The Second Parchment at Rennes-le-Chateau

and the intrusive W

by Ted Cranshaw

Abstract: We show that it is very unlikely that the second parchment was composed by de Cherisey, and somewhat unlikely that he forged the Stublein pamphlet. We suggest another area which should be searched.

Introduction

The central question in the Mysteries of Rennes-le-Chateau concerns the sudden acquisition of vast wealth by the parish priest, Berenger Sauniere. Between 1891 and 1901, it is said he spent about 2 million gold francs, having been previously a poor man living on the contributions from the villagers. Such a transformation of status is bound to attract interest, speculation, myth, and fraud.

One story, which might be called the standard model of these events, has been told by Lincoln (1-3), Fanthorpe (4), Mizrach (5), Nullens (6), de Sede (7) and others, and tells of parchments discovered at the Bibliotheque Nationale by de Sede and Lincoln bearing coded messages, which were allegedly found by Sauniere and his workmen in a wooden tube, or a bottle, during the reconstruction of the parish church. The key to the decryption of the messages was leaked to de Sede, and thence to Lincoln, and involved the use of the epitaph on the tombstone of Marie de Blanchefort in the churchyard at Rennes-le-Chateau as a keyword in the well-known method of encryption invented by Vigenere. The gravestone is at present defaced, probably by Sauniere, in order, it is suggested, to preserve its secrets. However, this intention was thwarted by an antiquarian Eugene Stublein, who had made a drawing of the gravestone, and published it in a pamphlet called Engraved Stones of the Langue Doc. The message after decryption is only partly comprehensible, but hints at hidden treasure and ways in which it might be found.

New information is constantly being added to this story, and a BBC TV Timewatch programme by Lincoln (3) broadcast in July 1996 significantly enlarged our knowledge. The parchments, we learned, were deposited at the Bibliotheque Nationale by two rogues, Pierre Plantard and Philippe de Cherisey, and the latter has made a written claim to be their author. They are currently in the possession of a writer, Jean Luc Chaumeil, who received them from de Cherisey. He also possesses a pamphlet called Stone and Paper, which surely should be made public, written by de Cherisey in which, we are told, he gives an explanation of the composition of the parchments, and how they can be decoded.

Furthermore, the story of the fortuitous survival of the Marie de Blanchefort epitaph thanks to its appearance in Engraved Stones of the Langue Doc is cast in doubt by the discovery by Pierre Jarnac of a Guide to Pyreneean Thermal Establishments by a Eugene Stublein, whose signature is utterly different from that of the author of Engraved Stones. The producers of the Timewatch programme seemed persuaded that the pamphlet, the parchments and their discovery were all the work of de Cherisey.

In what follows I try to examine closely and critically the steps the cryptographer actually took to produce the inscriptions on the second parchment, and to see whether this throws any light on the credibility of the story as we know it at present.

The second parchment

The second parchment carries a copy of the text of the Gospel of St John, Ch XII, verses 1-11. According to the standard model of the story, 140 letters have been inserted into the text, and these are extracted. 12 letters, spelling AD GENESARETH are removed, leaving an array of 128 letters. A second array consists of 119 letters said to be engraved on the gravestone of Marie de Blanchefort, who died in 1781, with the addition of the letters PS PRAECUM, which are also found elsewhere in the story, forming another array of 128 letters, which is used as a keyword in the decipherment. After an elaborate process, involving two transformations by the ciphering method suggested by Vigenere and a reordering of the letters by using a Knight's Tour on a chessboard, we arrive at a final array, the alleged plaintext.

The plaintext turns out to be an anagram of the second array, the epitaph of Marie de Blanchefort. This fact comes as rather a surprise as we follow the steps of the decipherment, and is regarded by Lincoln as evidence of the correctness and authenticity of the result, but by others, notably Townsend in the book by Fanthorpe (4), as showing that the whole elaborate construction is a game played by someone fascinated by letters and ciphers. In favour of the first view is the fact that the unclear message does establish some connection with the painter Poussin, whose picture "The shepherds of Arcadia" seems to be also implicated in the story, and in favour of the second view is the fact that in practice, the message has so far led nowhere.

About ciphers in general

The greatest number of users of ciphers must surely be amongst the military. Julius Caesar is said to have used a simple cipher in which each letter is replaced by the one which follows it in the alphabet. Throughout history, the successful interception and decipherment of messages intended to be secret, has had a huge influence on the outcome of conflicts. Only slightly less extensive is their use in commerce, where one must assume that a comparable effort is invested. We will be concerned only with cryptographic methods prior to the computer age.

