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

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From the course by University of Maryland, College Park

Cryptography

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

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

Maryland Cybersecurity Center

[SOUND] In this lecture we're going to begin our treatment of number theory.

Â Now you might actually be surprised that until this point in the class we

Â haven't had to rely on or to learn any number theory.

Â And the reason you might find that surprising is because my impression is

Â that when people think about cryptography popularly,

Â the first thing that comes to their mind is exactly the number theory.

Â That's kind of the quote, un-quote sexy part of cryptography.

Â However as we've seen in this class so far number theory is not really needed for

Â a lot of important cryptographic applications.

Â And in particular, all of practical private key cryptography is based on

Â things like stream ciphers, block ciphers, and hash functions, that can be

Â constructed analyzed, and reasoned about, without any mention of any number theory.

Â And so I think the point here is that number theory, is, of course, going to be

Â needed, for public key cryptography, but it's not needed for all of cryptography.

Â There's a lot of interesting things you can do without any number theory at all.

Â Having said that of course, it turns out that some kind of

Â number theory does seem to be inherent for any sort of public cryptography.

Â Now we haven't talked about public-key world cryptography yet.

Â We won't talk about it until next week, but this is just sort of a primer for

Â what we're going to need in that setting.

Â The point of this week is to really cover the basics of number theory, very quickly.

Â That is, the goal here is not to give a survey of number theory.

Â It's certainly not to give a comprehensive course on number theory.

Â But only to present the bare minimum that we need for

Â the applications that we're going to study.

Â And for that reason I'm not going to focus particularly on the proofs of a lot of

Â un-underlying concepts and results unless I

Â think that the proof sheds some insight on understanding what's going on.

Â For those that were interested I can recommend several textbooks that can be

Â used to supplement the material and

Â in particular a lot of what we cover here with appropriate proofs.

Â And also some additional details is available in the Cat Lyndel text book.

Â What we're going to talk about in this lecture is algorithmic number theory.

Â And this is actually interesting because very often from a mathematical point of

Â view or from a mathematician's point of view I should say,

Â they're interested in the existence of numbers with certain properties, or

Â in proving that some number has a particular property.

Â But there are very often less interested in the algorithmic efficiency of

Â determining whether or not some number has a given property.

Â In our setting, the computer scientist setting where we're

Â going to ultimately be interested in implementing cryptography.

Â We do have to also take care to think about the algorithmic

Â complexity of determining various properties of numbers, and

Â of computing various interesting things about those numbers.

Â And ultimately, if you're interested in particular in implementing very

Â fast cryptography, a good understanding of

Â these kind of algorithmic considerations is very important.

Â Now one thing that's important to stress,

Â is that when we talk about the efficiency of number theoretic operations,

Â we're always going to be interested in asymptotic complexity, of course.

Â But the asymptotic complexity will be measured in

Â terms of the length of the input.

Â That's always true throughout computer science but

Â it's important to stress it here because it's very often easy to

Â get confused between the magnitude of an input and it's length.

Â And in particular if we have some input integer A then

Â the magnitude of A is itself.

Â If A is equal to 1000 then the magnitude of A is 1000.

Â But the length of that input,

Â the length of A, is the length of the binary representation of A.

Â And I'm going to denote that by these two vertical bars surrounding the integers.

Â So, it's sort of like an extended absolute value sign.

Â So we have these the length of A.

Â The binary representation of a, and

Â the binary representation of an integer is the log of the magnitude of the integer.

Â Right, the number of bits that you need in order to represent some number,

Â is the logarithm of that number's magnitude, and conversely,

Â the magnitude of a number is exponential in its length.

Â And again it's very easy to get these concepts confused when thinking about

Â algorithms but we're going to try very carefully to keep them separate.

Â Now in terms of our study of algorithmic number theory, we're not going to

Â dwell extensively on different algorithms for different computational problems.

Â There's a lot of interesting things to be done there, and as I said it is

Â very important when it comes to efficient implementation of cryptography.

Â But for our purposes what we only want to do, is to

Â be able to distinguish between problems that are easy and problems that are hard.

Â Where for us, easy means that it can be solved in polynomial time.

Â And hard, at least for

Â now, is going to mean that it can't be solved in polynomial time.

Â And so beyond making that kind of a very gross distinction we're not going to

Â be extremely interested in fine tuning the, algorithms that we give or

Â that we discuss, or, the problems we consider.

Â And again, there are several textbooks you can use to supplement the material.

Â Several references you can use if you're ever implementing cryptography and

Â want to find out about the most efficient algorithms for some problem.

Â Before I jump in, I want to just review some notation,

Â regarding modular arithmetic, some of which is a bit idiosyncratic.

