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In cryptography, encryption is the process of transforming information (the plaintext) using an algorithm (the cipher) and a secret (the key) into something unreadable to anyone except those possessing the key (the ciphertext).
What it comes down to is that information is scrambled and that some other person can only unscramble it if he/she knows the key used to scramble it.
Cracking the encryption is figuring out how the information was scrambled. Encryption can be done both by hand and with a computer. Note that doing encryption by hand has some limitations, it is impossible to encrypt large amounts of information due to the time it would take and one could not use every encryption, because some calculations might be too difficult to do by hand.
Symmetric vs. asymmetric encryption
There are two main forms of encryption today: symmetric and asymmetric.
Symmetric encryption is the classic form of encryption as is known today. The plaintext is encoded into ciphertext using a secret key; the recipient, to decode the message, must know the secret key that was used to encode it. The name refers to the fact that encryption and decryption are inverse functions and both use the same key to work (thus symmetrical). This was the first form devised — primitive versions (simple substitutions of one letter for another) go back to ancient times, with increasingly sophisticated variations being developed as the arts of codemaking and codebreaking advanced. Eventually, the algorithms became so complex that machines (such as the Enigma device used in World War II) were required to encrypt and decrypt messages with reasonable speed and accuracy.
Asymmetric encryption is a newer form of encryption, devised in The Seventies; in this form, the key used to encrypt the message and the one used to decrypt it are not the same. In an asymmetric cipher, each party has a pair of keys: a public key and a private key. If Alice wants to send Bob a message, she uses Bob's public key to encrypt the plaintext, and Bob uses his private key to decrypt it. Public keys, as the name indicates, are not required to be secret; private keys are. In short, encryption and decryption are not inverse to each other in asymmetric encryption schemes, hence the name.
The advantage of asymmetric encryption is that there is no need for the sender and recipient to know a shared secret key. Suppose you wanted to send an encrypted message to somebody, and you tried to do so using a symmetric cipher. How would you send them the secret key if you're concerned that somebody might eavesdrop? To send them the key, you need to use a special, secure channel that is resistant to eavesdropping—for example, an inperson meeting.
Another advantage of asymmetric encryption is that it can be used in reverse, encrypting a message with your private key to create a ciphertext that anyone can decrypt using your public key. This gives the message a digital signature that proves that the private key owner wrote it, because it was encrypted with a key nobody else knows. The two processes can be combined, so that Alice can send Bob a message that nobody else could have written and nobody else can read.
The biggest practical disadvantage of asymmetric encryption is that you need to "trust" that what you think is the recipient's public key really is theirs, and that their private key has not been disclosed — I can pretend to be the President of the United States and send you a public key, and if you mistakenly believe me, you might unwittingly send your top secret messages to me instead of the President and accept my digitally signed messages as if they came from the President. The normal method to verify this is for a thirdparty public key repository to digitally sign and store usernames and public keys. This does require a third party trusted to be impartial and an accurate recordkeeper. Another alternative is a "web of trust", in which people sign each other's keys, so that (for example) Alice can verify that Carol and Dave have signed Bob's key, vouching for the fact that it actually belongs to Bob. Alice then decides whether or not to take their word for it (much as she would if Carol and Dave were vouching for Bob in person).
Another disadvantage of asymmetric encryption is that it is more computationally expensive than symmetric systems. Most secure encrypted channel schemes get around this problem by using the asymmetric encryption solely to transmit a randomly generated onetime symmetric encryption key, then switch to symmetric encryption for the bulk of the transaction.
Security certificates — the bits of bits that tell us that individuals online are who they say they are — use the above described digitalsignature technique to generate a "seal of approval" that can be read by everyone, but only manufactured by the issuing authority.
Onetime pad
The onetime pad is a special kind of cipher that is completely unbreakable if used correctly—but very weak if used incorrectly, and also very impractical. The trick is that the secret key must be as long as the plaintext, must be completely random, and must never ever be reused.
The reason onetime pads are unbreakable is that for any conceivable plaintext, there exists a possible key that would produce that plaintext from the encrypted message. This means that if you try to guess what the key is, there are exponentially many more false positives than the real message, and no way to tell a false positive from a true positive.
But if the users of a onetime pad get sloppy and reuse a key for more than one message, it becomes trivial to break. If the keys are not truly randomly generated, it can be broken too. A number of historical codebreaking successes resulted because somebody tried to use onetime pads but either reused the keys or generated them in a nonrandom fashion.
Then there is also the problem of communicating the keys, which is even harder than in the normal case because (a) you need as many keys as messages, (b) the keys are as long as the messages.