In these practical, every day cases, the messages are many, and the plaintext long, or sometimes very long. The key is relatively short, and the same key is used repeatedly. The provision of the key to all intended recipients of the messages while maintaining its secrecy, commonly called the "distribution problem", obviously presents many difficulties. The loss of security of the key is likely to be catastrophic.

This is the world of professional cryptology, employing thousands of people, where keys may be distributed by face to face contact, or in locked boxes, literal or metaphorical, which themselves can only be opened by a secret key.

Since the era of the Gothic novel, children, and many adults, have been fascinated by the idea of secret messages. Lewis Carroll, (1832-1898), writing for children, gave a humorous account of the failure of the schoolboy use of Caesar's method, and suggested what is essentially Vigenere's method as an improvement (1868). Adding to the romantic allure is the seductive association with buried treasure and adventure. R.L. Stevenson (1850-1894) for example, wrote stories of adventure, such as New Arabian Nights (1882), and Treasure Island (1883), which capitalise on this new fashion.

Most famously, Edgar Alan Poe (1809-1849) published a collection of stories called Tales of Mystery and Imagination (1841). In the Gold Bug, the hero, Legrand, is supposed to have found a parchment on which the pirate, Captain Kidd has written notes to remind himself of the site where he has buried a quantity of treasure at some time when he was fearing discovery. The parchment attracted his attention because, after suitable heat treatment, which he at first administered accidentally, he could see that it was headed with a skull and crossbones, and at the foot was a drawing of a kid. In between, it was covered with symbols. To safeguard the treasure against the possibility of an outsider finding his notes, Kidd had written them in a cipher, in which each letter is replaced by a symbol. Legrand analyses the frequency distribution of the occurrences of the symbols, and matches them against the known frequency distribution of the use of letters in English. This immediately gives him the symbol for the letter e, and the observation of three letter words gives him several other letters. Continuing in this way, he soon deciphers the whole plaintext and finds the treasure. In another story, The Purloined Letter, we find the chief of the Surete boasting of the thoroughness of his men in searching for documents. They regularly look for hollowed-out chair legs which could hide a secret paper.

This is cryptology in the world of fiction. If the story of the Gold Bug is examined literally, several questions must occur to the reader. For example, while granting the need to make the notes unclear to a finder, was it really necessary for Captain Kidd to use a cipher? If he was afraid of mislaying his notes, how did he ensure the security of the key to the cipher? He can hardly have expected to remember the key, if he was in danger of forgetting the burial site. The loss of the key would mean the loss of the treasure, unless he was as clever as Legrand, and could crack his own cipher. If the parchment was a note to himself, why did he need to head it with a skull and crossbones, and sign it with the drawing of a kid? This was surely calling attention to something which he hoped would remain private, and gave no advantage to himself.

It will surely be admitted that the ingredients of the Rennes-le-Chateau story already exist in fiction. Is it possible that the recipe could have inspired a parish priest in a Pyreneean village, or other prankster to play games with letters, either for fun, or perhaps for profit? If so, it is certainly a fact that the tricks which he has devised easily eclipse those in their source of inspiration.

If the documents are not game-playing, we are obliged to ask questions similar to those which occur to the readers of fiction. Who wrote the messages, and why? Who was the intended recipient? How was he supplied with the key needed to decipher the message? Were there, perhaps, a large number of people sharing some common belief who might be expected to guess at the key? The decipherment involves the use of a Knight's Tour of a chessboard, of which there is known to be a very large number. How could the intended recipients know of the particular tour, or where to fit it in the deciphering process?

To pursue this investigation further requires some knowledge of the elements of the science of cryptology, and these we set out in the next section. A more detailed account can be found, for example, in the article Cryptology in the Encyclopaedia Britannica.

Cryptology

Methods of encryption.

The methods of enciphering a message can generally be divided into two types, Transposition ciphers, and Substitution ciphers, and in the Rennes-le-Chateau documents both methods are used. In transposition ciphers some rule known to the intended recipient, is used to change the order of the letters, and similarly, in substitution ciphers, a rule is used to replace the characters of the plaintext by other symbols. Thus in the simplest case of a transposition cipher, we provide an array of symbols, containing as many symbols as there are letters in the alphabet. To encrypt a letter in the plaintext which is the n'th letter in the alphabet we write the n'th symbol in this array.

