 So number theory will, just a number of concepts about addition, subtraction, multiplication, division, which you all know, but we'll adjust it a little bit from what you, what you commonly think about and then see a few equations which were useful and some other properties of numbers which are useful when it comes to public key cryptography. We'll start very simple. We'll talk first about divisibility in prime numbers and I think you learn even in primary school now that we can talk about one number divides another number. So we can say B divides A if there's some M, some integer M such that when we multiply by B we get A. That is if we can divide and there's no remainder we say B divides A and we talk about divisors. So we can say B is a divisor of A if we can take A and divide by B with no remainder we say B is a divisor of A. Sometimes we may write that in a short form of B vertical bar A. So we know about divisors. We can then also, given a number has a set of divisors we can talk about the greatest common divisor amongst two numbers. So A has a set of divisors, B has a set of divisors, what's the greatest of that set? The greatest common divisor or GCD and there are algorithms for finding the greatest common divisor. We'll go through a couple of short examples and a new thing that you may not have heard of or you can't remember. If two integers A and B have a greatest common divisor of one we say those two integers are relatively prime. A couple of examples. What are the divisors of 16? It's that simple. Two, is there a number before two? One, alright. We can divide by two. We cannot divide by three. Okay, so simply the divisors. Easy. Twenty-four, we can divide by three. Four by six, three by eight, two by twelve, one by twenty-four. So that's the divisors. Greatest common divisor of sixteen and twenty-four, well the greatest value there is eight. That's all. We've done this many times in high school, primary school maybe. Fifteen divisors, one, three, five and fifteen. Greatest common divisor of sixteen and fifteen is one. If we look at those sets, one. So we say there that fifteen and sixteen are relatively prime. They're not prime numbers, we'll see them later, but we say that those two numbers are relatively prime to each other. Fifteen is relatively prime to sixteen and vice versa. Because the greatest common divisor is one. And another thing you probably know is that we can often write, well, that's all we've gone there. The next thing we know is about prime numbers. So instead of writing the divisors, we often think about the prime. So what are the primes on the slides? What's a prime number? Any integer greater than one, if and only, if it's only divisors, a plus or minus one and plus or minus p itself. It can be divided by one and itself only. And we can say any integer can be factored into the prime. So the divisors can be written as a set of primes multiplied together. So the divisors, we also say the factors, the factors of fifteen, one, three, five and fifteen. So we can talk about prime factors of a number. So some integer a can be written as multiplying a set of primes together, prime numbers only. Sometimes there will be multiple primes multiplied together. So we can talk about the prime factors of a number. What are the prime factors of sixteen? How can we write sixteen as multiplying primes together? And let's forget about one. So what's the first prime? Two. Two is a prime. It can only be divided by one and itself. One, our definition we consider greater than one. So two, two to the power of three, two to the power of four, sorry. Sixteen is two times two times two times two. So its prime factors are two and so the factor is two but it's four times there. 24 is what? What are the prime factors of 24? One's not, six is not prime, two and three, two to the power of three is eight times by three is twenty-four. So two and three are the prime factors. We can say 24 is, so in fact what we can generally do, we can do it for fifteen as well, three and five are the primes there. In general any integer can be expressed as multiplying primes together and sometimes we just list it in a general form such as here. And there's a mistake here, it's not P2, it should be P1 as the first subscript. That is some integer is prime one to the power of some positive integer, zero, zero one or above times the second prime, so prime one is two, the second prime is three, the third prime is five and so on. So we can multiply the primes and give some exponent and that will create any integer. So we'll often come back to prime factors. Some number can be written as multiplying primes together or the other way, given a number we can find those factors. What are the prime factors of a number? It will come up as an important problem. What's the prime factors of twenty-two? Two and eleven. Hundred and forty-five, there are two as a hint, five and five times what? Twenty-nine, five and twenty-nine, so a little bit more work in the head but we can find prime factors and there are algorithms for doing so and we'll come back to it but an important point for security is that for a very large number, not one hundred and forty-five but maybe a number with one hundred and forty-five digits, a very large number, it takes a long time to find the two prime factors. That is if we multiply two primes together to get a very large number, given that large number it's hard to work out what the primes are and that will be a feature of some of the encryption algorithms. What are