A substitution cipher takes each character (sometimes groups of characters) in a message and replaces it with a different character according to fixed rules.
Every occurrence of one character will be substituted with the same replacement character.
An encrypted message can then be decrypted with another substitution cipher, this time set to substitute each character with the one that it originally replaced.
It is a very simple system which, as we'll find out, makes it a very insecure method of encryption!
In this section, we will be looking at a simple substitution cipher called Caesar cipher.
Caesar cipher is over 2000 years old, invented by a guy called Julius Caesar.
Before we go any further, have a go at cracking this simple code.
If you're stuck, try working in a small group with friends and classmates so that you can discuss ideas.
A whiteboard or pen and paper would be helpful for doing this exercise.
DRO BOCMEO WSCCSYX GSVV ECO K ROVSMYZDOB,
KBBSFSXQ KD XYYX DYWYBBYG.
LO BOKNI DY LBOKU YED KC CYYX
KC IYE ROKB DRBOO
LVKCDC YX K GRSCDVO.
S'VV LO GOKBSXQ K BON KBWLKXN.
Once you have figured out what the text says, make a table with the letters of the alphabet in order and then write the letter they are represented with in the cipher text.
You should notice an interesting pattern.
Given how easily broken this cipher is, you probably don't want your bank details encrypted with it.
In practice, far stronger ciphers are used, although for now we are going to look a little bit further at Caesar cipher, because it is a great introduction to the many ideas in encryption.
When you looked at the Caesar cipher in the previous section and (hopefully) broke it and figured out what it said, you probably noticed that there was a pattern in how letters from the original message corresponded to letters in the decoded one.
Each letter in the original message decoded to the letter that was 10 places before it in the alphabet.
The conversion table you drew should have highlighted this.
Here's the table for the letter correspondences, where the letter "K" translates to an "A".
It is okay if your conversion table mapped the opposite way, i.e. "A" to "K" rather than "K" to "A".
If you were unable to break the Caesar cipher in the previous section, go back to it now and decode it using the table.
For this example, we say the key is 10 because keys in Caesar cipher are a number between 1 and 25 (think carefully about why we wouldn't want a key of 26!), which specify how far the alphabet should be rotated.
If instead we used a key of 8, the conversion table would be as follows.
What is a key?
In a Caesar cipher, the key represents how many places the alphabet should be rotated.
In the examples above, we used keys of "8" and "10".
More generally though, a key is simply a value that is required to do the math for the encryption and decryption.
While Caesar cipher only has 25 possible keys, real encryption systems have an incomprehensibly large number of possible keys, and preferably use keys which contains hundreds or even thousands of binary digits.
Having a huge number of different possible keys is important, because it would take a computer less than a second to try all 25 Caesar cipher keys.
In the physical world, a combination lock is completely analagous to a cipher (in fact, you could send a secret message in a box locked with a combination lock).
We'll assume that the only way to open the box is to work out the combination number.
The combination number is the key for the box.
If it's a three-digit lock, you'll only have 1000 values to try out, which might not take too long.
A four-digit lock has 10 times as many values to try out, so is way more secure.
Of course, there may be ways to reduce the amount of work required – for example, if you know that the person who locked it never has a correct digit showing, then you only have 9 digits to guess for each place, rather than 10, which would take less than three quarters of the time!
Try experimenting with the following interactive for Caesar cipher.
You will probably want to refer back to it later while working through the remainder of the sections on Caesar cipher.
Before, we looked at how to crack Casear cipher – getting the plaintext from the ciphertext without being told the key beforehand.
It is even easier to decrypt Caesar cipher when we do have the key.
In practice, a good encryption system ensures that the plaintext cannot be obtained from the ciphertext without the key, i.e. it can be decrypted but not cracked.
As an example of decrypting with Caesar cipher, assume that we have the following ciphertext, and that the key is 6.
ZNK WAOIQ HXUCT LUD PASVY UBKX ZNK RGFE JUM
Because we know that the key is 6, we can subtract 6 places off each character in the ciphertext.
