This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

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Cryptography

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This course will introduce you to the foundations of modern cryptography, with an eye toward practical applications.

From the lesson

Week 1

Introduction to Classical Cryptography

- Jonathan KatzProfessor, University of Maryland, and Director, Maryland Cybersecurity Center

Maryland Cybersecurity Center

[SOUND] In the previous lecture we talked

Â about defining what we mean by secure encryption.

Â And we reached an informal description of the security we were after.

Â Recall that what we said is that we should consider an encryption scheme secure

Â if regardless of any prior information the attacker has about the plaintext,

Â observing a ciphertext should reveal no additional information to

Â the attacker about the plaintext.

Â Remember also that we say we were going to be considering the threat model

Â of a ciphertext only attack, meaning that the attacker only observes ciphertext,

Â passively, but doesn't actively interfere with either the sender or receiver.

Â And that we would be considering the simplest threat model which is that

Â the attacker observes only a single ciphertext.

Â We can adapt our definition to more complicated cases, but

Â this is the one we're going to be considering for now.

Â Now what we want to do in this lecture, is to define this informal definition of

Â security that we have, formally, rigorously and mathematically.

Â To do that we're going to need to start talking about probability and so

Â I just want to go over some very basic concepts in probability that I assume you

Â have encountered before in one form or another before.

Â Again we are not going to need a very formal view of probability.

Â That formal view is, there it's present.

Â We can define everything completely rigorously if we wanted to but

Â I just want to assume here that you seen these concepts before, that you

Â are familiar with them, whether or not you have seen them in their complete rigor.

Â So we're going to use the, terminology of a random variable.

Â Now again, this has a formal definition in probability theory.

Â For our purposes, it's enough to just think of a random variable as a variable

Â that takes on certain values with certain specified probabilities.

Â For our purposes we're only going to be concerned with discrete random variables.

Â That means that there's some finite set of values that the variable can

Â possibly take.

Â And so we don't have to worry actually about taking integrals or

Â about considering continuous parameters or

Â continuous distributions on those random variables.

Â So again, a random variable is just going to be a variable that takes on

Â certain values with some probabilities that we're going to specify.

Â And in particular, a probability distribution for

Â a random variable is going to be a specification of the probabilities

Â with which that variable takes on each possible value.

Â We know that these probabilities must be in the range between 0 and 1 inclusive.

Â And, furthermore, the sum of the probabilities with

Â which the variable takes on all these different values should equal 1, right?

Â The various probabilities should all sum to 1.

Â again, we'll also use the terminology of an event, which is something that has

Â a formal definition within probability theory, but for our purposes,

Â we can just think of, as a particular occurrence, in some randomized experiment.

Â And we'll write probability bracket E, to denote the probability of

Â some particular event E, the probability that some occurrence actually happens.

Â So a typical example of an event that we'll be concerned with

Â is the probability that some random variable takes on some value.

Â We then have the notion of conditional probability which is the probability

Â that one event occurs, assuming that some other event has occurred.

Â And we'll write this as the probability of A Bar B.

Â That is the probability of event A, conditioned on the occurrence of event B.

Â And by definition the probability of A conditioned on

Â B is equal to the probability that both events A and

Â B occur, divided by the probability that event B occurs.

Â We're assuming here that event B occurs with non zero probability so

Â we don't have any division by zero, okay?

Â So this is both an informal definition of what we mean by conditional probability as

Â well as a formal definition of what the notation probability of

Â A conditioned on B actually means.

Â With this definition of conditional probability we can now define formally

Â what it means for two random variables X and Y to be independent.

Â And actually I'll point out here some notation that I'll generally be

Â consistent with which is to use capital letters for random variables and

Â lower case letters for values that those random variables can take.

Â Okay so here we have random variables capital X and capital Y,

Â and we'll say that they're independent, if for all possible values X and

Â Y, the probability that X, takes on the value little x,

Â conditioned on the event, that random variable Y, takes on value lowercase y.

Â Is exactly equal to the probability that random variable X takes on the value X in

Â the first place, i.e.

Â The fact that the random variable Y takes on any particular value is irrelevant as

Â far as determining the probability with which X takes on some particular value.