the primes? You should start remembering some of them, not all of them of course but for the simple problems you can remember the first five or ten, so there are lists of primes you can find, you'll quickly remember them. Alright, easy, we can divide numbers, we know something about primes, let's do some arithmetic, some addition, subtraction, multiplication and division. Of those four operations, well are there any other operations in arithmetic? Maybe we can expand exponentiation, we add numbers, subtract, multiply, divide, raise to the power, exponentiation, A to the power of B and the opposite of raising to the power, logarithm. The logarithm of a number is the opposite of exponentiation. So think of those six operations and you know they're related so they all go in pairs. Subtraction is really addition. Subtraction in our arithmetic is adding a negative number. Alright, to subtract a number we add the negative. The negative of a number we will say is the inverse, the additive inverse to be specific. We'll list that in a moment but we'll introduce and say if we add the negative of a number we get subtraction. So subtraction is just addition where we add the additive inverse of a number. And similar, multiplication and division go together. Division is just the inverse of multiplication, that is to divide we multiply by the inverse of the number. Ten divided by three is ten times one divided by three, one over three, the inverse of three. So we can talk about multiplication and division is just multiplication times by the multiplicative inverse. So we'll introduce that and similar, their exponentiation and logarithms go together. And that will become important again when we look at cryptographic algorithms. So I know you can do all those six operations. Now we're going to do it mod n, modular arithmetic. And most of it's the same as your normal arithmetic but we just mod by n at the end. Well we know what mod n means, well sometimes it's not so clear. Do we have a positive or a negative number? So we will say a mod n, a is an integer, n is a positive integer, is the remainder when a is divided by n. We refer to n as the modulus. We can say that two integers a and b, we can think are equivalent. We call it congruent modulo n if those numbers mod n are the same. Twelve mod ten, two mod ten, twenty two mod ten are all congruent modulo ten. They're all the same in mod ten. Two mod ten is two, twelve mod ten is two, twenty two mod ten is two. So we say that congruent modulo n and we'll try and write that as three lines in the equals. So we'll think of mod n as an operator, some number mod n. And it returns an answer in the set of zero up to n minus one. So mod ten, the answer was always between zero and nine. No negative values, always positive, zero to n minus one. So we'll sometimes write that as the set z n. When we perform modular arithmetic, we do addition, subtraction, multiplication, division, exponentiation, logarithms, where the answers or the operations are within the confines of that set z n. So we'll go through some examples of those six operations mod n. Start simple. Just yell out the answer so we can move along quickly. There's no need to follow the lecture notes, just some examples today. And to go through these examples, how about the five or six students at the back come down to the front so we can all here come down to the front seats. Let's turn off these two monitors. Plenty of spare seats down the front. Then I can hear you when you tell me the answer. We know 13 mod ten, the remainder is three when we divide by ten. 13 divided by ten, the remainder is three. We can say 13 is congruent modulo three, congruent modulo ten, they are equivalent. In mod ten, 13 is the same as three. In the set z ten, the answers go up from zero to nine. So when we're doing mod ten, the answer is always between zero and nine. So let's do some addition. And in these examples, we'll do everything in z ten. Everything's mod ten. I'm not going to write the mod ten. So some examples, addition, what's four plus three? Easy. Four plus seven, not eleven. Eleven mod ten. So everything's mod ten. So in addition, it's easy. We just do our normal addition and then mod the answer by ten. So four plus seven is one. So addition in modular arithmetic is very simple. It's natural for our perspective. We just mod the answer by our modulus n. Subtraction. Well, let's do subtraction in, over here I'll write the normal, normal arithmetic. That is our no modulus, just normal arithmetic. In normal arithmetic, what's seven minus three? Four or seven plus minus three. Subtraction is just addition, where we add the negative number. So seven minus three is really seven plus minus three, where minus three we say in our normal arithmetic minus three is the additive inverse of three. The additive inverse of a number is the number we add to it such that we get zero. What's the additive inverse of three? Three plus minus three gives us zero. So the additive inverse of three is minus three. This is our normal arithmetic. Additive inverse since, and I'll write the sign as well, plus three, add minus three equals zero. Therefore, just in brief, the additive inverse, A i, additive inverse of three is minus three. The point being is that we don't need to do subtraction, we just do addition. Subtraction is addition plus the additive inverse. So we need to find the additive inverse to do subtraction. That's normal arithmetic. Let's return to Z ten, everything mod ten. What's the additive inverse