The cipher used in The Gold Bug is clearly a substitution cipher, and because each character in the plaintext has one unique corresponding symbol in the ciphered message, it is called a monoalphabetic cipher. As the example in the Gold Bug shows, the cipher can easily be cracked for even short pieces of plaintext simply by finding the frequency distribution of the appearance of letters, and matching it to the known distribution which is found in normal text. If the division into words is maintained in the encrypted text, bigraphs and trigraphs can also be informative.

Vulnerability to this mode of attack can be greatly reduced by using a polyalphabetic cipher, invented in 1586 by Blaise de Vigenere (1523-1596), a court functionary and an alchemist. The idea here is to use a multiplicity, m, of arrays of symbols for substitution, and to have a key word containing m letters which tells us which array to use. Thus typically, we write the keyword repeatedly under the plaintext, letter by letter, and to encrypt a letter in the plaintext which is the n'th letter in the alphabet, we note the letter in the keyword which is below it, and substitute the n'th symbol in that array. Vigenere formalised the method in an efficient manner known as the Tableau de Vigenere, which will be familiar to readers of the books of Lincoln or Fanthorpe. We write a square of letters, of which the first row is the alphabet of 25 letters. The second row is the alphabet starting with b, c.... and ending with .....z, a. We proceed in this way until we have the last row, starting with z, and ending with y. We now label the columns with the letters of the alphabet, from a to z, and similarly we label the rows with the letters from a to z, as shown below.

A B C D E F G H I J K L M N O P Q R S T U V X Y Z

A A B C D E F G H I J K L M N O P Q R S T U V X Y Z

B B C D E F G H I J K L M N O P Q R S T U V X Y Z A

C C D E F G H I J K L M N O P Q R S T U V X Y Z A B

D D E F G H I J K L M N O P Q R S T U V X Y Z A B C E E F G H I J K L M N O P Q R S T U V X Y Z A B C D

F F G H I J K L M N O P Q R S T U V X Y Z A B C D E

G G H I J K L M N O P Q R S T U V X Y Z A B C D E F

H H I J K L M N O P Q R S T U V X Y Z A B C D E F G

I I J K L M N O P Q R S T U V X Y Z A B C D E F G H

J J K L M N O P Q R S T U V X Y Z A B C D E F G H I

K K L M N O P Q R S T U V X Y Z A B C D E F G H I J

L L M N O P Q R S T U V X Y Z A B C D E F G H I J K

M M N O P Q R S T U V X Y Z A B C D E F G H I J K L

N N O P Q R S T U V X Y Z A B C D E F G H I J K L M O O P Q R S T U V X Y Z A B C D E F G H I J K L M N

P P Q R S T U V X Y Z A B C D E F G H I J K L M N O

Q Q R S T U V X Y Z A B C D E F G H I J K L M N O P

R R S T U V X Y Z A B C D E F G H I J K L M N O P Q

S S T U V X Y Z A B C D E F G H I J K L M N O P Q R

T T U V X Y Z A B C D E F G H I J K L M N O P Q R S

U U V X Y Z A B C D E F G H I J K L M N O P Q R S T

V V X Y Z A B C D E F G H I J K L M N O P Q R S T U

X X Y Z A B C D E F G H I J K L M N O P Q R S T U V

Y Y Z A B C D E F G H I J K L M N O P Q R S T U V X

Z Z A B C D E F G H I J K L M N O P Q R S T U V X Y

Now, with the aid of this diagram, let us encrypt the example given by Fanthorpe, in Rennes-le-Chateau. The plaintext is Tresor est a Rennes, and the keyword is Sauniere. The first letter is T, and the cipher letter is found by tracing the T-column down to the S-row, where we find the letter M. Proceeding in this way, we obtain the series

Plaintext T R E S O R E S T A R E N N E S

Keyword S A U N I E R E S A U N I E R E

__________________________________________

Cipher M R Z G X V V X M A M R V R V X

and this is the enciphered message. To decipher the ciphered message, the rule is to find the letter naming the column containing the intersection of the cipher diagonal and the row named by the keyletter and this process returns the plaintext, TRESORESTARENNES. We note that this is not the order of actions taken by Fanthorpe.

The manipulation of the letters is somewhat cumbersome, and instruments were soon devised to make it easier. In France, the most popular instrument consisted of two slides, rather in the manner of a sliderule, with two alphabets along their length. In England, the alphabets were engraved round the circumference of two disks of slightly different radii, so that the two alphabets could be seen simultaneously. The disks were held together by a bearing at the centre, allowing them to be rotated independently.