Â Finally we're going to use in this lecture and some of the examples the law

Â of total probability which is just a very convenient way to break up and

Â therefore calculate the value of some probability.

Â So let's let E1 through EN,

Â be events, that partition the space of all possibilities.

Â This means, that the Ei, are all, pairwise impossible.

Â That means we can't have it be the case that both Ei, and Ej occur for

Â any i and j.

Â So that is if E1 occurs, then it's impossible for E2 to occur also, and

Â if En minus 1 occurs, then it can't be the case that E3 occurs.

Â Okay, so these events are, are pairwise impossible and

Â moreover they partition the space of all possibilities in the sense that at least

Â exactly one of E1 through En have to occur, so some event Ei has to occur.

Â Then for any other event A we can express the probability of event A as

Â the sum over all i of the probabilities that A occurs and also Ei occurs.

Â Right, this completely partitions the possibilities for

Â when A occurs because some Ei has to occur.

Â Exactly one of the Ei's has to occur.

Â And so by summing over the probabilities of A and each of those events Ei,

Â we recover the total probability that event A occurred.

Â And then we can break this up further using our definition of

Â conditional property to be equal to the sum over all i of the probability of

Â A conditioned on event Ei times the probability of event Ei.

Â Think, now again, I do assume that you've seen or

Â encountered all of these concepts before.

Â This was just meant to be a brief review.

Â If you haven't, then I would recommend you look at some

Â introductory probability text.

Â The first chapter or the first chapter and a half perhaps.

Â For more, thorough coverage of these different term, terms and definitions.

Â So let's get back now to cryptography.

Â So remember the formal definition of a private key encryption scheme, okay?

Â A private key encryption scheme is defined by a message base, bold M, all right,

Â that's the set of all possible messages that are allowed to be or that can be

Â encrypted by the scheme, as well as three algorithms, Gen, Enc and Dec.

Â Gen was the key generation algorithm.

Â This is a probabilistic algorithm that generates a key,

Â that Iâ€™ll denote by lower case k.

Â The encryption algorithm takes as input a key k, and a message M,

Â in the message base and outputs some ciphertext E.

Â And just remember that we're allowing the encryption algorithm to

Â be randomized meaning that if you run it several times, even using the same inputs,

Â k and m, you can potentially get different ciphertext out.

Â 'Kay, we haven't seen an example of such a scheme yet, but we're,

Â we're allowing that in our definition.

Â Finally, the decryption algorithm takes as input a key k and

Â a ciphertext c and outputs a message m.

Â And we have the correctness condition that decryption of a ciphertext encrypted,

Â that the encryption is the message, where you encrypt and

Â decrypt using the same key, should recover the original message.

Â Again, I haven't put that on this slide.

Â Now by way of some more notation, I want to define in parallel to

Â the notion of using bold face m for the message space.

Â I want to define the notation of a bold face k to denote the key space.

Â Thatâ€™s just the set of all possible keys, that is the set of

Â all possible things that can be output by the key generation algorithm.

Â We use also a bold face C, for the ciphertext space, that is the set of

Â all possible ciphertexts that can be output by the encryption algorithm,

Â taken over all possible keys that can be output by the key generation algorithm and

Â all possible messages that are in the message space.

Â So now what we want to do is look at probability distributions

Â over the messages that can be sent.

Â So we'll let capital M without the bold face, okay,

Â capital M be a random variable denoting the value of

Â the message that the sender wants to communicate to the receiver.

Â So this is a random variable that can potentially take on values in

Â the entire message space.

Â Right, so a priori, the set of possible values for

Â what the message can be are exactly those values that are in the message space.

Â We're not allowed to encrypt anything outside the message space but

Â we are potentially allowed to encrypt anything we like in the mess,

Â message space.

Â Now, the random variable M is meant to reflect the likelihood of different

Â messages being sent by the parties, and this takes into

Â account anything that the attacker might know about what those messages might be.

Â So this distribution, this random variable, M,

Â over the messages that can be sent is meant to reflect, not only the different

Â probabilities with which the sender might want to transmit a certain message.

Â But also something about, perhaps, the attacker's external knowledge that may

Â have some influence or some bearing over what the sender is sending.

