by R P Nathan
That one piece of knowledge, easily achieved by a series of counts on the coded text will by itself place you on the edge of cracking the code. And why? Because every language has its own characteristic structure, its fingerprint based on the frequency of occurrence of its letters. In English the letter E appears most frequently. If you take a sample of English you would normally expect to come across the letter E 12.7% of the time, so roughly one letter in every eight will be an E. T appears 9.1%, A 8.2% and so on.
So all that is needed is to count the letters in the coded text and the most frequent letter is likely to represent E, the second most a T and so on. The method is called frequency analysis. Of course there are subtleties since in a given passage there might be a preponderance of other letters because of particular words or names or just because the sample is small; and of course in this case I was dealing with Italian which had its own frequency fingerprint. But I had confidence because it was a powerful method and simple and for me even fun; and best of all I had a laptop to help me.
I fetched my computer, plugged it into the phone line and got back into bed. I switched on, my fingers drumming against the case as I planned my course of action. The first thing I needed was to find the frequency of letters in Italian. I got onto the internet and did a search for the information and got back the ordering of the most common letters: A E I L O... as opposed to E T A O I... for English. But it didn’t give me the frequency percentages. So I went into an online version of the Corriere della Sera copied a few pages of text and dumped them into a word processing document and used a word count on each letter in turn to produce an approximate frequency table. A bit rough and ready I thought but it would do.
In order to use the method I first needed to count how many times each letter appeared in the coded text. Rather than do this by hand I decided to type in the six pages, one hundred and forty-five lines, five thousand six hundred and twenty-five letters of the code and then get the computer to do it for me. I could normally type nearly fifty words a minute, maybe 250 characters but here with no word structure, no aid to memory, and with constant checking necessary to ensure I had made no mistakes, I was much slower and it was nearly one o’clock by the time I had typed it all in. But I didn’t mind because that was where using a computer now came into its own. With all the data in its memory I simply ran another word count on each letter until I had a table showing me the relative frequencies. And that was when I sat back in surprise.
For instead of a table showing letter frequencies similar to the one I had made from the Corriere della Sera, a kind of mountain range with peaks at A and E and I and L; instead of that, five letters didn’t appear at all, and the frequencies for the rest were practically identical.
Chapter 27
I was exhausted the next day and sat in my office staring at the file in front of me. I had slept only fitfully, my mind awhirr, trying to make sense of the code and the results of the frequency analysis.
The first part had proven to be simple: the five letters which did not appear at all were J, K, W, X, and Y. Although these letters were used in modern Italian they were only found in foreign words that had been introduced to the language. Thus the core alphabet consisted of only 21 letters.
However, the second issue was more troubling. All the letters appeared with roughly the same frequency. What did this mean? Obviously the comparison with the frequency analysis tables was pointless but what did it mean? What structure did it imply for the original text? What clues did it throw up?
All night the ideas had gone through my mind. All night I had gone through the options and proposed ideas and counter ideas to impose order on the seemingly random characters. These thoughts had prevented me sleeping, the problem had taken over, but so quickly that already it was a part of me, occupying a part of my brain so that even while performing other conscious actions it was there whirring away in the background and yet when the switch to subconscious was needed it sprang forward, it took over, and the blank screen of my closed eyes became back projected with V E L G A S A G A I I...
Already it was ingrained. It flowed where my blood flowed, it spread into the spaces between cells, it moved like lymph. It had taken over so quickly that I was shocked and however hard I shook my head and tried to get on, it returned, a voice to the letters, a picture to the text, patterns forming where there were none.
What did it mean? It meant it was not a simple monoalphabetic cipher. Of that I was sure. But then what? What were the options out there? What else could this code be? How to explain the even distribution of the letters?
Before I went further forward I went back. I revisited the Caesar shift grid altering it for the shorter alphabet: could this perhaps be the answer after all? And where before I would have been disappointed with a Caesar shift solution now I would have been mighty relieved for the initial shot of pleasure I had got from the code had faded leaving only the aftertaste of compulsion, a tang of obsession which lingered and grew.
So I reset the grid, but all I got was:
And I did what I’d done before and circled what I thought might be clues and hints but were in fact nothing but the coincidences that language throws up. None of these lines was recognisable Italian. They meant nothing. So it was not a Caesar cipher. There could be no more doubt about that.
How then to proceed?
I sat at my desk. Open in front of me was the file of accounts and Derek’s last few questions which needed to be cleared by tomorrow’s meeting. But as I looked down at the notes to the accounts showing director’s remuneration and depreciation and tax, I thought only of the code, V E L G A S A G A I I...
What did it mean? It meant the code was more complex than I’d first thought. But from the nebulous possibilities open to me only one seemed likely, possible, probable. A code so simple yet powerful that it sent my head spinning, running desperate for alternative solutions rather than this one, this code, the thought of which made my stomach cramp and my breath get short. For I knew the code they must have used, in my gut I knew it, but it made me scared even to think it. For I believed the code was the Vigenère cipher; so powerful when it was first devised in the sixteenth century that it was known as Le Chiffre Indéchiffrable, the undecipherable cipher.