of three? Write it down. Additive inverse of three. This is in Z ten. Z ten means the numbers, the answers between zero and nine. So don't give me a negative answer. What's the additive inverse of three? What number do we add to three such that we get zero? Same as here in the normal arithmetic. What number do we add to three such that we get zero? Negative three. Three plus seven equals zero in mod ten. Three plus seven is ten, mod ten. So yes, the additive inverse of three is seven because in mod ten, of course, I haven't written. Three plus seven, mod ten is zero. So we can talk about an additive inverse. If we can find the additive inverse then we can do subtraction. Another one. What's four minus seven? Four minus seven is seven. Four minus seven is four plus the additive inverse of seven. What's the additive inverse of seven? What number do you add to seven to get zero? Well, actually we just did that. Three, but it's the opposite. That is, the additive inverse of three is seven. Three plus seven equals zero. Therefore, seven plus three also equals zero. So the additive inverse of seven is three. Four minus seven is four plus the additive inverse of seven. Subtraction is just addition where we add the additive inverse. We know the additive inverse of seven is three. So it becomes four plus three. And now we just have normal addition. Mod ten, seven. Four minus seven is seven. So in modular arithmetic, addition is natural to us. That's easy I think for most people. Subtraction is a little bit different from how we think. We need to find the additive inverse. But it's in fact the same principle of our normal arithmetic. Two minus six, find the answer. Two plus the additive inverse of six. Six plus four equals zero. So it becomes two plus four equals six. Five minus three. Five plus the inverse of three. Five minus three. We found that before. The additive inverse of three is seven. So it becomes five plus seven. Two. Everything's mod ten. Five minus three is two. That makes sense. Two minus six is six in mod ten. Any questions so far? Yep. There are no negative numbers. All the set z ten, mod ten means answers are always between zero and nine. So this is the difference between our modular arithmetic and our normal arithmetic. We define everything is within that set of zero to nine. So there are no negative numbers. Back to the additive inverse. Let's consider all possible values in our set. We're in z ten. All possible values. A can be zero. The other possible values that we can operate with is z ten. What are the additive inverses? Zero plus what gives us zero? Mod ten. These are easy. So in this case, every number in our set has an additive inverse. And in general, in any set and any mod n, every number has an additive inverse. Doesn't matter at z ten, z eight, z one million. That is whatever the set, whatever the modules, will always have an additive inverse of each number. Meaning we can always do subtraction. Because subtraction is adding the additive inverse. Any questions before we move on to the next operations? Addition and subtraction. Easy so far. What's next? What's the next operation? If we can add numbers, we can subtract numbers. We need to multiply numbers. Multiplying is what? Adding multiple times. So there's a relationship between multiplication and addition. So let's look at multiplication and then division. And let's do it in z eight. Just to make it more fun. Three times two. Are you sure? What's the answer? Six. Okay, good. Three times four. Three times four in z eight is four. Remember everything's mod eight. Remember everything, we're just omitting writing the modulus. And the same with three times two equals six. Mod eight, we still get six. Multiplication is easy in modular arithmetic. It extends upon addition. We just do naturally our normal multiplication mod by our modulus n. What's the next operation then? We can do multiplication. Next is division. What is division? Multiply by the inverse. Is that? In our normal arithmetic, we say ten divided by three is ten times one divided by, one over three, we can think. The inverse of three. Not the additive inverse, but we say the multiplicative inverse. So we introduce the multiplicative inverse. Again, this is our normal arithmetic. What we know. Division eight divided by three, in fact, is the same as multiplication, eight times one over three. So we can convert division to multiplication where we say since three times one over three equals one, therefore the multiplicative inverse of three is one over three. This is our normal arithmetic. No modulus yet. The multiplicative inverse is defined as the number we multiply by such that we get one. Three times one over three gives us one. So three and one over three are the multiplicative inverses of each other in our normal arithmetic. So division divided by three is eight times the multiplicative inverse of three. Eight times one third. So that holds also in modular arithmetic. Back in Z eight, five divided by three, what do you get? Find five divided by three in Z eight. Fine calculator may not have this answer. These are the ones you need to do in your head. Five divided by three, five times the multiplicative inverse of three. Let's do the full steps. Division is multiplying by the multiplicative inverse. The multiplicative inverse is the number that we multiply by such that the answer is one. What times three gives us one in mod eight? So three times three gives us nine. Mod eight gives us one. So the answer is three. The