Â By a bracket a mod N,

Â I'm going to mean the remainder of the integer a when divided by the integer N.

Â So this means you just take, a divided by N.

Â You compute some quotient, which here we don't care about.

Â And you take the remainder.

Â And note, of course, that bracket A mod N,

Â has to lie strictly between 0 and N minus 1.

Â I want to contrast that with the more standard notation that you might be

Â familiar with, namely that A ie equal to B mod N.

Â So A being equal to B mod N means exactly that A and

Â B have the same remainder modulo N.

Â But, the point is that the notation on the left,

Â A equals B mod N, is different from the notation a mod N inside the brackets.

Â Because again, A equals B mod N is expressing something about

Â the relationship between two integers.

Â And A mod N inside the brackets refers to a specific integer in the range of

Â 0 to N minus1, which is the remainder of A when divided by N.

Â Okay. [SOUND] So

Â let's jump in with a our discussion of algori, algorithmic number

Â theory by just briefly noting some basic facts about what can be done efficiently.

Â So if we look at different standard arithmetic operations,

Â we can see that their's actually can be done efficiently.

Â So integer addition for

Â example, can be done efficiently using the standard grade school algorithm.

Â And if you think about how that works, you'll see that it runs in time,

Â essentially linear, in the length of the inputs and the same for subtraction.

Â Multiplication can also be done using a grade school algorithm.

Â The running time now is quadratic rather than liner.

Â And I'll mention that better algorithms are known, but, again, for

Â our purposes, we're just going to be happy with the grade school algorithm because it

Â does run in polynomial time.

Â A little bit more difficult to see, but perhaps not very surprising,

Â is that division with remainder, can be done efficiently.

Â And this in particular means that we can compute bracket A mod N

Â efficiently by just performing the divisions and taking the remainder.

Â For that reason similarly, we can do modular addition,

Â subtraction, multiplication, and reduction efficiently as well.

Â So for example, modular multiplication, would just mean that we have two integers.

Â We want to compute their product, modulus if any N, modulus N if any.

Â [COUGH] And what we can do there is to simply perform the multiplication over

Â the integers and then reduce mod N.

Â And because of those component operations can be

Â done efficiently we know that the entire computation can be done efficiently.

Â What about a more complicated operation, in particular, modular exponentiation?

Â And I focus here on modular exponentiation really for two reasons.

Â First is that it's an important computation that's going to be

Â used all the time in cryptography, and second,

Â because I think it gives a very good illustration of a non trivial algorithm.

Â That is one where the naive algorithm as we'll see does not run in polynomial time,

Â but with a little bit of cleverness you can come up with an algorithm that

Â does run in polynomial time.

Â So lets look at the problem of exponentiation.

Â First of all think about exponentiation over the integers.

Â So here's there's no modular reduction.

Â Let's say we want to compute A to the B, all right, the integer A to the Bth power.

Â Well, let's first consider the size of the result.

Â So the number of bits that we need to specify the result,

Â A to the B, is going to be the order of the logarithm of that,

Â which is the order of the magnitude of B, times the length of A.

Â And what you can see, here, is just writing down the answer

Â can require exponential time, exponential in B, that is.

Â Because the running time, sorry the, the number of bits we need to specify

Â the answer, is going to be linear in the magnitude of B, rather than length of B.

Â So we don't really have any hope of computing integer exponentiation in

Â polynomial time.

Â Fortunately in cryptography we never need to compute integer exponentiation,

Â what we do need instead is modular exponentiation, where we're interested in

Â computing the result of A to the B modulo sum, given modulus N.

Â And note here we don't run into the problem we had before,

Â because the size of the answer here is going to have to be at most the size of N,

Â because the answer has to lie in the interval 0 to N minus 1.

Â So how do we do it?

Â Well the first thing to observe is that a strategy that will not work.

Â If the compute a to the b over the integers.

Â And then reduce the result mod N.

Â That won't work because as we just noted, the the length of the intermediate result,

Â A to the B over the integers cannot even be written down in polynomial space.

Â So, that's not the approach we're going to want to use.

Â Let's consider a, maybe a naive algorithm that we would want to use for

Â computing modular exponentiation.

Â So here, I'm just writing a simple function that takes three arguments, A, B,

Â and N and is meant to compute the result, A to the B, mod N.

Â And we'll assume that B is greater than or equal to 0.

Â We won't worry about negative exponents, for now.

Â They can be handled but it's not important for our purposes.

Â So one natural approach is to just start off with a variable called ans that's

Â going to hold the answer, initialize that to 1.

Â And then for I ranging from 1 to B, what we'll do is we'll simply multiply

Â the current value of the answer by A and then reduce mod N.

Â So this of course is not valid C code.

Â This is meant to just be pseudo code.

