And if you see that the block ciphers are

considerably slower than stream ciphers. But we'll see that we can

do many things with block ciphers that we couldn't do very efficiently with,

constructions like RC4. Now my goal for this week is to show you how block ciphers work,

but more importantly I want to show you how to use block ciphers correctly,

for either encryption or integrity or whatever application you have in mind.

So to show you how to use block ciphers correctly, it actually makes a lot of sense

to abstract the concept a little bit so we have a clean, abstract concept of a block cipher

to work with. And then we can argue and reason about what constructions are correct and

what constructions are incorrect. And so an abstraction, it's a very elegant abstraction of a

block cipher is what's called a pseudorandom function, a pseudorandom

permutation. So let me explain what these things are. So a pseudorandom function

basically is defined over a key space, an input space, and an output space.

And all it is, is basically a function that takes a key and an input as inputs and outputs

some element in the output space. Okay, so it takes an element in K, an element in X,

and outputs an element in Y. And the only requirement essentially, is that there's

an efficient way to evaluate the function. For functions we're not requiring that

they be invertible, we just need them to be evaluatable, given the key and the input x.

Now, a related concept that more accurately captures what a block cipher is

it's called a pseudo-random permutation. So a pseudo-random

permutation is, again, defined over a key space, and then just a set X. And what it

does is it takes an element in the key space, an element of X, and outputs,

basically, one element in X. Now, as usual, the function E should be easy to

evaluate. So there should be an algorithm to evaluate the function E. But more

importantly, once we fix the key K, it's important that this function E be one-to-one.

In other words, if you think of the space X as the set here, and here's the

same set X again, then basically the function E, what it does, is, it's a one-to-one

function, so every element in X gets mapped to exactly one element in X.

And then because it's one to one, of course it's also invertible. So given some

output there's only one input that maps to that output. And the requirement is that

there is an efficient inversion algorithm, call it D, that given a particular output,

will output the original preimage that mapped to that output. So really, a

pseudorandom permutation captures very accurately and syntactically what a block

cipher is, and often I'll use the terms interchangeably, either a block cipher or

a pseudorandom permutation. I'll use whichever term depending on the context

where we're discussing things. Okay, so we have two examples, as we said, of

pseudorandom permutations, triple DES and AES, say for AES-128. The key space would

be 128 bits and the output space would be 128 bits. For Triple DES, as we said, the

block size is only 64 bits. And the key size is 168 bits, okay. So we'll use

these running examples actually throughout, so whenever I say a PRP, concretely

you should be thinking AES or triple DES. Now one thing that I

wanted to point out is that in fact any pseudo-random permutation, namely any block

cipher, you can also think of it as a PRF. In fact a PRP is a PRF that has more structure.

In particular, a PRP is a PRF where the input space and the output space are

the same, so X is equal to Y, and in fact is efficiently invertible once you

have the secret key k. Okay. So in some sense a PRP is a special case of a

PRF, although that's not entirely accurate, and we'll see why in just a second.

So, so far, we just described the, kind of, the syntactic definition of what

is a pseudo random permutation and a pseudo random function? So now, let's talk

about what it means for a PRF or a PRP to be secure. And this concept will

essentially capture what it means for a block cipher to be secure, okay? So this

is why I wanted to show you these definitions, before we look at actual

block cipher constructions, so at least it's clear in your mind what it is we're

trying to construct. Okay, so here we have a PRF. And I'm gonna need a little bit of

notation, not too much though, so I'm gonna need to define the set "Funs of X, Y".

This is basically the set of all functions from the set X to the set Y, denoted here as a

big circle, Funs[X, Y]. Now this set is gigantic. Its size is basically, you know,

the size of Y to the size of X, so for example for AES, remember both X and Y

would be 2 to the 128, so for AES the size of the set is enormous. It'll be

2 to the 128 times 2 to the 128. So it's kind of a double exponential.

So this is a gigantic number, this is more particles than there are in the universe.

But regardless, we can kind of think of this set abstractly. We never have to kind of

write it down, we can just keep it in our head and not worry about computing on it.

So this is a particular gigantic set of the set of all functions from X to Y.

Now we're gonna look at a much smaller set of functions, namely I'll call this set

S sub F, and that's gonna denote the set of all functions from X to Y that are

specified by the PRF as soon as we fix the particular key k. Okay so,

we fixed the key k, we let the second argument float, and that basically defines

a function from X to Y. Then we're going to look at the set of all such functions

for all possible keys in the key space. Okay, so, if you think about it again,

for AES, if we're using 128-bit keys, the size of this, I'll say S-AES, is basically going to be

2 to the 128, so much, much, much smaller than the set of all possible

functions from X to Y. And now, we say that a PRF is secure, basically if a

random function in, from X to Y. So we literally, we pick some arbitrary function

in this gigantic set of all functions from X to Y. And we say that the PRF is secure,

if, in fact, a random function from X to Y is indistinguishable from a pseudo-random

function from X to Y. Namely, when we pick a random function from the set SF.

Okay. So, more precisely basically again,

the uniform distribution on the set of pseudorandom functions

is indistinguishable from the uniform distribution on the set of all functions.