multiplicative inverse of three is itself because three times three equals one. Therefore, five divided by three continuing is five times the multiplicative inverse of three, which is also three, which is seven. Five times three mod eight. Remember all mod eight at the end. For multiplication and addition, just do the mod eight at the end. The easy to solve. Five divided by three equals seven. Any questions before a couple more examples? So think about additive inverse, the number we add to get zero, multiplicative inverse, the number we multiply by to get one. Subtraction is adding the additive inverse. Division is multiplying by the multiplicative inverse. Yes, this. This is from our definition. We want to find the multiplicative inverse of three. What does that mean? Similar to our normal arithmetic, it's the number that we multiply by that gives us one as the answer. Everything's mod eight, remember. Something times three equals one in mod eight. So something times three mod eight equals one. What is that something? It's actually three. It's itself. Three times three mod eight gives us one. So we say three is a multiplicative inverse of itself. Yes. What other answers are there? Do we need to define this though? Right. The answers, remember, also always within the set z eight. So three times three is one mod eight. There's another answer which is, can I think of it, something times three mod eight. Yes, there are numbers greater than eight that will give the answer, and I can't think of one right now, but there are. But if those numbers are mod by eight, it will come back to three. So the answers or the numbers are always within the set z eight, zero to seven in this case. Three times 11 is 33, mod eight is one. Right, you found one. 11, three times 11. But what is 11? It's three in z eight. In mod eight, 11 and three are the same. They're congruent modulo. Okay. So yes, three, 11, probably 19 and so on. I think when we mod by eight. The reason we're doing this is we're working our way up. Addition and subtraction are not so, we don't see them so often in cryptographic operations, but we've moved from addition and subtraction up to multiplication and division, because they are based upon addition and subtraction. And then the next step is the exponentiation and logarithm, and we'll see that they are commonly used in cryptographic operations, public key cryptography. So that's why we need to learn about the principles here. Do you need to show the operations? Yes, sometimes there's a small question solved by hand, one of these, but usually not such small ones like addition and multiplication. Six divided by four, find six divided by four. There are no fractions, there are no negative numbers. The answers are between zero and seven. The integers between zero and seven. How do you find the answer? Well, six times the multiplicative inverse of four. So we need to find the multiplicative inverse of four. Something times four, mod eight, we'll write it down in full here, we're still in z eight, equals one. Where that something is zero to seven. No answer. There is no such number. So four does not have a multiplicative inverse. There's no number between zero and seven when we multiply by four and mod by eight and get one. So there is no answer. Not every number has a multiplicative inverse. Every number has an additive inverse, but not every number has a multiplicative inverse. Four does not. So we cannot do that. There's no answer. There's no multiplicative inverse of four. Staying in z eight, so let's list them. What are the multiplicative inverses of the eight numbers? Zero times something equals one. There's no such number. One times something equals one. Nine is not in our set. Remember our answer is zero to seven, so one times one. One is its own inverse. Two times something equals one. Well, we can't get an odd number when we multiply by an even number here, so no such answer. Three we saw is itself. Four we don't have an answer. Five, five times five. Mod eight is one. Six won't work when we mod eight. Seven, seven times seven. Mod eight is also one. In z eight, it turns out one, three, five and seven are multiplicative inverses of themselves. But it's not always the case. Let's go back to z ten to show another case. Zero cannot be multiplied by anything to get one. This is mod ten. One will always be a multiplicative inverse of itself. Doesn't matter what the modulus is. Two, two won't work. Two times something, mod ten gives us a remainder of one and even multiplied by something else will always give us an even number. We can't get a remainder of one. So the even numbers are not going to work. Three, three and seven. Three, seven, twenty one. Mod ten is one. Five, no. There won't be a remainder of one when we mod by ten. Seven, we've got that one. Nine, nine, nine's 81. So just a different set of multiplicative inverses. In z eight, they were inverses of themselves. But that doesn't have to be the case. As we see with z ten, three and seven are multiplicative inverses. And another important point, not all numbers have a multiplicative inverse. So we can't divide by just any number, only by sum. Which numbers do have a multiplicative inverse? When the remainder is one, if the two numbers are relative, if the number is relatively prime with the modulus, it will have a multiplicative inverse. What's the greatest common divisor of one and ten? One. One and ten? The greatest common divisor is one. Three and ten? The greatest common divisor is one. Seven and