contains an
even number of disjoint cycles of the same lengths.
*

Proof. Let
*X
* and
*Y
*
stand for the permutations to be multiplied and let their degree be
*2n
*.
If in the permutation
*X
* a transposition identical with a transposition
in
*Y
*, e.g.
*(ab)
*, incidentally occurs, then in the product
*XY
*
a pair of single-letter cycle
*(a)(b)
* will be observed. With respect to
transpositions, identical in the two permutations, the theorem is thus true.
After rejecting identical transpositions we can assume, without loss of
generality, that the follow transpositions occur:

In
permutation
*X
* in permutation
*Y
*

*
(a
_{1
}
a
_{2
}) (a
_{2
} a
_{3
})
*

*
(a
_{3
}
a
_{4
}) (a
_{4
} a
_{5
})
*

*
............. .............
*

*
(a
_{2k-3
}
a
_{2k-2
}) (a
_{2k-2
} a
_{2k-1
})
*

*
(a
_{2k-1
}
a
_{2k
}) (a
_{2k
} a
_{1
})
*

Indeed, the initial
letter
*a
_{1
}
* must finally appear in the permutation

*
(a
_{1
}
a
_{3
} ... a
_{2k-3
} a
_{2k-1
}) (a
_{2k
} a
_{2k-2
}
... a
_{4
} a
_{2
})
*

If in this way not all letters of the permutation are exhausted, we continue our procedure to exhaust all the letters.

Simultaneously we note that:

(1)
the letters of a given
transposition are always observed in two different cycle of the same length in
the permutation
*XY
*;

(2)
if two letters appearing
in two different cycles of the same length in the permutation
*XY
*belong
to the same transposition, then their neighbouring letters (the left neighbour
and the right one) belong to the same transposition.

The reverse theorem is particulary important:

*
If any permutation of
even degree there appears an even number of disjoint cycles of the same length,
then the permutation can be regarded as a product of two permutations each of
which consist only of disjoint transpositions.
*

There is neither a need
do develop a proof to the quoted reverse theorem nor a formula for the number
of possible solutions for
*X
* and
*Y
*. It suffices to mention that
this theorem – applied to the products
*AD
*,
*BE, CF
* – provides for
each of the expressions
*A, B, C,
* depending on the form of the products,
over ten or several tens possible solutions, while the permutations
*D, E, F
*
are uniquely determined by them. For the whole characteristic set of three
equations we get several thousands or several tens of thousands of possible
solutions, thus choosing the actual one would be a difficult task.

The theorem on the product of transpositions does not lead us to the point we are aiming to get at, it brings us, however, to the proximity of it.

Let us assume, for
example, that we know that the cryptographers prefer the same three letters,
e.g.
*jjj
*, as the initial keys. If
*xqr gve
* are the initial key of
a ciphered message, then making use of the characteristic set (1) and of the
assumption that these letters mean
*jjj
* in the plain text we conclude,
e.g. that the letters
*nfa qqb
* and
*eug imf
*, as the beginnings of
messages, mean the letters
*ppp
* and
*zzz,
* respectively.

In this way, an accurate knowledge of preferences of the cryptographers together with the theorem of the product of transpositions enables us to find the only actual solution. Finally, the left-hand sides in the set of equations

*
A = SPNP
^{-1
}QPN
^{-1
}P
^{-1
}S
^{-1
}
*

*
B = SP
^{2
}NP
^{-2
}QP
^{2
}N
^{-1
}P
^{-2
}S
^{-1
}
*

(2)
* .......................................
*

*
E = SP
^{5
}NP
^{-5
}QP
^{5
}N
^{-1
}P
^{-5
}S
^{-1
}
*

*
F = SP
^{6
}NP
^{-6
}Q
^{1
}P
^{6
}N
^{-1
}P
^{-6
}S
^{-1
}
*

can be regarded as known.

If the enquiring reader asks how the cryptologist knows – before breaking the cipher – the preferences of the cryptographers, we can reply that the cryptologist does not know these preferences but he tries to compensate his ignorance with long tests, imagination, and sometimes with an once of luck.