When we compare this with the accounts given of the decipherment of the second Rennes parchment through two successive Vigenere transformations, two discrepancies appear. First, the procedure which we have described in accordance with the usual convention, as encryption, is there described and used as decryption. Second, each use of the Tableau de Vigenere is followed by another step which changes each letter in the message into the one following it in the alphabet. Now, of course, if we were using the Tableau method, it would be possible to relabel the rows in the Tableau so that this step was accomplished automatically, but if we were using one of the instruments mentioned above, which greatly reduce the labour involved, this step has to be taken separately, and the usefulness of the instrument is partly nullified. We may also note that in the account by Lincoln, the step is taken twice, once after each use of the Tableau, whereas it would be perfectly possible to take the step once only, but to move two places down the alphabet.

When this decipherment was next examined by Fanthorpe in Rennes-le-Chateau, the "position value" method was used, which happens in fact to be the natural way to program a computer to carry out the Vigenere transformations. Each letter is given a numerical value corresponding to its position in the alphabet. For encryption, the numbers corresponding to the first letter of the text and the first letter of the keyword are added together, and the letter corresponding to this number (subtracting 25 if necessary) gives the first letter of the encrypted text, and so on. For decryption, the number corresponding to the letter in the keyword must be subtracted (adding 25 if this number is zero or negative) from the position number of the letter in the text. Rather unexpectedly, we find that the use of this method automatically includes the one step shift down the alphabet which had to be made once for each Vigenere transformation. This is because it is natural in the position value method to make the position values run from 1 to 25, whereas the values implicit in the Vigenere transformation run from 0 to 24, which is to say that the a alphabet in the Tableau corresponds to no change to the text letter. This quirk in the position value method avoids the step which needs to be made in the Tableau method if we are to follow the cryptographer. It is most unlikely that our cryptographer used a digital computer, but this is rather strong evidence that he did use the position value method instead of the Tableau. The arrays of letters would be first converted into arrays of numbers, a process called Gematria by occultists, the necessary additions made, and the resulting arrays converted back into letters.

We are still left with the question why Fanthorpe and apparently our cryptographer have used what would conventionally be called encryption for decryption. Strangely, in Fanthorpe's book, Paul Townsend makes a spoof set of documents in which he reverts to the original convention, and this inconsistency seems to have passed unnoticed by both Fanthorpe and Townsend.

It may be worth noting that because encryption is a purely additive process, there is a symmetry between text and keyword. It does not matter which is which. For decryption, on the other hand, there are two different outcomes if text and keyword are interchanged.

The Keyword

The function of the keyword is to minimise or destroy that information which can be obtained from an encrypted message by comparing the frequency distribution of letters in the message with the known frequency distribution of the letters of the alphabet in normal text. In a monoalphabetic cipher, the frequency distribution of the encrypted symbols gives, as in Poe's story, an immediate indication of the code symbol for the most commonly used letters, but in a system such as we have been describing, a letter in the plaintext may be represented by many other letters, according to the operation of the keyword. The level of success achieved by the keyword in hiding this information can be measured by a well-known statistical test known as the chi-squared test. Ideally, the distribution of occurrences of the letters of the alphabet in the encrypted message would be the same as would be obtained by taking letters at random from the alphabet, i.e. all letters would have the same chance of appearing in the encrypted message. Obviously in that case, no information could be obtained by studying the distribution of letters in the message. The chi-squared test tells us what is the chance that a given array of letters could be a random selection of letters from the alphabet. The effectiveness of the keyword can thereby be measured, and we use this test later on..

A few fairly obvious remarks about keywords can be made.

1) In general, a longer keyword is more effective than a short one.

2) In general, a keyword containing a wide range of letters is more effective than one containing many repetitions of a letter.

3) It is possible to make successive transformations with different keywords. If the numbers of letters in two keywords are n and m, their combined effect is about as strong as a keyword of length n x m.

4) If the keyword is as long as the plaintext, and of sufficiently varied letters, it is impossible to break the code.

5) It is always possible to find a keyword of length m which will transform any arbitrary array of length m into any other arbitrary array of length m. In fact, if we find such a keyword being used, we would probably suspect that this was its purpose.

Following the account of the decipherment as given by Lincoln (1,2), and presumably by de Cherisey in Stone and Paper, what can be said of the work of our cryptographer in the light of these remarks?

First, we notice that he uses two Vigenere transformations, and the first keyword in the decryption process (presumably, the last keyword in the encryption process) consists of two words, mort and epee. It would have been more effective to have used them in successive transformations, but he chooses to use them together as one word. The significance of the word mort is not easy to guess. Perhaps he is making some comment on earlier methods of encryption. On the other hand, the choice of the word epee is interesting, because this is by no means the first time that it has figured in the history of French cryptology. It is the only word in the French language of the form xyxx, so if a would-be code-cracker spotted such a form in his text, he would know the word must be epee. In the twentieth century, epee is a fairly rarely used word, but it was probably not always so. Of all four letter words in the French language, it is the only one which has three letters the same. Considering 2 above, it follows that of all possible four letter keywords in the French language, epee is the worst. Can this be a coincidence? We comment on this question again later.