For example, the letter 6 places before "Z" is "T", 6 places before "N" is "H", and 6 places before "K" is "E".
From this, we know that the first word must be "THE".
Going through the entire ciphertext in this way, we can eventually get the plaintext of:
THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG
The interactive above can do this process for you.
Just put the ciphertext into the box on the right, enter the key, and tell it to decrypt.
You should ensure you understand how to encrypt messages yourself though!
Decrypting a Caesar cipher
Decrypt the following message using Caesar cipher.
The key is 4.
Encryption is equally straightforward.
Instead of rotating backwards (subtracting) like we did for decrypting, we rotate forwards (add) the key to each letter in the plaintext.
For example, assume we wanted to encrypt the following text with a key of 7.
HOW ARE YOU
We would start by working that the letter that is 7 places ahead of "H" is "O", 7 places ahead of "O" is "V", and 7 places ahead of "W" is "D".
This means that the first word of the plaintext encrypts to "OVD" in the ciphertext.
Going through the entire plaintext in this way, we can eventually get the ciphertext of:
OVD HYL FVB
Encrypting with Caesar cipher
Encrypt the following message using Caesar cipher and a key of 20:
JUST ANOTHER RANDOM MESSAGE TO ENCRYPT
Why is using a key of 26 on the following message not a good idea?
USING A KEY OF TWENTY SIX IN CAESAR CIPHER IS NOT A GOOD IDEA
ROT13 Caesar cipher
The Caesar cipher with a key of 13 is the same as an approach called ROT13 (rotate 13 characters), which is sometimes used to obscure things like the punchline of a joke, a spoiler for a story, the answer to a question, or text that might be offensive.
It is easy to decode (and there are plenty of automatic systems for doing so), but the user has to deliberately ask to see the deciphered version.
A key of 13 for a Caesar cipher has the interesting property that the encryption method is identical to the decryption method i.e. the same program can be used for both.
Many strong encryption methods try to make the encryption and decryption processes as similar as possible so that the same software and/or hardware can be used for both parts of the task, generally with only minor adaptions.
A substitution cipher simply means that each letter in the plaintext is substituted with another letter to form the ciphertext.
If the same letter occurs more than once in the plaintext then it appears the same at each occurrence in the ciphertext.
For example the phrase "HELLO THERE" has multiple H's, E's, and L's.
All the H's in the plaintext might change to "C" in the ciphertext for example.
Caesar cipher is an example of a substitution cipher.
Other substitution ciphers improve on the Caesar cipher by not having all the letters in order, and some older written ciphers use different symbols for each symbol.
However, substitution ciphers are easy to attack because a statistical attack is so easy: you just look for a few common letters and sequences of letters, and match that to common patterns in the language.
So far, we have considered one way of cracking Caesar cipher: using patterns in the text.
By looking for patterns such as one letter words, other short words, double letter patterns, apostrophe positions, and knowing rules such as all words must contain at least one of a, e, i, o, u, or y (excluding some acronyms and words written in txt language of course), cracking Caesar cipher by looking for patterns is easy.
Any good cryptosystem should not be able to be analysed in this way, i.e. it should be semantically secure.
What do we mean by semantically secure?
Semantically secure means that there is no known efficient algorithm that can use the ciphertext to get any information about the plaintext, other than the length of the message.
It is very important that cryptosystems used in practice are semantically secure.
As we saw above, Caesar cipher is not semantically secure.
There are many other ways of cracking Caeser cipher which we will look at in this section.
Understanding various common attacks on ciphers is important when looking at sophisticated cryptosystems which are used in practice.
A frequency analysis attack involves looking at how many times each letter appears in the encrypted message, and using this information to crack the code.
A letter that appears many times in a message is far more likely to be "T" than "Z", for example.
The following interactive will help you analyze a piece of text by counting up the letter frequencies.
You can paste in some text to see which are the most common (and least common) characters.
The following text has been coded using a Caesar cipher.
To try to make sense of it, paste it into the statistical analyser above.