**
5. Connections of the drum
N
.
**
It is not known to the author whether the
set of six equations (2) with three unknown permutations

Actually, the necessary
supplementary data were received in another, much shorter way. In December
1932 the French Bureau of Ciphers supplied to the Polish Bureau of Ciphers
confidential material containing the tables of German keys to Enigma including
the plugged connections
*S
* to the plugboard. It was possible to transfer
the permutation S as known to the left-hand side of the set of equations

*
S
^{-1
}AS = PNP
^{-1
}QPN
^{-1
}P
^{-1
}
*

*
S
^{-1
}BS = P
^{2
}NP
^{-2
}QP
^{2
}N
^{-1
}P
^{-2
}
*

*
.......................................
*

*
S
^{-1
}ES = P
^{5
}NP
^{-5
}QP
^{5
}N
^{-1
}P
^{-5
}
*

*
S
^{-1
}FS = P
^{6
}NP
^{-6
}QP
^{6
}N
^{-1
}P
^{-6
}
*

In this way we obtained
the set of six equations with only two unknown permutations
*N
* and
*Q
*.
As we will show, this set is solvable, but needs yet certain transformations.
Before doing the transformations we expalain a problem of theory of
permutations.

If we have three permutations
*G,
H, T
*and

*
G = T
^{-1
}H T,
*

Then we say that
*the
permutation G is transformed from the permutations H by the permutation T.
*

As is proved in the
theory of permutations, there is no need to multiply the permutation
*H
*
by
*T
^{-1
}
* from the left and next by

*
U = P
^{-1
}S
^{-1
}ASP = NP
^{-1
}QPN
^{-1
}
*

*
V = P
^{-2
}S
^{-1
}BSP
^{2
} = NP
^{-2
}QP
^{2
}N
^{-1
}
*

*
.......................................
*

*
Z = P
^{-6
}S
^{-1
}BSP
^{6
} = NP
^{-6
}QP
^{6
}N
^{-1
}.
*

Next we calculate the proucts:

*
UV = NP
^{-1
}(QP
^{-1
}QP)PN
^{-1
}
*

*
VW = NP
^{-2
}(QP
^{-1
}QP)P
^{2
}N
^{-1
}
*

*
WX = NP
^{-3
}(QP
^{-1
}QP)P
^{3
}N
^{-1
}
*

*
XY = NP
^{-4
}(QP
^{-1
}QP)P
^{4
}N
^{-1
}
*

*
YZ = NP
^{-5
}(QP
^{-1
}QP)P
^{5
}N
^{-1
}
*

Eliminating the common expression
*
QP
^{-1
}QP
*
we get a set of four
equations wit only one unknown

*
VW = NP
^{-1
}N
^{-1
} (UV)NPN
^{-1
}
*

*
WX = NP
^{-1
}N
^{-1
} (UW)NPN
^{-1
}
*

*
XY = NP
^{-1
}N
^{-1
} (WX)NPN
^{-1
}
*

*
YZ = NP
^{-1
}N
^{-1
} (XY)NPN
^{-1
}
*

Proceeding accordingly to
the method given earlier, we get from the first equation several tens of
possible expression for
*
NPN
^{-1
}
*
depending on the form
of permutation

**
**

**
6. Concluding remarks.
**
The description of the machine presented at
the beginning was simplified in order to illustrate the process of
reconstructing the connections of the drum
*N
*. Actually, the machine and
its operations were much more complicated. For example, besides of the three
ciphering drums and the reverting drum, Enigma had also an entry drum which
complicated greatly breaking the cipher. Moreover, the rings of the drums
carryng the letters of the alphabet could be shifted with respect to the rest
parts of drums, so that the knowledge of the basic position brought no
information on the actual position of the inner part of drums. Not only the
drum
*N
* could rotate, but – at a smaller rate – also the drums
*L
*
and
*M
*, which caused an additional complication. Finally, it was possible
to change the sequence of ciphering drums and due to that the number of
possible combinations increased six times. However, this last complication gave
an effect not forseen by the designers. It caused that each of the three
ciphering drums was placed from time to time at the right side of the set of
drums. So the method described for the reconstruction of the drum
*N
*
could sequentially be applied for each of the drums, and in this way the entire
reconstruction of the inner structure of the Enigma ciphering machine was
possible.