Let me be just a little bit more precise,

just to give you a little bit more intuition about what I mean by that and then we'll

move on to actual constructions. So let me a bit more precise about what it means for

a PRF to be secure. And so what we'll do is basically the following. So we have our

adversary, who's trying to distinguish truly random function from a pseudo-random

function. So what we'll do is let them interact with one of the two. So here in

the top cloud, we're choosing a truly random function. In the bottom cloud,

we're just choosing a random key for a pseudo-random function. And now what this

adversary's gonna do is he's gonna submit points in X. So he's gonna submit a bunch

of Xs. In fact, he's gonna do this again and again and again. So he's gonna

submit X1, X2. X3, X4, and then for each one of, of those queries, we're gonna give him

either the value of the truly random function at the point X, or the value of

the pseudorandom function at the point X. Okay, so the adversary doesn't know which

ones he's getting. By the way, for all queries, he's always getting either the truly

random function, or the pseudorandom function. In other words, he's either

interacting with a truly random function for all his queries, or he's interacting

with a pseudorandom function for all his queries. And we say that the PRF is

secure if this poor adversary can't tell the difference. He cannot tell whether he's

interacting with a truly random function or interacting with a pseudo random

function. Okay, and we're gonna come back actually later on and define PRFs more

precisely but for now, I wanted to give you the intuition for what it means for PRFs to be secure

so you'll understand what it is that we're trying to construct when we construct

these pseudorandom functions. And I should say that the definition for a

pseudorandom permutation is pretty much the same, except instead of choosing a

random function, we're going to choose a random permutation on the set X. In other

words, a random one-to-one function on the set X. The adversary can either query this

random function on the set X, or he can query a pseudorandom permutation, and the

PRP is secure if the adversary cannot tell the difference. Okay, so again the

goal is to make these functions and permutations look like truly random

functions or permutations. Okay. So let's look at a simple question. So suppose we

have a secure PRF. So we know this PRF F. Happens to be defined in the set X. And

it so happens, you know, it outputs 128 bits every time. It so happens that this

PRF cannot be distinguished from a truly random function from X to {0,1} to the 128.

Now we're gonna build a new PRF. Let's call this PRF G. And the PRF G is gonna be

defined as follows. We say, if x is equal to zero, always output zero. Otherwise,

if x is not equal to zero, just output the value of F. So, my question to you is,

do you think this G is a secure PRF?

Well, so, the answer, of course, is that its very easy

to distinguish the function G from a random function. All the adversary has to do

is just query the function at X=0. For a random function, the probability

that the result is gonna be zero is one over 2 to the 128. Whereas for the

pseudo-random function, he's always gonna get zero. Because at zero, the function is

always defined to be zero, no matter what the key. And so all he would do is he

would say, hey, I'm interacting with a pseudo-random function if he gets zero at X=0,

and he'll say I'm interacting with a random function if he gets nonzero at X=0.

So it's very easy to distinguish this G from random. So what this example shows

is that even if you have a secure PRF, it's enough that on just one known input

the output is kinda not random, the output is fixed, and already the PRF is broken,

even though you realize that everywhere else the PRF is perfectly

indistinguishable from random. Okay, so let's just show you the power of PRFs.

Let's look at a very easy application. I want to show you that in fact pseudorandom functions

directly give us a pseudorandom generator. Okay. So, let's assume we have

a pseudo random function. So this one happens to go from N bits to N bits. And then,

let's define the following generator. Its seed space is going to be the

key space for the PRF, and its output space is gonna be, basically, t blocks of

n bits each. Okay, so you can see the output is a total of n times t bits for

some parameter T that we can choose. And it turns out, basically, you can do this

very simple construction, this is sometimes called counter mode,

where essentially, you take the PRF and you evaluate it at zero, you evaluate it at one,

you evaluate it at two, at three, at four, up to T, and you concatenate all these values.

That's the generator, okay? So we basically took the key for the PRF

and we expanded it into n times t bits. Okay. A key property of this generator is

that it's parallelizable. What I mean by that is if you have two processors or two cores

that you can compute on, then you can have one core compute the even entries of the

output, and you can have another core compute the odd entries of the output.

So basically, if you have two cores, you can make this generator run twice as fast as

it would if you only have a single core. So the nice thing about this is, of

course, we know that pseudo-random generators give us stream ciphers, and so

this is an example of a parallelizable stream cipher. And I just wanted to point out

that many of the stream ciphers that we looked at before, for example, RC4,

those were inherently sequential. So even if you had two processors, you couldn't make the

stream cipher work any faster than if you just had a single processor. Now the main

question is why is this generator secure? And so here I'm only gonna give you a

little bit of intuition and we're gonna come back and argue this more precisely

later on. But I'll just say that security basically falls directly from the PRF property.

And the way we reason about security, is we say, well this PRF by definition

is indistinguishable from a truly random function on 128 bits.

So in other words, if I take this generator and instead I define a generator using a truly

random function, in other words, I'll write the output of the generator as

f(0) concatenated f(1), and so on and so forth, using a truly random function,

then the output of the generator using the truly random function

would be indistinguishable from the output of the generator using

a pseudorandom function. That is the essence of the security property of a PRF. But with

a truly random function, you notice that the output is just truly random. Because

for a truly random function, f(0) is a random value. f(1) is an independent

random value. f(2) is an independent random value, and so on and so forth.

So the entire output is a truly random output. And so with a truly random function,

a generator produces truly random outputs, and is therefore a perfectly secure

generator. And so you see how the PRF security property lets us argue security.

Basically, we argue that when we replace the PRF with a truly random function, the

construction is necessarily secure, and that says that the construction with a

pseudorandom function is also secure. Okay? And we're gonna see a couple more

examples like this later on. So now you understand what a block cipher is, and you

have intuition for what security properties it's trying to achieve.

And in the next segment, we're gonna look at constructions for block ciphers.