"E" is the most common letter in the English alphabet.
It is therefore a reasonable guess that "J" in the ciphertext represents "E" in the plaintext.
Because "J" is 5 letters ahead of "E" in the alphabet, we can guess that the key is 5.
If you put the ciphertext into the above interactive and set a key of 5, you will find that this is indeed the correct key.
The message you should have decrypted is:
A LONG MESSAGE CONTAINS LOTS OF
STATISTICAL CLUES THAT CAN BE
USED TO ANALYSE WHAT THE MOST
FREQUENT LETTERS ARE, AND EVEN
THE MOST COMMON PAIRS OR TRIPLES
OF LETTERS CAN HELP TO BREAK
As the message says, long messages contain a lot of statistical clues.
Very short messages (e.g. only a few words) are unlikely to have obvious statistical trends.
Very long messages (e.g. entire books) will almost always have "E" as the most common letter.
Wikipedia has a list of letter frequencies, which you might find useful.
Put the ciphertext into the above frequency analyser, guess what the key is (using the method explained above), and then try using that key with the ciphertext in the interactive above.
Try to guess the key with as few guesses as you can!
The letter E isn't always the most common letter...
Although in almost all English texts the letter E is the most common letter, it isn't always.
For example, the 1939 novel Gadsby by Ernest Vincent Wright) doesn't contain a single letter E (this is called a lipogram).
Furthermore, the text you're attacking may not be English.
During World War 1 and 2, the US military had many Native American Code talkers translate messages into their own language, which provided a strong layer of security at the time.
The Vigenere cipher
A slightly stronger cipher than the Caesar cipher is the Vigenere cipher, which is created by using multiple Caesar ciphers, where there is a key phrase (e.g. "acb"), and each letter in the key gives the offset (in the example this would be 1, 3, 2).
These offsets are repeated to give the offset for encoding each character in the plaintext.
By having multiple Caesar ciphers, common letters such as E will no longer stand out as much, making frequency analysis a lot more challenging.
The following website shows the effect on the distribution: The Black Chamber - Vigenere Strength
However, while this makes the Vigenere cipher more challenging to crack than the Caeser cipher, ways have been found to crack it quickly.
In fact, once you know the key length, it just breaks down to cracking several Caesar ciphers (which as you have seen is straightforward, and you can even use frequency analysis on the individual Caesar ciphers!).
Several statistical methods have been devised for working out the key length.
Attacking the Vigenere cipher by trying every possible key is hard because there are a lot more possible keys than for the Caesar cipher, but a statistical attack can work quite quickly.
The Vigenere cipher is known as a polyalphabetic substitution cipher, since it is uses multiple substitution rules.
Another kind of attack is the known plaintext attack, where you know part or all of the solution.
For example, if you know that I start all my messages with "HI THERE", you can easily determine the key for the following message.
AB MAXKX LXVKXM FXXMBGZ TM MPH TF MANKLWTR
Even if you did not know the key used a simple rotation (not all substitution ciphers are), you have learnt that A->H, B->I, M->T, X->E, and K->R.
This goes a long way towards deciphering the message.
Filling in the letters you know, you would get:
AB MAXKX LXVKXM FXXMBGZ TM MPH TF MANKLWTR
HI THERE _E__ET _EETI__ _T T__ __ TH______
By using the other tricks above, there are a very limited number of possibilities for the remaining letters.
Have a go at figuring it out.
The above message is...
The deciphered message is:
HI THERE SECRET MEETING AT TWO AM THURSDAY
A known plaintext attack breaks a Caesar cipher straight away, but a good cryptosystem shouldn't have this vulnerability because it can be surprisingly easy for someone to know that a particular message is being sent.
For example, a common message might be "Nothing to report", or in online banking there are likely to be common messages like headings in a bank account or parts of the web page that always appear.
Even worse is a chosen plaintext attack, where you trick someone into sending your chosen message through their system so that you can see what its ciphertext is.