*
*

PART II - THE INITIAL KEYS

*
*

**
1.Cyclometer.
**
The reconstruction of the machine was a necessary
condition for breaking the Enigma cipher and a continous deciphering, but it
was not sufficient. Methods should be devised to reconstruct quickly the daily
initial keys. In other words, the problem to be solved was the reverse one that
described in Part I. While then the task involved a reconstruction of the
machine if the initial keys were known for a certain period (from the French
confidential material), in the next step it was necessary to reconstruct the
initial keys if the machine was reconstructed. Again the theory of permutation
was helpful.

As follows from formulas (1), the
permutation
*S
* transforms only letters within cycle which appear in the
permutations
*AD, BE, CF
* and leaves unchanged the form of the cycles. The
permutations
*AD, BE, CF
* have a charcteristic form (see the example
following formulas (1)) and a set of three such permutations of the same form
of cycles does not appear very frequently.

The three ciphering drums can be put of the
axle in six different positions and the drums themselves can take 26
×
26
×
26 = 17576 different positions. If it there possible
to find a device which for each position of drums gave the length and the
number of cycles in the characteristics, and if the lengths and numbers of
cycles were catalogued, then it would be sufficient to compare the products
*AD,
BE, CF
* for a given day with the products of the same form in the catalogue
to obtain immediately the proper sequence of drums and the permutation
*S
*,
while the remaining elements of the daily initial key could be reconstructed
by another method.

Such a device, called
*cyclometer
*,
was really found and its unusual simplicity of design was striking. The
cyclometer is illustrated in Fig. 3. Its main part consists of two assemblies
(I and II) of ciphering drums connected with leads through whichh current
flows, the drum
*N
* of the assembly II being shifted by three letters with
respect the drum
*N
* of the assembly I, while the drums
*L
* and
*M
*
of the assembly II are always in the same position as the drums
*L
* and
*M
*
of the assembly I. Fig. 4 illustrates the principle of operation of the
cyclometer.

For sake of simplicity the sequence of
drums in the assembly II is reversed which, however, does not change the
matter. The reverting drums are denoted by
*Q
*. They are equivalent (in
the diagram only) to the drums
*R, L
* and
*M
*. Between the assemblies
I ad II there is a system with bulbs and switches. If for any of the bulbs,
e.g.
*l
*, the source of current (denoted by
¸
in the diagram) is switched on, then the current flows
alternately through the assemblies I and II, and after a certain number of
turns it comes back to the bulb
*l
*. All the bulbs in the circuit are
then shining simultaneously. Their number, always even, is equal to the doubled
number of letters in one of the cycles of permutation
*AD
*. After
switching the source of current and closing in this way the circuit of another
bulb not shining yet, further bulbs will shine, the number of which allows us
to calculate the length of the next cycles in the permutation. In this manner,
by rotating successively the drums and counting the number of lighting up
bulbs, we can determine the length and the number of cycles in the
characteristics for all 17576 drum positions for a given sequence. Since there
are six possible sequences, the catalogue of characteristics include 6
×
17576 = 105456 positions altogether. The
cyclometer was equipped with a variable resistor, since – due to incessantly
changing number of shining points – bulbs had not lit up or they would blow.

**
2. Perforated charts.
**
The cyclometer, or rather the catalogue of
characteristics based on it, accomplished its task till September 15, 1938, and
since that date in all of German units using Enigma, with the exception of SD
(Sicherheitsdienst), quite new regulations realted to ciphering the initial
keys of messages got mandatory. Since that time the Enigma operator had to
choose himself the basic position which was different for ciphering the
individual key of each message and this basic position was placed without
ciphering (as a plain text) in the message heading. The individual key of a
message was, as previously, ciphered twice. In this way the first letter of the
individual key meant as before the same as the fourth, the second as he fifth,
etc., but the basic position now known to the cryptologist was different for
each message. Now, for a given day there are no characteristics products
*AD,
BE, CF
*, the form of which could be found in the catalogue, but the
relations between the first and the fourth, the second and the fifth, the third
and the sixth letter of the key still existed and this had be of use. If, e.g.,
the individual key after ciphering had the form
*pst pwa
*, i.e. the first
letter was the same as the fourth (or the second as the fifth, otr the third as
the sixth), then in terms of permutations this meant that in the produ.cts
*AD,
BE
* or
*CF
*(if they existed) there appeared one-letter cycle, called
*fixed
point of permutation
*. Since the length of the cycles in the products
*AD,
BE
*,
*CF
*is invariant with respect of the transoformations by the
permutation
*S
*, the presence or absence of the fixed points in the
products is invariant with respect to those transoformations.