It was no longer a monoalphabetic cipher but polyalphabetic: each original letter could be represented by more than one character. So A might be represented as G and M and Z. But it was more powerful than that because G might not always represent A, sometime it might be K or L or F or even G itself. There was no longer a one-to-one or even a one-to-many relationship between letters of the original text and the code letters. The relationship was many-to-many. And this was what made the code so hard to crack since if you found three occurrences of the letter P in the code text you simply had no idea whether they all represented the same letter in the original. Hence frequency analysis on its own would not work and if tried would throw up a roughly even distribution of letters.
Yet the Vigenère cipher was also beautifully simple to use so that coding and decoding text was a relatively easy process. All that was needed was a key word or phrase, which was known only to the sender of the message and the recipient. I sketched out an example on the back of the draft accounts as I sat there. The keyword was PIZZA and the message was Io sono Italiano. I wrote the keyword above the message, repeating it as required:
For each letter in the message which had a P above it a particular code alphabet would be used to encrypt it, for each with an I a different code. Thus the letter O would be encoded in three different ways even in such a short message. The code alphabets could be chosen arbitrarily but to make things simpler they were normally taken from a so-called Vigenère square. I wrote out the square for the 21 letter Italian alphabet which was simply made up of successive Caesar shifts of the entire alphabet.
To encode the first letter of the message I moved across the top of the square until I got to the column starting with the letter to be enciphered, I. Then ran d
own the left hand side of the square till I got to associated keyword letter, P. Then read off the code letter from where the two lines crossed, A:
Repeating for the rest of the message would give:
So for example in the final coded message there are two S’s but one represents an I and the other a T; and of the four O’s in the original message, two were encrypted as N’s, one as a Z and one as an E.
I doodled on the back of the accounts, drawing a swirl of letters running this way and that, interlocking and interlacing. It was still possible to break this code. The easiest way would be to find the keyword or phrase, but there seemed no chance of that as it could literally be anything. However it might be possible to find the length of the keyword. If the keyword was, say, five letters long then the 1st, 6th, 11th etc. letters would all be encoded using the P line in the Vigenère square. The 2nd, 7th, 12th etc. letters would be encoded using the I line and so on. Thus the polyalphabetic cipher would have effectively been broken into five monoalphabetic ciphers upon which frequency analysis could be used.
How though to find the length of the keyword or phrase? Again it all came down to the structure of the underlying message and, in particular, repetition. Short words in Italian like una, sono, con would be expected to appear frequently in any text. Because they are short it is quite likely that sometimes the reference letters used to code them will be the same as earlier in the document and so the same word could get encoded identically:
So when looking at the coded text it should be possible to spot sequences of characters which repeat again and again. This is an indication that a word has been encoded identically. Crucially it gives a clue that the code breaker can use. If counted from the start of one repeated phrase to the next that number will have to be a multiple of the length of the keyword or phrase. In the example I was playing with it was 15 characters between the repeated phrase. And so the key word or phrase must be either 3 or 5 or 15 letters long. By doing this for other repeated fragments it would be possible to find other examples which could narrow things down so that the length of the phrase could be ascertained exactly. And then frequency analysis could be used to do the rest.
And so what I needed to do now was to analyse the text in detail to find those repeated fragments to see if they would reveal the length of the key. But the code was at home in my flat and I knew I would have to put it out of my mind for the rest of the day and get on with finishing off the accounts for Derek. But when I looked down at the figures all I could see before me were the letters of the code, V E L G A S A G A I I... I wanted to start the analysis now. I wanted to search for the repetitious structure which would solve the riddle for me.
I stood up abruptly and paced around the small office. This was ridiculous. I had work to do. I could not let something like this take control. To occupy my thoughts so entirely that it stopped me functioning. I was stronger than that. I sat down again at my desk and took a deep breath. Then picked up a pen and looked again at the accounts. I saw the doodles I had made on the back and flipped the pages over. I needed to concentrate now, just a final read-through for mistakes, to make sure nothing was missing, or that none of the accounting notes had been repeated.
Repeated.
Repetition was the key I was sure it was. If only I had the text in front of me now I could probably solve it in half an hour.
I threw down my pen. My palms were sweating and my face felt hot. I needed to do something active. I couldn’t read at the moment, my mind was buzzing far too strongly for that. I needed some alternative problem to occupy my mind. There was a trainee called Simon working on the directors’ remuneration note. I could help him with that. There were other things I needed to do as well, but it was still only three o’clock. If I spent an hour with Simon it would calm me down. Get me back on track. I found myself breathing shallow and fast. I got up again and felt dizzy. I went to the closed door and stood there for a moment to regulate my breathing. To give it more structure. Structure was the key. What was the key? If only I could guess that then the code would unfold in a moment—
I stepped outside into the open plan office. Smiled at my secretary Susan. She had a message for me, she said. The closeout meeting was tomorrow at 12 o’clock in Derek’s room. The finance director and MD would be coming in for it and she had booked us in for lunch afterwards. Derek wanted to sit down at 11 to run through the final points.