ten? One. Nine and ten? One. Eight and ten? It's not one. Six and ten is not one. Five and ten is not one. That is, a number has a multiplicative inverse in mod n if that number is relatively prime with n. Relatively prime means the greatest common divisor is one. That is, we're not listing them all, but the greatest common divisor of one and ten, we can list them all three and ten, seven and ten. They're all one. That is, all those numbers are relatively prime with ten, and therefore they have a multiplicative inverse in mod ten. So when we want to do division, we need to know which numbers have an inverse, and we can find them by checking if they're relatively prime with a modulus. We can do addition, subtraction, multiplication, division. We can't do division in all cases, only when we have a relatively prime number to the modulus. Let's come back to the slides, see what else we've said. The properties or the laws of our normal arithmetic, there are similar laws of modular arithmetic, and some of them are obvious. The three, for example, a plus b mod n is the same as a mod n plus b mod n, all mod n. And the third one is especially useful, a times b mod n, we can expand out, the a mod n times b mod n, and then all mod n. Remember, if you mod a number mod n multiple times, if you add a mod n mod n, all mod n is the same as a mod n. If you add another mod n at the end, it makes no difference. 12 mod ten is two. Mod ten again is still two. Mod ten again, it's still two, so we can add mod n's at the end, making no change to the answer. This third one is sometimes used to simplify multiplication. When we have large numbers, we can expand out two large numbers mod n multiplied together, gives us a very large number, then mod n. An easier way to solve that is to take those two large numbers, a and b, and mod n first, which makes them smaller numbers. Multiply two smaller numbers is easier, and then mod n. So that's a common way to both in our heads do calculations of multiplication, but also computers can use algorithms or laws like that to speed up the calculation. And there are other laws there. Let's see an example of this third rule, and then we'll move on to some other concepts. Staying in z eight, a simple case, we could probably do it in our head, but let's see if you can expand to solve it a little bit easier. 132 mod eight, well you could try and calculate what times eight was the range, but the property that we can take advantage of to make it a little bit easier, if we recognize anything about 132, what are the divisors of 132? Something times something is 132, maybe if you know your times tables. There are small numbers, but 12 times 11, there are other ways to do it, but 132 is 12 times 11, so we can expand that. 12 mod eight times 11 mod eight, all mod eight. 12 mod eight is four, 11 mod eight is three, 12 mod eight, four. That's just an example of doing this expansion to maybe make it a little bit simpler. Now these numbers you could do it either way in your head, but as the numbers get larger, especially even for computers, it makes it easier if you expand because these numbers become smaller before we multiply. Try that one, a couple of minutes, try to expand and do it without a calculator. No calculator, solve that. 11 to the power of seven, mod 13, and try and use the expansion to solve it, just to see how powerful it is. So one approach to this is think, right, let's make this number smaller. 11 to the power of seven is 11 to the power of four times 11 to the power of three, or even better. Now let's stick with that. 11 to the power of three, I'm not so good at my multiplication, so even that's too big for me. Maybe we can split that up. We'll come across here. I know my squares, I know 11 squared, but I don't know 11 to the power of three, so we can think of that as, 11 to the power of four is 11 squared squared, mod 13 times, 11 to the power of three, I don't know, but I know that's 11 squared mod 13 times 11 mod 13, 11 to the power of one, and let's not forget mod 13 at the end. Some of these numbers I can do in my head now. 11 squared, 121, mod 13, can someone do that for me? Four, 11 mod 13, 11 squared mod 13, 11 squared is, mod 13 is four, and then square that again is 16, mod 11, let's leave it in here just to be complete. Four, 11 squared mod 13 gives us four, squared mod 13, 16 mod 13, we get three times four times 11, we could probably do that in our head. 12 times 11, actually that was before, 132 mod 13, divide by 13, the remainder is going to be two. 10 times 13 is 130, right? The idea of this is that this expansion makes large calculations much simpler. Not just for you in a quiz or an exam, but for a computer that has to deal with very large numbers. Very large numbers, maybe hundreds of digits. Not hundreds, but hundreds of numbers is the length of the number. Multiply them together, calculate the modulus, takes time, so break them into smaller numbers. What's next? That was an example of this property that we could use, and that's the main one we'll see in the cryptographic algorithms. Division we've talked about, we've defined additive inverse, multiplicative inverse. Some number has a multiplicative inverse in n if it is relatively prime to n. So we can do division. In the last 10 minutes, before we won't introduce the theorems, we'll do that next week, let's consider another thing, relatively prime. How many numbers are relatively prime to 10? Back to our examples. How many numbers are relatively prime with eight? One, two, three