Cryptanalysis
The act of analyzing the cipher and the ciphertext in order to retrieve the original plaintext. It is not true that any ciphertext can be cracked. Using a wrong key can sometimes result in a validlooking plaintext that is in fact not the correct plain text (onetime pads work this way).
To recover a plaintext from a ciphertext, the key and the algorithm used are required. Having only the ciphertext is the hardest problem: the cryptanalist must guess both the algorithm and the key. This is called a ciphertextonly attack and it requires the experience and the intuition of the analyst, knowledge of the circumstances, the sender, the receiver, current events, etc... While statistical analysis of the ciphertexts could provide information about the algorithm, it requires plenty of ciphertexts or it doesn't give any meaningful information. With modern encryption algorithms, ciphertextonly cryptanalysis is basically impossible no matter how much data you have.
If the algorithm is known, the recovery can be easier: only the key (usually a password, though other things can be considered as keys) is required. When evaluating the security of an encryption system, it is prudent to assume that the attacker knows the algorithm (a dictum known as Kerckhoffs's principle, named after cryptographer Auguste Kerckhoffs).
The simplest method of cracking a password is known as brute force: trying every possible password. The problem with this is that it can take a very long time to find the right password. The number of possibilities for a password increases with every character added to the length of the password and every character added to the range of options. For example, if you wanted to to find a password that was six (uppercase only) letters long, you might have to try 26^6 = 308,915,776 possible passwords. At the rate of a thousand guesses per second, it would take three and a half days to run through the list. Trying every sevenletter password at the same rate would take three months. If, instead of uppercase letters only, the passwords use lowercase letters, uppercase letters, and digits (26 + 26 + 10 = 62 options for each character), a sixcharacter password requires 1.8 years to exhaustively search at this rate, and a sevencharacter one requires 111.5 years.
The problem for the user is that memorizing a truly random string of characters is very difficult. It's easier to use actual words as passwords. However, this is more vulnerable to bruteforce attack: the number of words in the dictionary is much smaller than the number of random combinations of characters. Using "leetspeak" substitutions of other characters for letters and using unusual words may defeat a dictionary attack; however, sophisticated attackers will use an exhaustive vocabulary and try leetspeak as well as standard spellings.
It is possible to combine randomness and easy memorization using tricks such as remembering a phrase and using the first letter of each word (e.g. "This website will ruin your life" becomes "Twwryl"). Another option is to use a password manager program to store an encrypted database of passwords; the user then only needs to remember one master password to access all the others.
Of course, if the encryption algorithm itself is weak, even an unguessable password won't help you. Cryptographers consider an algorithm broken if there is a way to figure out the key faster than brute forcing it. Sometimes, this is only of theoretical interest (for example, even with the speedup it would still take longer than the age of the universe). Othertimes the algorithm is so broken that the key can easily be recovered in a practical amount of time. There are a large variety of attack techniques using advanced math, and new cryptosystems are expected to show evidence of resistance to them. If after years of analysis by expert cryptographers there aren't any practical attacks discovered, then it's considered probably secure. That little code you created yourself, however, doesn't stand a chance.
As mentioned above, the key doesn't have to be a password. For example, in Cryptonomicon, two people communicate using the "Solitaire cypher". The cypher uses a deck of cards; their initial arrangement is the key leaving 54! (54 factorial, 54×53×52×...×2×1 = about 2.3 times 10 to the 71st power) possible keys and no dictionary to use.
The knowledge of the plaintext or parts of the plaintext (socalled cribs) can make a cryptanalysis problem exponentially easier. The plaintext  or parts of  could be acquired by oldfashioned spying or, more inventively, by feeding the mole. This is called a known plaintext attack.
And then (as the xkcd comic at the top of the page illustrates) there's the ageold standby of rubber hose cryptanalysis — beating/torturing the key out of a holder. (The name comes from the rather vivid image of the keyholder being beaten across their bare feet with a rubber hose). This does not have a direct counter, but many applications (such as Truecrypt) allow a defense based on plausible deniability for an encrypted volume to decrypt to a 'decoy', which hides a second encrypted volume with a different key. Thus, someone coerced into giving up a key can reveal one secret while hiding a bigger one. The interrogator may suspect the presence of a hidden inner volume, but its existence can not be proved or disproved.^{note }
Of course, no encryption could protect you from stupidity. If you ever find yourself in a situation where the secret service is digging through your trash and anything you say might spell your doom if it ever gets in the wrong hands (because, be honest, who doesn't get into situations like this?), remember the following:
