Main content

## Computer science

### Unit 2: Lesson 1

Ancient cryptography- What is cryptography?
- The Caesar cipher
- Caesar Cipher Exploration
- Frequency Fingerprint Exploration
- Polyalphabetic cipher
- Polyalphabetic Exploration
- The one-time pad
- Perfect Secrecy Exploration
- Frequency stability property short film
- How uniform are you?
- The Enigma encryption machine
- Perfect secrecy
- Pseudorandom number generators
- Random Walk Exploration

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Perfect secrecy

Claude Shannon's idea of perfect secrecy: no amount of computational power can help improve your ability to break the one-time pad. Created by Brit Cruise.

## Want to join the conversation?

- Wouldn't it be better to choose his card, and put a decoy card inside the box?(192 votes)
- I agree, a decoy would have been a good idea, but if they knew a "decoy" existed, then the odds of her picking the right card just got better.(180 votes)

- What exactly is the difference between the key space and the other two? I guess I just do not really understand the definitions of all the spaces.(18 votes)
- Suppose Alice wants to send a letter to Bob.

We call the contents of the original letter the**Message**or**Plain Text**

If Alice wants to prevent someone from being able to understand the message while it is being sent to Bob, Alice will want to turn the original**Message**into text that others can not understand. The text that others can't understand is called**Cipher Text**. The process of turning a**Message**into**Cipher Text**is called**encryption**.

Bob will then need to turn the**cipher text**back into**plain text**. The process Bob uses is called**decryption**.

When you**encrypt**a message you choose a**key**to use. The**Cipher Text**that you get after encrypting a**Message**depends on what**key**you use

i.e. if we have a message and encrypt one copy using key1 to get a cipher text 1

and we encrypt another copy using a different key, key2, to get a cipher text 2

cipher text 1 and cipher text 2 will usually be different**Message Space**- all of the possible messages Alice could send to Bob**Key Space**- all of the possible keys Alice could use to encrypt a message**Cipher Text Space**- all of the possible cipher texts that Alice could generate by encrypting messages

Example:

Alice wants to send a message to Bob.

The only messages that Alice wants to send are "Yes","No" answers to 2 questions**The Message Space**= { (Yes,Yes),(Yes,No), (No,Yes), (No,No) } it has a size of 4

Suppose Alice uses a method that only allows her to choose from the following keys to encrypt her message:

key 1: results in the following encryptions (Yes,Yes)-> (W,W) ,(Yes,No)->(X,Z), (No,Yes)->(W,X), (No,No)->(X,X)

key 2: results in the following encryptions (Yes,Yes)-> (Z,W) ,(Yes,No)->(W,W), (No,Yes)->(Z,X), (No,No)->(W,X)

Alice and Bob need to have agreed on which key they are going to use.**The Key Space**= {key1,key2} it has a size of 2

The only possible CipherTexts that Alice could generate are (W,W),(X,Z),(W,X),(X,X),(Z,W),(Z,X)**The Cipher Text Space**= { (W,W),(X,Z),(W,X),(X,X),(Z,W),(Z,X)} it has a size of 6

Note: This example does not show perfect secrecy.

e.g. A cipher text of (X,Z) reveals that the message must have been (Yes,No).

e.g. A cipher text of (W,W) reveal that the 1st answer was Yes (Message was either (Yes,Yes) or (Yes,No) )

Hope this makes sense.(54 votes)

- Couldn't you test all the possible combinations using a computer and then search for a common, almost always used word like "the"? That would limit the number of possibilities drastically, and by running all of those messages through a spell/grammar check program, you could limit those possibilities further. Lastly, if you included spaces in the number of letters, you could search first for the message that had the nearest average number of letters per word; for example, ten letter words are rather rare, so you could generally eliminate those. Lastly, you could find the message that makes the most sense given the identity of the sender as well as the situation. Although this would not entirely eliminate "junk" messages, it would give you a much better idea of what they are messaging each other about.(8 votes)
- Great question, the answer is no - and it's important to see why (see other questions on this). If *
**every*** outcome is equally likely how would you eliminate anything? Remember, the encrypted text will be indistinguishable from a random string.(5 votes)

- But Eve could figure out his card, if she randomly guesses. I mean, if you randomly pick a number between one and ten and your friend guesses which number you picked, he could guess right. Is there some way to make SURE that Eve CANNOT guess the card?(3 votes)
- No, there will never be any way to make sure that Eve cannot guess correctly because there always is some correct answer and guessing can alway work. Even so, if the only way to find the answer is to guess randomly, this is considered "perfect secrecy."

The problem is that there are only 10 possible choices in this example so guessing isn't difficult but imagine instead I pick a number between one and a sextillion (1,000,000,000,000,000,000,000). Now the chance of Eve guessing correctly is mind-bogglingly low. If Eve could guess 60 numbers a second for as long as it takes, it would take her on average 264,248,266,531 years to guess the correct number, or about 19 times longer than the universe has existed. And that's only a message of about 15 letters. Yeah.(7 votes)

- I disagree with Brit. There's still a chance of her guessing the card, even if it is a small one.(0 votes)
- The key point is that Eve has no way of knowing if she has the right card. Among the cards are all possible messages, not just a lot of noise surrounding one sensible message. If the message is "My name is Alice", then Eve may very well guess that. But the message could just as well be "My name is Bob". Eve cannot tell which is the right message, and therefore has no information about the content of the message. Eve has the same amount of information about the content of the message as she did before the message was sent.(16 votes)

- I disagree that given a cyphertext, every messagetext is equally likely. Assuming the message is a message, and not just a string of letters, there are messagetexts that can be eliminated as the right answer.