Â So just as a, as a, as one particular example,

Â we might have the following distribution over random variable M,

Â that the probability that the message is equal to attack today Is 0.7 and

Â the probability that the message is don't attack is 0.3.

Â And everything else in the message space has a probability of zero.

Â So the only possibilities are either attack today or

Â don't attack and those are going to occur with respective probability 70% and 30%.

Â Okay, now in other situations of course, it might be different.

Â Maybe again, those are the only two possible messages, but

Â maybe the attacker has no idea whether the sender wants to attack or not.

Â And therefore the probabilities would each be 0.5 or maybe there are some other

Â messages that are possible that can occur with different probability what have you.

Â You're right, you have different possibilities of what those can be, but

Â here we're just fixing some particular distribution for

Â this random variable M as specified, okay?

Â Now we're going to let capital K be a random variable denoting the key.

Â The values that this random variable can take are exactly those in

Â the key space, right?

Â Only those values in the key space are possibilities for the key.

Â And actually any prob, any value in that space is a possible value for the key and

Â so is a possible value that this random variable K can take.

Â Now one important thing is that if we fix the encryption scheme, or

Â fix some encryption scheme, Gen, Enc and Dec, then the key generation algorithm

Â itself defines for you the probability distribution for K, right.

Â The probability that the random variable K, takes on some particular value,

Â lowercase k or

Â the probability that the key is equal to some particular value K is, by definition,

Â equal to the probability that the key generation algorithm outputs that key.

Â The probability that the key generation algorithm outputs the particular key,

Â given by this value lowercase k.

Â So the encryption scheme defines your distribution over the random variable k.

Â Now we're going to make the very crucial and

Â important assumption that the random variables M and K are independent.

Â Right, this means that the probabilities with which the message takes on some value

Â are independent of the probabilities that the key takes on some particular value.

Â Now this is actually a very reasonable assumption,

Â because the messages that a party wants to send to another party should not

Â depend on the key that's used to encrypt that message, right?

Â We imagine that we externally to the encryption scheme,

Â have some message that they want to communicate.

Â And then we're sampling a key and

Â using some encryption scheme to encrypt that message.

Â But the details of the encryption scheme and the particular key we ended up

Â choosing shouldn't influence the message that we're sending.

Â This is a crucial assumption, and one that we're going to make throughout the class.

Â It is possible to concoct scenarios where this assumption doesn't hold.

Â But in that case, the definitions we're giving do not apply, and

Â much more complex definitions need to be considered.

Â But traditionally we, the assumption is made that M and K are independent, and

Â this is the assumption we are going to make here.

Â Moreover, this is the reasonable assumption that should hold except in

Â some very unusual circumstances.

Â So, let's fix some encryption scheme, given by Gen, Enc and

Â Dec, and fix some distribution for the message.

Â Some distribution on capital M on the message that the sender is going to

Â send to the receiver.

Â So we have now fixed by the encryption scheme the distribution over capital K,

Â and we've fixed by assumption a distribution over

Â the message space given by capital M.

Â Consider now the following randomized experiment.

Â What we'll do, is first, chose a message little m,

Â choose a particular message m, according to the given distribution, right?

Â So, going to back to the example before,

Â that means we might pick the message attack now with probability 0.7 and

Â the message don't attack, with probability 0.3.

Â So we sample it according to those probabilities and

Â now we've chosen our message and let's call that little m.

Â We then generate a key, K,

Â using the key generation algorithm, i.e we sample the key k

Â according to probability distribution on the random variable capital K.

Â And then what we're going to do is kind of couple those together

Â by encrypting the message we've chosen using the key that we've chosen, and

Â the encryption algorithm that we're given having fixed the encryption scheme.

Â This gives us a ciphertext that we'll denote by little c.

Â The key point here is that this defines a distribution on the ciphertext, right?

Â This process of sampling a random message, sampling a key and

Â then encrypting the message using that key defines some distribution on

Â the ciphertext, and we can denote the distribution on the random variable

Â that denotes the ciphertext by the variable capital C.

Â 'Kay, so we'll let capital C be the random variable denoting the value

Â that the ciphertext takes on after this experiment.

Â I think it's helpful here to walk through an example of how this works out and

Â do an explicit calculation.