Â But what I've indicated here is that we just multiply a into the current answer

Â and then increment B.

Â And we're going to avoid the problem that we had on the previous slide.

Â Of first computing A to the B and then reducing modular N.

Â Because you'll notice here that we're reducing modular N at every step.

Â So every time we multiply, we're going to then reduce.

Â And so the value of Ns at any point during the computation,

Â is going to be strictly in the range of 0 to N minus 1.

Â So we do this for a total of B iterations of the loop, and

Â then we return the answer.

Â This will indeed give the right result, but

Â what's the running time of the algorithm?

Â Well, it's easy to see here that the inner loop is going to be running for

Â a total of B times.

Â And that is not, efficient.

Â That is not polynomial time.

Â Because we would like an algorithm that runs in time polynomial in the length of B

Â not in the magnitude of B.

Â And again here we have a loop running for a number of steps linear in

Â the magnitude of B, so this is not going to qualify as an efficient algorithm.

Â What kind of approach could we use to get a more efficient algorithm,

Â to get in particular a polynomial kind of algorithm?

Â Well it's easiest to think about the problem if we assume for

Â simplicity that B is a power of 2.

Â So let's say that B equals 2 to the K for some integer K.

Â If you kind of unwrap the proceeding algorithm or

Â maybe just view it in a different way.

Â You can see that the preceding algorithm roughly corresponds to

Â performing B multiplications of A.

Â That is, to computing A times A, times A again, times A again, et cetera, for

Â A total of 2 to the K minus 1 multiplications.

Â But a better way to do the computation.

Â Would be to first square A, right to just compute A times A,

Â take that result and square it again.

Â Take that and square that, and keep going until you've done a total of K squarings.

Â Or K minus 1 squarings I guess.

Â So, that would give a much better algorithm.

Â Then the former one because we have k

Â squarings versus two to the k minus one multiplications.

Â And again, the important thing to note here is that k is order of log of B,

Â which means that K is order of the length of B.

Â So what we have here in the second case is an algorithm that's running in

Â time linear, in the length of B.

Â So of course this only works when B is a power of two.

Â But as we'll see in the next slide we can modify this approach even when B

Â is not a power of 2.

Â So here is the algorithm.

Â So again we're going to have an algorithm taking as input A, B, and N.

Â And we're going to again assume that B is greater than or equal to 0.

Â What we'll do here is to initialize two variables.

Â One called X and one called T.

Â We'll initialize X to a and T to 1.

Â And then while B is greater than 0 we do the following.

Â So first of all we check if B is odd.

Â If B is odd we then compute a within a set T equal to T times X mod N.

Â And we reduce B by 1.

Â Then we compute X equals X squared And B is equal to B divided by 2.

Â So note that in this point the algorithm must be even because if it

Â were odd at the beginning of the loop then in the previous step we checked that it

Â was odd and so we subtracted 1.

Â We keep doing this as long as B is greater than 0, and then finally return t.

Â Now it's a bit difficult, perhaps, to be convinced that this is working.

Â But you can prove that the algorithm works by looking at, or

Â by showing that the algorithm satisfies the following invariant.

Â And the invariant is, that the answer, the correct answer that you're looking for

Â is always equal to T times X to the B amount N.

Â And that's certainly true at the beginning of the algorithm when we

Â initialized the variables because then X is equal to A and T is equal to 1.

Â And indeed A to the B amount N is exactly the answer we're looking for.

Â And then you can check throughout the loop.

Â That if B is even for example,

Â then what we end up doing is squaring the value of X and reducing B in half.

Â And so X to the b mod N, or t times x to the b mod N,

Â is of course equal to T to the X squared, to the B over 2 mod N.

Â Just algebra.

Â And similarly, if B is odd, we then shift a factor of X into T, and reduce B by 1.

Â And you can write down on paper and easily verify that the invariant is satisfied.

Â At the end of the algorithm, B is equal to 0, and

Â so the correct answer is given by exactly T.

Â And T is a value that we return.

Â What's the running time here?

Â Well the important thing to note is that in every execution of the inner loop.

Â The value of B is decreased by at least half.

Â And that means that the number of iterations of the inner loop is going to

Â be 0 of log B, which is linear in the length of B.

Â In each iteration of the inner loop,

Â we perform some computation that can be done in time polynomial, in, A, B and N.

Â And so without getting into too much of a fine grained analysis we can see that

Â the overall running time of the algorithm is indeed polynomial in the length

Â of it's three inputs.

Â In the next lecture we'll spend some more time on modular arithmetic.

Â And we'll also introduce the very important notion of an abstract group

Â that will allow us to think in a more clean away about various of

Â the manipulations that we're going to be doing and

Â that we're going to use for crypt, for cryptography.

Â See you next time.

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