Second, we now have another transformation in which the keyword has the same length as the text. This seems like overkill unless the cryptographer is making use of 5 above. To pursue this examination further, we need to have before us the arrays of letters which figure in the decipherment.

The Arrays of Letters

Here are the arrays of letters. For convenience, we will name each array by its first four letters. Below each array we have given the value of chi-squared for the distribution, and the likelihood that such a value of chi-squared could arise from a random selection of letters.

Array vcps. The letters inserted into the text of St John's Gospel.

Chi-squared = 23, likelihood is 51%.

Array jrin. After the first Vigenere transformation and step down alphabet.

Chi-squared = 39, likelihood is 3%.

Array muce. The Marie d'Hautpoul tombstone with the addition of

pspraecum, reversed, as required in the decipherment.

Chi-squared = 119.7, likelihood less than one in a hundred million million.

Array xnls. After second step. Chi-squared as for muce.

Array berg. The alleged plaintext after Knight's Tour. An anagram of xnls and muce.

We are now in a position to discuss the composition of the parchment inscription. In Fanthorpe's book, Rennnes-le-Chateau, Townsend has covered some of the same ground, and we will make continuous comparison with his account.

Step 1.) The first step is to construct the anagram muce to berg. Townsend invented a tombstone inscription, and used a computer anagramming program to produce, in a few hours, a "message", which has about the same level of sense and incomprehensibility as is found in berg. This shows that the labour involved in generating the Rennes-le-Chateau parchment was not small, but on the other hand, it is also true that some individuals have a remarkable faculty for handling anagrams. Some mediaeval authors have used quite long anagrams to reveal their authorship to the initiated. The scientist Hooke (1635-1703) published a meaningless anagram of the letters sic tensio ut extensio to establish his priority as the discoverer of the laws of elastic deformation. Townsend suggests that the author of the Rennes-le-Chateau documents may have written one of the arrays, presumably the gravestone array, muce, on a piece of paper, cut out the letters, and then rearranged them to make another message, berg. Another possibility is that he first cut the text into lines, and then sometimes used groups of two, three, or more letters as well as single letters to devise the anagram. Testing this hypothesis statistically is not simple, because some particular bigraphs and trigraphs appear frequently in normal texts and complicate the situation. We show below the two arrays as they might have appeared to the composer.

ct git noble m bergere pas de

arie de negre tentation que

darles dame poussin teniers

dhaupoul de gardent la clef

blanchefort pax dclxxxi

agee de soix par la

ante sept ans croix et ce

decedee le cheval de dieu

xvii janvier jacheve ce daemon

mdcolxxxi de gardien

requies catin a midi

pace ps pommes

praecum bleues

We have written the letters in italics for groups greater than three which appear in both texts. At first sight, the appearance of a group of five letters in both texts might seem worth noting, and even more remarkable when we see that these five letters come from the date, mdcolxxxi. Indeed, if we ignore the 'o' in this train of letters, on the grounds perhaps that it is part of the keyword, mortepee, we have a string of seven letters common to both texts. Whoever wrote the gravestone text, it would be bound to have the date of death, 1781, in it. It has been suggested by others that the phrase pax 681 might be significant, but it may be simply that the anagramatist found distributing the large number of x's difficult. We note that the letters poul de in the gravestone text appear as pou and l de in berg, suggesting that they may have been separated by scissors.

It might also be thought worth noticing that no group of three or more letters occurring in the gravestone text is split by the lines, consistent with copying the text on to paper, and subsequent separation of lines by cutting between them with scissors. However, the statistical significance of this statement is quite weak, because such a splitting only occurs once in the Townsend spoof, and it is unlikely, though not impossible, that the computer program took note of the grouping of letters into words and lines in the spoof gravestone text. Moreover, that text is nearly twice as long as the Blanchefort epitaph so that the chance of such splitting is greater.

Step 2.) The order of the letters is now transformed by using the Knight's Tour, changing berg into xnls. This presents no problem. The choice of Tour is discussed later on in this paper.

Step 3.) This is perhaps the most interesting step, and one which is not quite reproduced in Townsend's spoof. Our cryptographer chooses to use muce as a keyword. It is this which causes the shock of surprise at the end of the decryption, because the plaintext message is found to be an anagram of the gravestone text. Thus we have three arrays, anagramatically related. muce is the gravestone inscription reversed, berg is an anagram of the gravestone inscription, devised by the cryptographer, and xnls is an anagram of berg brought about by the Knight's Tour. Now jrin is the result of using muce as a keyword on xnls. Because of the symmetry between text and keyword noted above, we could equally well say that jrin is the keyword that converts xnls into muce, i.e. that converts one array into another that is its anagram.(See 5 in the comments on keywords). This has an interesting consequence when the frequency distribution of the letters in the arrays is not a random selection from the alphabet, but has the distribution corresponding to normal text, as in the present case. Consider the most frequently occurring letter, e. Every occurrence of e in one text has about a one tenth chance of coming opposite another e in the other text. This corresponds to no change in the text, i.e. to the letter z in the keyword. The same argument applies to the second most populous letter, further increasing the population of z in the keyword. Such an enlargement is precisely what we find in the array jrin. There are 14 occurrences of z in jrin, far above the expectation for either a random selection from the alphabet, or a selection from normal text. The chi-square test shows that jrin has only a 3% chance of being a random selection from the alphabet.

It is a moot point whether or not our cryptographer noticed this weakness in his encryption. He might well have noticed the large number of z's in jrin, and realized that it was a consequence of the trick he wished to play. A reasonable way out of the difficulty would be to use another Vigenere transformation, and this is the course he chooses. Since the improvement needed in the degree of concealment is now minimal, the efficiency of the keyword is of little consequence. He has a free hand. His choice of the relatively inefficient mortepee was good enough to make his final text a good approximation to a random selection from the alphabet, and secure from attack by frequency analysis. Perhaps this should make us wonder whether the words mort epee themselves have some significance.

The Knight's Tour

The puzzle of how to make a tour of the Chessboard visiting every square in 64 Knight's moves may be an ancient one known to the Arabs, but in relatively recent times (about 1720) it was suggested by Brook Taylor (1685-1731), known to every schoolboy mathematician for Taylor's Theorem. The credit for the first published solution may be in dispute, but it is generally awarded to de Moivre (1667-1754), known to the schoolboy for de Moivre's Theorem.

Fig 1. Montmort Fig 2. Mairan

Fig 3. de Moivre Fig 4. CSA

In Figs 1- 6, we present six figures, giving six solutions of the Knight's Tour problem. The first three, roughly contemporaneous, were given by the mathematicians Montmort, Mairan, and de Moivre. The fourth, labelled CSA, is a computer solution generated by the method of Computer Simulation of Annealing. Let sixty four numbered atoms be placed at random on an 8x8 lattice. When consecutively numbered atoms are separated by a distance whose square is 5 (i.e. a Knight's move) the energy of this pair is said to be zero. For any other separation, the energy is given some positive value. A computer program calculates the total energy of the system summed over all pairs. Pairs of atoms are now taken at random, and tentatively exchanged. If the new value of the energy for the system is unchanged or reduced, the exchange is accepted and the configuration becomes the starting configuration for another tentative exchange. Otherwise the exchange is rejected. The process is repeated until the energy of the system is zero. The resulting tour is thus a random walk on the 8x8 lattice in which all the steps are Knight's moves.

The CSA method is clearly impractical without a computer. Human beings attempting a solution have usually tried to simplify the problem by dividing the area into two sub-areas which can be solved separately, always bearing in mind that a step from one sub-area to the other must be provided. Inspection of Montmort's solution shows that an attempt was made to divide the board into an upper and lower half. There are distinct resemblances between the paths in the two halves. Within each half, a shallow zigzag path from one side of the board to the other is evident. In Mairan's solution, the same zigzag pattern is evident, but this time, taken round the board. Anyone following the path cannot fail to be aware of some guiding principle being at work, in contrast to the CSA solution in Fig 4, which shows no trace of human design. The possible recognition of a guiding principle gives hope that we might identify the author of a particular tour.

Of the four solutions in Figs 1-4, it will surely be agreed that de Moivre's is the most elegant, amply justifying its popularity. De Moivre has evidently divided the board into an outer ring of 48 squares, and an inner block of 16 squares. He would certainly know that no tour exists for a square of side 4, and would therefore accept two squares from the outer ring into the inner block. Then there exist several solutions for the remaining 14 squares. It has been suggested that de Moivre went round the outer ring of 50 squares, keeping as close as possible to the outer wall. A closer inspection shows that in fact he tried to keep within the outer ring, and to maintain a steady progression round the board. This fact is significant when we consider the tour used in the Rennes-le-Chateau documents shown in Fig 6. This tour uses 54 of the 64 moves in the de Moivre tour, and where there are possible options, uses the de Moivre choice. This leaves little room for

Fig 5. de Moivre made reentrant Fig 6. Rennes-le-Chateau

doubt that the composer of the Rennes tour used the de Moivre tour as a basis for his own.

It will be noticed that the Rennes-le-Chateau tour is reentrant, i.e. the last square is a Knight's move from the first. It is not clear why the composer of the documents chose to use a reentrant tour. So far as the cryptographic aspects are concerned, very little advantage is gained. It is true that there are 64 possible starting points for a reentrant tour, which would make cryptanalysis marginally more difficult, but equally, the key to the whole puzzle, which has to be made known to the intended recipient, is made more cumbersome. Further, if he had simply required a reentrant tour, he could have used rules devised in 1759 by Euler (1707-1783), the inventor or discoverer of the quantity e, by which the de Moivre tour can very simply be converted into a reentrant tour. An example is shown in Fig 5, in which only one extra step is made between the inner and outer areas.

Perhaps the reason for the choice of Fig 6 lies in the highly symmetrical design made between squares 1 and 10. A star shaped pattern appears in the nine squares surrounding square 36, which seems to be picked out by a many-sided polygon drawn by an approximation to a star of David. There are even hints of the construction of a regular pentagon. The tour turns through the angle 37 degrees at squares 4, 6 and 8, close to the pentagon angle of 36 degrees. The motive for this design might be aesthetic or possibly magical. In either case, if it is deliberate, it seems to demonstrate a considerable mastery in the manipulation of the Knight's moves. The appearance of the names of so many eminent mathematicians in the history of the puzzle must make us look for a mathematician as the author of this tour.

The intrusive W

We now turn our attention to the strange case of the intrusive W. We will start by quoting from Lincoln (1) in The Holy Place, p 162, where he is recounting the history of the decipherment of the parchments.

'When de Sede sent the decipherment, he employed a normal 26 letter alphabet and it was this method which was demonstrated in the 'Chronicle' films. I must thank a number of television viewers who wrote to me to point out that the letter W was not commonly incorporated in the French alphabet during the eighteenth century when, it is assumed, the cipher was devised. The removal of the letter W from the alphabet produces a slight simplification in the decipherment process in its latter stages.' And again, three pages further on, 'Stage Four, like Stage Two, is a one-letter shift down the alphabet. (De Sede's original 26 letter alphabet version necessitated a two-letter shift at this stage.)'

It sounds too good to be true that such a radical error in the keyword can be corrected simply by making an extra shift down the alphabet. And, of course, it isn't true. Here is the result of the decipherment with a 26 letter alphabet and an extra shift down the alphabet. It may be compared with the true decipherment, berg, on page 9.

We see that about a quarter of the letters are wrong, being displaced up or down the alphabet by one step. Also, some w's have appeared in the text. It is clear that a 'slight simplification' is a gross understatement of the effect of the removal of the W from the alphabet. What has gone wrong?

A possible sequence of events might be as follows. When Lincoln received the comments from viewers concerning the letters of the eighteenth century French alphabet, he went back to de Sede for advice. De Sede has probably never undertaken the tedious business of decipherment himself (there has never been any mention of the gadgets for simplifying the process), and so went to his source, presumably de Cherisey, for his comments. De Cherisey found himself in a difficult position (no doubt not for the first time). He had already written, or was about to write, the 44 page pamphlet Stone and Paper, in which he claims to be the author of the documents. He may have been uneasily aware that his description of the cryptographic method as a combination of the Tableau de Vigenere and a step down the alphabet was somewhat deficient, as the actual, and simpler, method appears to have been the position value method. He may even have noticed the confusion between encryption and decryption, which has not previously been remarked upon. Now he is faced with a real blunder, because he has told de Sede, and presumably committed it to paper in Stone and Paper, that he used a 26 letter alphabet, when only a 25 letter alphabet will work. What should he do? He decides to brazen it out. He asserts that it is really a trivial matter, and that a simple extra step down the alphabet clears up the problem. He gambles that no one will carry out the chore of checking his assertion. And this gamble he very nearly won, because in almost twenty years, no one has.

De Cherisey has made so many mistakes in his account of the working of the cipher that it is impossible to take his claim to authorship seriously. We are forced to conclude that he acquired the arrays of letters from somewhere else, and decided to pass them off as his own. As M. Chaumeil remarks, 'They have earned thousands in royalties.'