Â So let's consider the shift cipher.

Â We said that the the fixing an encryption scheme defines the distribution over

Â capital K.

Â And indeed in the shift cipher, the key is chosen uniformly from the set of

Â all characters, or from the set of all numbers between 0 and 25, i.e.,

Â the distribution on capital K is the one where the probability that the key

Â takes on, any particular value between 0 and 25, is exactly 1 over 26.

Â Right, every key in the range of 0 to, 0 to 25 is chosen with equal probability.

Â Now let's assume we'll take a simple example of

Â a distribution over the message space.

Â Assume that the probability that the message is the single character a is 0.7,

Â and the probability that the message is the single character z is 0.3.

Â Well, let's calculate now

Â the probability that the ciphertext is the single character b.

Â Well, there are only two possibilities.

Â Either the message is equal to a and the key is equal to 1.

Â Or, in that case we shift our message by one position and indeed get

Â the ciphertext b, or the message is equal to z and the key is equal to 2.

Â Those are the only two ways that we can end up with a ciphertext being equal to

Â the character b.

Â And now we just have to compute these probabilities.

Â So the probability that the ciphertext is equal to the character b is equal to

Â the probability that the message is equal to a and the key is equal to 1,

Â plus the probability that the message is equal to z and the key is equal to 2.

Â Because the message and

Â the key are independent, those become the, just the products of the two terms.

Â So we have the product sorry the, we have the probability that M equals a times

Â the probability of K equalling 1, plus the probability that M is equal to z,

Â times the probability that K is equal to 2.

Â And now we just plug in the known values for those different probabilities.

Â Right, the probability that M, is equal to a, we said was 0.7.

Â The probability that k is equal to 1 is 1 over 26 and

Â the probability that m is equal to z is 0.3.

Â The probability that the key is equal to 2 is 1 over 26.

Â Adding that all together, we see that the total probability with which

Â the ciphertext is equal to the character b is exactly 1 over 26.

Â 'Kay, nothing magical, but just going through the steps,

Â I think clarifies a little what we mean, what we meant on the previous slide.

Â Taking another example, a slightly more complicated one,

Â we'll again consider the shift cipher, but

Â now let's look at a different distribution over the message space.

Â So now we'll assume that we have two possible messages that can be sent,

Â like before, but they're now longer than one character each.

Â So we have that with probability one half, the message is equal to the string O-N-E.

Â And with probability one half, the message is equal to the string T-E-N.

Â Right? One or ten.

Â What's the probability that the ciphertext happens to

Â turn out to be equal to the string rqh?

Â Well, if we break it up as before, what we have is that this is equal to,

Â using the law of total probability, the probability that C is equal to rqh

Â conditioned on M being equal to one, times the probability that M is equal to one,

Â plus the probability that C is equal to rqh conditioned on M being equal to ten,

Â times the probability that M is equal to ten.

Â And we can compute that as follows, so conditioned on the message,

Â being equal to one, the cipher text turns out to be equal to rqh if and

Â only if we have a shift of three i.e.,.

Â the key was equal to three.

Â Right in that case we have o going to r,

Â we have n going to q and we have e mapping to h.

Â So now the probability that the ciphertext is equal to

Â rqh conditioned on the fact that the message is equal to

Â one is exactly the probability that the key is equal to three, which is 1 over 26.

Â We multiply that now by the a priori probability with which

Â the message was equal to one, which we're given as one half, and

Â now we come to a more interesting calculation, right?

Â What's the probability that the ciphertext is rqh conditioned on

Â the message being equal to ten?

Â Well, if you think about it a little bit, or if you try to exhaust over all

Â possible keys, you can see actually that there's no possible way

Â the ciphertext can be equal to rqh, given that the message is equal to ten.

Â And the reason for that is simply that there is no shift that will

Â simultaneously shift t to r, e to q, and n to h.

Â There's just no way for that to happen.

Â So the probability that the ciphertext is equal to rqh

Â conditioned on M being equal to ten is 0.

Â We can write one half for the probability that M equals ten.

Â It's not going to matter in the end anyway because it's going to go to zero,

Â but there you have it.