Instead of a catalogue of cycle lengths in products a catalogue of fixed points of a all 17576 possible products (for each sequence of drums in the set) had to be elaborated to enable a comparison with the fixed points in the individual keys of messages of a given day. There was, however, a difficulty in performing such a comparison. The basic positions for each key have been known, as now the cryptographer had to write them as a plain text in the message heading, however, since the rings at the drum circunferences could be rearranged, actually only the relativity distances of fixed points displayed in the given daily keys were available.

The fixed point occurred in the catalogue in approximately 40% of all permutation products and perforated on a long tape would form a certain pattern. The fixed points of the keys of a given day perforated on another tape according to their basic positions would give also a certain pattern and the task consisted of the determination of the place at which all the fixed points of the second tape would coincide with those of the first tape. But this task presented, at least at that time, great technical difficulties. Moreover, the first tape should have double length to enable sliding the second tape over it. However, another method has been found by H.Zygalski.

For all 26 positions of the drum
*L
*,
paper sheet (rather thick), denoted by
*a
* to
*z
*, were prepared and
a square divided into 51x51 smaller squares was drawn on each sheet. Along the
sides of the square (or a rectangle) the letters from
*a
* to
*z
* and
from
*a
* to
*y
* were placed. It was a kind of coordinate system in
which the abscissae and ordinates of points denoted possible positions of the
drums
*M
*and
*N
*, respectively, while the small square denoted the
permutation corresponding to these positions with o without the fixed points.
The cases with fixed pionts were perforated. Such a sheet (in reduced scale)
was like that in Fig. 5.

We see that each fixed point had to be perforated even four times. It was an enormous work. When the sheets, according to a prescribed program and in a proper sequence with proper relative position were placed one upon another, the number of transaprent holes diminished gradually. If a sufficient number of data was available, at the end a single hole remained corresponding probably to the good case, i.e. to the solution. From the position of the hole the sequence of drums and the position of rings could be calculated so that, by comparing the letters of the keys with the letters in the machine, the permutation

**
3. Concluding remarks.
**
Besides the two methods of reconstructions
of the keys, some other simple methods were in use, e.g. a so called
*grate
method
*, together with mechanized and more expensive ones such as, e.g, the
*cryptologic
bomb
*. They were used accordingly to the needs in different circumstances
and time intervals, frequently treated as complementary to the cyclometer or to
the perforated sheets. Different techniques and strategies were elaborated,
with restricted range, but enabling to spare a lot of time and effort, such as
a so-called
*clock method
* of J. Rozycki. As the German ciphering service
introduced new and new obstacles to upset reconstruction of keys, it was
necessary to counteract. So, on November 1, 1937, the reverting drum was
changed to another, the number of cables in the plugboard increased gradually
from 6 to 13 pairs, and on December 15, 1938, the number of the enciphering
drums was increase from 3 to 5. The number of communication nets using the same
enigma but with different keys was also gradually increased.

In September 1939 almost the whole equipment and the majority of files of the Ciphering Bureau were destroyed before and during the evacuation. However, after a meeting of the delegates of the Polish, the French and the British Ciphering Bureaus, held in Warsaw on July 25, 1939, the Polish side made available all its methods and the equipment for the Enigma deciphering to the allies together with copies of the German ciphering machine reconstructed in Poland with the use of theoretical investigations.

**
Notes
**

**
(
**
1)
See, e.g., C. Jordan
*, Traitè des substitutions
* , Paris 1870.

**
References
**

**
[
**
1]
Gustave Bertrand
*, Enigma ou la plus grande énigme de la guerre 1939-1945
*
, Paris 1973.

[2] Wladyslaw Kozaczuk,
*Zlamany szyfr
*,
Warszawa 1976.

[3] Wladyslaw Kozaczuk,
*Wojna w eterze,
*Warszawa
1977.

[4] F.W. Winterbotham,
*The Ultra Secret
*,
London, 1974.

Typesetted version by Enrico Grigolon (November 2002).