If you receive the cyphertext "gcy" and you know the messagetext is a single English word, you can throw out any solution that does not return at least one vowel.(3 votes)- "and you know the message text is a single English word"

Why do you know that? What if someone was applying a one-time pad to a Caeser cipher? The Caeser cipher would provide unreadable text and then you encrypt that with a one-time pad. Besides "Attack the East" would still be indistinguishable from "Attack at night" since they have the same length so which do you defend against?(3 votes)

- 2:28

How do shannons proofs pertain to information theory?(3 votes)- Great question! The Information Theory lesson will explain this eventually. (hint. randomness is difficult if not impossible to compress)(2 votes)

- I actually have an idea for a cipher that is pretty hard to break. I call it the fifth power cipher. This is the cipher:

A=A

B=F

C=I

D=J

E=E

F=B

G=K

H=H

I=C

J=D

K=G

L=L

M=M

N=N

O=S

P=V

Q=W

R=R

S=O

T=X

U=U

V=P

W=Q

X=T

Y=Y

Z=Z

I know that some of the letters are the same after encryption, but consider an example. Say I want to encrypt the phrase "The world is round." Plugging it into my cipher gives "Xhe qsrlj co rsunj." And if you apply a Caesar cipher of 13 and another fifth power cipher, it gives "Gur jbeyq vo ebhaq." Pretty hard, right?(1 vote)- This cipher can be easily broken by looking at the letter frequency fingerprint. Any method that directly maps letters to other letters will have this vulnerability.(5 votes)

- what if he doesn't lock the correct card(2 votes)
- I would put the card back in the deck and then lock the empty box(3 votes)

- Do we use the one time pad still?(2 votes)
- It's still used for super secure applications, but very rarely.

However, the idea is still used.

1) First we start with a master key of some size.

2) Then our messages will be broken up into small chunks of bits e.g. 64 bits.

3) Then for each chunk we generate a pseudorandom key, based on our master key. The pseudorandom key will be the same size as the chunk.

4) Then for each chunk it's pseudorandom key will be XOR'd with the chunk to encrypt it. So, it's the same as using the one time pad except the pseudorandom key is not truly random. Since the key is not truly random, it can still be broken (but it may be nearly impossible to break it).

This method sacrifices a little bit of security, but allows us to use a short key for long messages.

Hope this makes sense(2 votes)

## Video transcript

(tranquil music) - [Voiceover] Consider the following game. Eve instructs Bob to go into
a room. (door creaks shut) Bob finds the room empty,
except for some locks, an empty box, and a single deck of cards. Eve tells Bob to select a card from the deck and hide it as best he can. The rules are simple. Bob cannot leave the room with anything, cards and keys all stay in the room, and he can put, at most,
one card in the box. Eve agrees that she has
never seen the locks. He wins the game if Eve is
unable to determine his card. So what is his best strategy? Well, Bob selected a
card, six of diamonds, and threw it in the box. (box clicks shut) First he considered the
different types of locks. Maybe he should lock the
card in the box with the key. Though, she could pick locks, so he considers the combination lock. The key is on the back, so if he locks it and scratches it off, it
seems like the best choice. But suddenly he realizes the problem. The remaining cards on the table leak information about his choice, since it's now missing from the deck. The locks are a decoy. (metal jangles) He shouldn't separate
his card from the deck. He returns his card to the deck but can't remember the
position of his card. So he shuffles the deck to randomize it. Shuffling is the best
lock, because it leaves no information about his choice. His card is now equally likely
to be any card in the deck. He can now leave the cards
openly, in confidence. Bob wins the game, because
the best Eve can do is simply guess, as he has left no information about his choice. Most importantly, even if we give Eve unlimited computational power, she can't do any better than a guess. This defines what we
call "perfect secrecy." On September first, 1945,
29-year-old Claude Shannon published a classified paper on this idea. Shannon gave the first mathematical proof for how and why the one time
pad is perfectly secret. Shannon thinks about encryption schemes in the following way. Imagine Alice writes a message
to Bob, 20 letters long. (paper ruffling) This is equivalent to picking one specific page from the message space. The message space can be
thought of as a complete collection of all possible
20 letter messages. (paper ruffling) Anything you can think of that's 20 letters long is a page in this stack. Next, Alice applies a
shared key, which is a list of 20 randomly generated
shifts between one and 26. The key space is the complete collection of all possible outcomes,
so generating a key is equivalent to selecting a page
from this stack at random. When she applies the shift
to encrypt the message, she ends up with a cipher text. The cipher text space represents all possible results of an encryption. When she applies the key, it maps to a unique page in this stack. Notice that the size of the message space equals the size of the key space equals the size of the cipher text space. This defines what we
call "perfect secrecy," for if someone has access to
a page of cipher text only, the only thing that they know is that every message is equally likely. So no amount of computational power could ever help improve a blind guess. Now the big problem, you're
wondering, with the time pad, is we have to share these
long keys in advance. To solve this problem, we
need to relax our definition of secrecy by developing a
definition of pseudo-randomness. (white noise)