Â And so by summing these probabilities we get that the probability that C,

Â the ciphertext is equal to the string rqh, is exactly 1 over 52.

Â Now, recall the definition of perfect secrecy, from the earlier in the lecture.

Â All right, we said that regardless of any prior information the attacker has

Â about the plaintext, the ciphertext that the attacker observes should leak

Â no additional information about the plaintext.

Â What we're going to do now is equate the attacker's information about

Â the plaintext with the attacker known distribution over the plaintext.

Â So, what this means is that perfect secrecy will require that observing

Â the ciphertext should not change the attacker's knowledge about

Â the distribution of the plaintext at all.

Â Right, so the attacker has some knowledge about the distribution of the plaintext

Â before observing the ciphertext.

Â But that's just given by the distribution capital M,

Â the distribution over the possible messages that might be sent.

Â After observing the ciphertext the attacker can update its knowledge about

Â the distribution, and what we require is that a scheme is, is defined as,

Â to be perfectly secret, if, and only if those two distributions, right,

Â the a priori distribution before observing a ciphertext, and the a posteriori

Â distribution, after observing a ciphertext, should be exactly equal.

Â So more formally, what this means, is that an encryption scheme, defined by

Â the algorithms Gen, Enc, and Dec, and the message space bold face M is perfectly

Â secret, if for every possible distribution over the message, over the message space,

Â right, any possible distribution over the possible messages that might be sent.

Â Every message m and every ciphertext c that can occur with non zero probability,

Â it holds that the probability that the message was equal to m

Â conditioned on observing a ciphertext equal to lowercase c is exactly

Â equal to the a priori probability that the message was equal to m in the first place.

Â So we have here an expression, on the right hand side we have the a priori

Â probability with which the message takes on some value lowercase m.

Â Right, this is the attacker's knowledge about the distribution about the plaintext

Â before observing any ciphertext.

Â And we're requiring that to be equal to the left hand side,

Â which is the probability that the message was equal to some value M, conditioned on

Â observing that the ciphertext took on some particular value lowercase c.

Â Okay, so again this is our definition of perfect secrecy.

Â This is just a formalization, a mathematical description of our informal,

Â intuitive notion of security that we had on the previous slide.

Â Now I know the definition is a little bit difficult to get used to, and

Â so what I wanted to do was walk through just two examples of

Â calculations that pertain to that definition.

Â Let's go back to an example we had before, where we had the shift cipher.

Â And again,

Â the distribution where the message is equal to one with probability one half.

Â And the message is equal to ten with the probability one half as well.

Â And let's take as one particular message the string ten and

Â as one particular ciphertext the string rqh.

Â Well, we can then ask, what's the probability that the message was equal

Â to ten conditioned on the fact that we observed a ciphertext equal to rqh?

Â You might already be able to answer this question just based on intuition, but

Â I want to go through the explicit calculation.

Â Well, the reason, we, we can actually say quickly that this is exactly equal to 0.

Â Sorry, I didn't go through the calculation.

Â I'll do that in the next example.

Â Why is that?

Â Okay, well remember what we said before is that the condition on

Â the message being equal to ten.

Â It's simply not possible for the ciphertext to be equal to rqh.

Â There's no key that will map the message M onto the cipher text rqh.

Â What that means is that if we've observed the ciphertext rqh,

Â then it cannot possibly the case, be the case that the message was ten, right?

Â Because again, there's no possible way for

Â the message to be equal to ten and the ciphertext to be equal to rqh.

Â So therefore, the probability that the message is equal to ten,

Â conditioned on the ciphertext being equal to rqh is precisely 0.

Â It's simply not possible that the message was ten given that

Â we've observed the ciphertext rqh going across the channel.

Â The thing to note here is that this is not equal to the a priori probability that

Â the message is equal to ten right?

Â The a priori probability that the message was equal to ten is one half which is

Â not equal to 0.

Â And this shows actually that the shift cipher does not meet the definition of

Â perfect secrecy.

Â I'm going to turn next to a more complicated example, and

Â in analyzing that we're going to use Bayes's theorem.

Â This is something else that I'm assuming that you've seen before but

Â I just wanted to review it very quickly.

Â So Bayes' theorem is just a way of switching the order of two events in

Â a conditional probability statement.