I smiled and thanked her. My ears were buzzing like I was in a pressurised capsule and though I could hear her, she was indistinct, the nuance to her voice lost. But it was good to be out. Everyone saying hi and smiling; James, another manager, talking and laughing with me about... about what? The football? Yes. Talking about the football so that I could just nod at him and laugh with him as he talked about... what now? I moved on, touching his arm, saying Yes, I definitely would (would what?) and smiled at another secretary Julia who was my friend and liked me and she wanted to talk but I had to walk on. I just needed to find Simon so I could fill my mind, which was buzzing louder now, angry buzzing wasps and bees, V E L G A S A G A I I...
I stepped into the student area where the trainees hot-desked until they had passed their exams and been assigned a permanent place in the office. There was no sign of Simon but John was there.
“John?”
“Oh, hi Patrick. How’s it going?”
I nodded at him. “Yes fine, fine.” I was distracted and glanced back over my shoulder.
“You OK?”
“Yes, of course.”
“How was Julius?”
I frowned. The thing was that modified frequency analysis would not be easy. There were no guarantees.
“Patrick?”
I just nodded now, hoping that would answer his question and then laughed because people always want you to laugh at what they’ve said, don’t they. An appreciative audience. Well I could be one just as much as the next person. “So,” I said skilfully changing the subject. “Is Simon around?”
John was staring at me but then just shook his head. “He had to go to the dentist. But he said he’d left some files for you. I think with Susan. He said he’d finished everything.”
“Yes I’m sure.” I turned abruptly, needing to get back into my office. And yes there were the files sitting by Susan and I picked them up and went inside again and closed the door. Sat down. I suddenly felt very hot and was sweating profusely. I wiped a hand across my forehead and loosened my tie. Simon had finished the work. He had even left a yellow sticky to say it all tallied. So he didn’t need my help. Well that was good. All I needed to do was to look at those accounts. Just one last check before I spoke with Derek tomorrow morning. One last thing.
I got up again.
I had come to a decision.
I would do it tomorrow. I would get in early. It was only one last thing and would take no more than an hour if I looked at it with a clear head. I had no other meetings today. I deserved an early finish. I’d been working too hard. I felt ill.
I snapped my briefcase closed, grabbed my jacket, looked unseeing around my room, biting the inside of my lip, sweat pouring from my face. Go home. Do the code. Finish it off. Come back tomorrow morning. Fresh. Early.
I told Susan and she agreed. She said I didn’t look well. I said I was fine, just wanted an early night. She said I needed it. And John was there at the lift too and he smiled at me oddly in the way he can sometimes. He’ll need to lose that chippiness if he wants to progress, get anywhere, not stay at his level for the rest of his life and I just smiled back at him. Nothing to say. Nothing to say. Nothing to say.
Chapter 28
It had been light.
It had been light and now it was dark.
I sat on the floor, laptop by my side, printouts before me, around me, pages of letters with lines highlighted and fragments circled and underlined. Thirty, forty sheets of paper surrounding me, covering the floor. Where was the meaning in all this? Where was it leading? Where was the truth?
I had rung Julius late. Twelve o’clock. One o’clock. It was light and had got dark. But there were things I had to know.
“Julius, it’s Patrick.”
“Patrick? Patrick? What’s going on? I’m in bed. I’m heading off to Paris first thing—”
“I’ve been looking at the code.”
“Have you solved it?”
“Not yet, but I’m close. I need to know a few things.”
“Are you OK Patrick? You sound strange. And Christ, it’s one o’clock—”
“I need to know when was it written?”
“When—? I don’t know exactly. Between 1580 and 1600 maybe.”
“How can that be? Veronese died in 1588. I looked it up.”
“Oh. Of course.” He sounded confused. “I mean between 1570 and 1588.”
“Fine. That’s OK then.”
“What’s OK?”
“The cipher they used,” I said impatiently. “It would have been in existence then. But no one at the time could have broken it.”
“Not even the Council of Ten?”
“No one.”
“Then when you break the code you’ll be the first person to read what they wrote. What else do you need to know?”
“Nothing more. I’m on the verge. I made a partial breakthrough a couple of hours ago. I know the keyword is ten letters long. But I can’t get the frequency analyses to yield anything. Having ten of them makes it so much harder: the sample sizes become so small it’s hard to work out the distributions accurately.”
“I don’t really understand...”
“It doesn’t matter. I’ve got to go.” I hung up. So it was almost certainly a Vigenère cipher. So what? How did that help me with finding the key? I looked around me. I had ten frequency analyses to do. Ten different problems to make sense of. So much data. I was surrounded by data.