For this reason, it is essential for any good cryptosystem to not be breakable, even if the attacker has pieces of plaintext along with their corresponding ciphertext to work with.
For this, the cryptosystem should give different ciphertext each time the same plaintext message is encrypted.
It may initially sound impossible to achieve this, although there are several clever techniques used by real cryptosystems.
More general substitution ciphers
While Caesar cipher has a key specifying a rotation, a more general substitution cipher could randomly scramble the entire alphabet.
This requires a key consisting of a sequence of 26 letters or numbers, specifying which letter maps onto each other one.
For example, the first part of the key could be "D, Z, E", which would mean D: A, Z: B, E: C.
The key would have to have another 23 letters in order to specify the rest of the mapping.
This increases the number of possible keys, and thus reduces the risk of a brute force attack.
A can be substituted for any of the 26 letters in the alphabet, B can then be substituted for any of the 25 remaining letters (26 minus the letter already substituted for A), C can then be substituted for any of the 24 remaining letters, and so on.
This gives us 26 possibilities for A times 25 possibilities for B times 24 possibilities for C, all the way down to 2 possibilities for Y and 1 possibility for Z.
Representing each of these possibilities requires around 88 bits, making the cipher’s key size around 88 bits, which is below modern standards, although still not too bad!
However, this only solves one of the problems.
The other techniques for breaking Caeser cipher we have looked at are still highly effective on all substitution ciphers, in particular the frequency analysis.
For this reason, we need better ciphers in practice, which we will look at shortly.
Another approach to cracking a ciphertext is a brute force attack, which involves trying out all possible keys, and seeing if any of them produce intelligible text.
This is easy for a Caesar cipher because there are only 25 possible keys.
For example, the following ciphertext is a single word, but is too short for a statistical attack.
Try putting it into the decoder above, and trying keys until you decipher it.
These days encryption keys are normally numbers that are 128 bits or longer.
You could calculate how long it would take to try out every possible 128 bit number if a computer could test a million every second (including testing if each decoded text contains English words).
It will eventually crack the message, but after the amount of time it would take, it's unlikely to be useful anymore – and the user of the key has probably changed it!
In fact, if we analyse it, a 128 bit key at 1,000,000 per second would take 10,790,283,070,000,000,000,000,000 years to test.
Of course, it might find something in the first year, but the chances of that are ridiculously low, and it would be more realistic to hope to win the top prize in Lotto three times consecutively (and you'd probably get more money).
On average, it will take around half that amount, i.e. a bit more than 5,000,000,000,000,000,000,000,000 years.
Even if you get a really fast computer that can check one trillion keys a second (rather unrealistic in practice), it would still take around 5,000,000,000,000 years.
Even if you could get one million of those computers (even more unrealistic in practice), it would still take 5,000,000 years.
And even if you did have the hardware that was considered above, then people would start using bigger keys.
Every bit added to the key will double the number of years required to guess it.
Just adding an extra 15 or 20 bits to the key in the above example will safely push the time required back to well beyond the expected life span of the Earth and Sun!
This is how real cryptosystems protect themselves from brute force attacks.
Cryptography relies a lot on low probabilities of success.
The calculator below can handle really big numbers.
You can double check our calculations above if you want!
Also, work out what would happen if the key size was double (i.e. 256 bits), or if a 1024 or 2048 bit key (common these days) was used.
Big Number Calculator
Difficult calculations may take up to 5 seconds to complete.
Tractability – problems that take too long to solve
Brute force attacks try out every possible key, and the number of possible keys grows exponentially as the key gets longer.
As we saw above, no modern computer system could try out all possible 128 bit key values in a useful amount of time, and even if it were possible, adding just one more bit would double how long it would take.
In computer science, problems that take an exponential amount of time to solve are generally regarded as not being tractable – that is, you can't get any traction on them; it's as if you're spinning your wheels.
Working out which problems are tractable and which are intractable is a major area of research in computer science – many other problems that we care about appear to be intractable, much to our frustration.
The area of encryption is one of the few situations where we're pleased that an algorithm is intractible!