Â And it simply says that the probability of A conditioned on B

Â is equal to the probability of B conditioned on A times the probability of

Â A divided by the probability of B.

Â It's quite easy to derive this on your own from what we had on the previous slides,

Â but in any case, this is the statement that we're going to be using.

Â So, for the next example, we'll make it a little more complicated yet,

Â we'll take again the shift cipher as our example, but

Â now we'll consider a distribution over three possible messages.

Â Right, so there are three different messages that can potentially be sent.

Â We'll have that the message is equal to hi with probability 0.3,

Â the message is equal to no with probability 0.2,

Â and the message is equal to in with probability 0.5.

Â I just made these numbers up.

Â What's now the probability that the message is equal to hi,

Â conditioned on observing the ciphertext being equal to xy?

Â Well, we could rewrite this using Bayes' theorem in the following way.

Â It's just a probability that the ciphertext is equal to xy,

Â conditioned on the message being equal to hi, times the probability that the message

Â is equal to hi, divided by the probability that the ciphertext is equal to xy.

Â And now we just need to calculate each of those probabilities individually.

Â Well, the probability that the ciphertext is equal to xy

Â conditioned on the message being equal to hi is easy to calculate.

Â The prod, get, conditioned on the message being equal to hi,

Â there's exactly a single key that will result in a ciphertext xy.

Â I forget exactly what the shift is but

Â you can see just by exhausting over all 26 possibilities that there's

Â exactly one value of the key that will map the message M onto the ciphertext.

Â Sorry, that will map the message hi onto the ciphertext xy.

Â So therefore the probability that that key is the one that's chosen by

Â the key generation algorithm is exactly 1 over 26.

Â Because again the shift cipher chooses keys uniformly from the space of 26

Â possible keys.

Â More interesting or more complicated,

Â is the computation of the probability that the ciphertext is equal to XY.

Â Here, we'll just go back to the law of total probability that we used before.

Â So the probability that the ciphertext is equal to xy,

Â we can break that up as the sum of three different terms.

Â The first term is the probability that C equals xy conditioned on the message being

Â equal to hi times the probability of hi,

Â which is 0.3, plus the probability that c equals xy conditioned on M equals no,

Â times the probability that M equals no, which was 0.2.

Â Plus the probability that C equals xy conditioned on the event that

Â the message was equal to in times the probability of in which was 0.5.

Â Now the probability that the cipher text is equal to xy,

Â conditioned on the message being equal to hi is 1 over 26 as we said earlier.

Â The probability that the ciphertext is xy conditioned on

Â the message being equal to no is also 1 over 26.

Â There's again exactly one key that will map the message M onto

Â the ciphertext xy and the probability that that key is chosen is exactly 1 over 26.

Â On the other hand the probability that the ciphertext is equal to

Â xy conditioned on the message being equal to in is 0.

Â You can again check that there's no possible key that will

Â map the message in onto the ciphertext xy.

Â And so when we add these up we get a total probability of 1 over 52.

Â Finally, the probability that the message is equal to hi conditioned on C equals xy.

Â We can go back to our previous expression in terms of Bayes' Law,

Â plug in the values that we computed, and compute the value of 0.6.

Â So the probability that the message was equal to hi,

Â conditioned on the ciphertext being equal to xy, is exactly 0.6.

Â And the thing to note again, is that this is not equal to the a priori probability,

Â with which the message was equal to hi, which was 0.3.

Â So this demonstrates again that the shift cipher is not perfectly secret because we

Â found a particular distribution over the message space for

Â which the a priori probability that the message took on some

Â particular value given that the observed ciphertext took on some other value,

Â was not equal to the a priori probability with which the message took on that value.

Â So just to summarize, we see here two examples demonstrating that

Â the shift cipher is not perfectly secret, and the question that

Â remains is how can we construct a perfectly secret scheme, right?

Â We have this nice definition of what perfect secrecy means,

Â this nice definition that encapsulates our intuitive understanding or

Â our intuitive desire for what we want to achieve by an encryption scheme.

Â How can we build an encryption scheme and prove that it satisfies that definition?

Â And this is the problem we'll turn to in the next lecture.

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