Summary

1) We have pointed out several similarities between the Rennes-le-Chateau story and others occurring in fiction. There is a strong possibility that adventurers or fraudsters are taking advantage of a current taste for mystery and imagination.

2) Where in the world there are deliberate encrypted messages, there has to be some method of delivering the key to their intended recipients. None has been discovered at Rennes-le-Chateau.

3) In the standard model of the story, the process described as decryption is actually encryption. This mistake would certainly confuse its intended recipient, and is unlikely to be made by anyone other than a fraudster.

4) The encryption method used is usually described as the Tableau de Vigeneres, presumably following de Cherisey. In fact, it appears to be much more likely that the position value method was used. In this method each letter is replaced by the number of its position in the alphabet, and complicated manipulation of letters is replaced by simple arithmetic.

5) The choice of keywords in the two 'Vigeneres transformations' is puzzling. The first keyword consists of two words, mort and epee. The word epee is the weakest four letter keyword in the French language. The second has the same length as the message, and should therefore be more than sufficient, but is in fact also an anagram of the message, leading to an unusual consequence.

6) The Knight's Tour used to transform the order of the letters is almost certainly based on the tour produced by de Moivre, the first published solution. It has been modified to yield a reentrant tour, i.e. a tour in which the last point is a Knight's move from the first point, and it contains a pattern of unusual beauty, and perhaps, significance.

7) The first revelations of the decoding of the document to de Sede and Lincoln, presumably by de Cherisey, contains mistakes, particularly about the alphabet used. When these errors were pointed out to him, he attempted a coverup.

Conclusions

In the standard model of the story there are several points which must cause unease. None of the points made below is conclusive, but taken together they open new questions.

The serendipitous recording of the Blanchefort gravestone by Eugene Stublein was a story always rather hard to swallow. In the newer version, 'Stublein' is a nom-de plume, perhaps of de Cherisey. Yet why should he or anyone else have chosen this name? It is not hard to invent a name, or find a common one from a telephone directory. Instead, the forger chooses a rather rare name, as though inviting discovery. Was it necessary for him to sign his pamphlet, and, if so, why with a signature which is defiantly different from the other Stublein? He seems to have made no attempt to forge the original signature.

We have reached above some reasons for believing that de Cherisey was not the author of the parchments. We may add that he claimed that the parchments were written 'in his own hand'. To the untutored eye at least, they look like good examples of an antique calligraphy, and apart from a few dots, which might be the result of a spluttering pen. contain few or no mistakes. The writing in the Paper and Stone is quite undistinguished, and appears to contain a reasonable number of errors and corrections. It would need a handwriting expert to pronounce further on this point. Modern scientific investigation could probably date the parchment and the ink.

The near incomprehensibility of the deciphered message was assumed by Townsend, and also by the present writer, to be due to the difficulty of making an anagram of a message 128 letters long. Yet Chaumeil was able to account for nearly all the puzzling features as local references, puns on names, or masonic jokes, though admittedly, not always convincingly.

The basis of the Knight's Tour used in the decipherment is the tour devised by de Moivre, which is easily available in published works. The modified version seems very stylish, requiring a high degree of skill. On the other hand, it also could possibly have been published in books of mathematical games or puzzles, and adopted by the author of the documents.

If we are to look for another author, probably in the eighteenth century when the letter w had not entered the French alphabet, we might look for a mathematician to account for his knowledge of solutions of the Knight's Tour problem, and skill with the manipulation of the moves. He must probably have an interest in the occult, to account for his taste for the transformations between numbers and letters, called Gematria or Numerology, and the use of anagrams, particularly the anagram that consists in a reversal of the order of the letters, called Athlash. In the decipherment there are two instances of the reversal of letters, one of the gravestone text, which might be thought to be a deliberate desecration, and in the forward and backward use of the Knight's Tour. The pattern round square 36 in the Knight's Tour may also indicate an interest in the occult.

Figures.

Figure 1.

Figure 2.

Figure 3.

References

1. Lincoln, H. The Holy Place, Jonathan Cape, London, 1991

2. Lincoln, H. BBC TV 'Chronicle' 1979

3. Lincoln, H. BBC TV 'Timewatch' 1996

4. Fanthorpe, L. Rennes-le-Chateau, Bellevue Books, 1991

5. Mizrach, S. http://www.clas.ufl.edu/anthro

6. Nullens, G.C.H. http://www.isabel-uk.com

7. de Sede, G. Rennes-le-Chateau, Editions Robert Laffont, Paris, 1988