and four. How many? Just counting them. How many numbers were relatively prime with ten? One, two, one, two, three, and four? Four? It's not always four, but let's consider counting the number of numbers relatively prime with some number. Let's get a different colour. Start simple. For example, how many numbers are relatively prime with four? And less than four. We'd need to test three numbers. One, two, and three are one and four relatively prime? Two and four, three and four. So relatively prime, remember, means the greatest common divisor of four and one, four and two. So we'd like to count how many numbers less than four are relatively prime with it. Four and one, greatest common divisor is one, four and two, divisor is two. Four and three, divisor is one. These are relatively prime. This one's not. So two numbers less than four are relatively prime with four, the count of numbers. And we have a special name for this. The number of numbers less than n, which are relatively prime with n, is called Euler's totient. Euler's totient function is spelled E-U-L-E-R-S, the top part here. Euler's totient function. Written, so the totient we say, easier to say, the totient of n is the number of numbers less than n, which are relatively prime with n. The totient of four equals two. We did it before, the totient of eight, we had, from our previous example, we had four. The totient of ten, from our previous example, we counted four numbers. What's the totient of nine? Euler's totient function is defined as the count of numbers relatively prime with n and less with n. So the totient of nine, consider the numbers from one to eight, check if they're relatively prime with nine and count how many are. Well, let's, one and nine are one and nine relatively prime. Yes, one in any number is always relatively prime. Two and nine, greatest common divisor of two and nine is one. So yes, they are relatively prime. Three and nine, greatest common divisor of three and nine is three, so no. Four and nine, greatest common divisor of one, so yes, I'll give it a tick, it is relatively prime. Five and nine, six has a divisor of three, so it is nine, so no. Seven's okay, seven and nine, and eight is also okay. One, two, three, four, five, six numbers. Euler's totient of nine is six. Just to make sure you know what you're doing, let's try another number. We'll not write it down. One and thirteen, yes, one and n always relatively prime. Two and thirteen, think of the greatest common divisor, greatest common divisor between thirteen and two. Is it one? If so, relatively prime. Two and thirteen are relatively prime. How many numbers less than thirteen are relatively prime with it? All of them, twelve. Okay? Because, because of what? Thirteen is a prime number. By definition then all of the numbers must have a greatest common divisor with that number, the prime number of one. So the numbers less than thirteen, which are relatively prime with thirteen, one through to twelve, so there are twelve numbers. So that's a nice shortcut. The Euler's totient of a prime number is the prime number minus one. If you recognize twenty-nine, it is a prime number. Totient is twenty-eight. So if we know n is a prime number, we can quickly calculate the totient. Very easy. If it's not a prime number, then, well, we saw the algorithm on our small numbers, we could check the numbers. We go through and check them. If it's a prime number, we don't have to check them. We know straight away that if twenty-nine is prime, the answer of Euler's totient of twenty-nine is twenty-eight. But if it's not a prime number, we need to check. And it turns out that there are no fast algorithms for checking. The algorithms that people know of for checking these, basically you need to try them all. And that takes a long time when n is very large. That will become another security property we're using cryptography. A couple more to finish for today. Totient of seven, it is prime. Totient of five, five is prime. Totient of thirty-five, thirty-five is a prime? No. So what are you going to do? You could try numbers one compared to thirty-five, two, three, up to thirty-four and compare them all. But we do notice thirty-five has two prime factors of seven and five. And it can be easily shown that the quotient, Euler's totient, not quotient, the totient of two prime factors multiplied together is the totient of those numbers multiplied, running out of space. Totient of seven we know is six. Totient of five is four. The answer is twenty-four. So we have n, n is thirty-five. If we can factor it into the two primes, in this case seven and five, then it's easy to calculate and you can check the math. It's not hard to show that the totient of two numbers multiplied together is the same as the totient of each number and then multiply. And since it's easy to find the totient of the two primes, seven and five is six and four, it's easy to find the totient of seven times five of thirty-four, thirty-five. This is a property that will make use of insecurity as well. And it will come down to how easy is it to find the totient of a number, a large number. If it's a prime number, yes, very easy, just the prime number minus one. If the number is made up of multiplying two prime numbers, also very easy. In this case, it's made up from multiplying two prime numbers so we can quickly find the totient. If not, it's very hard. We basically need to try all the numbers. And if we have to try a large number of numbers, it will take too long to try them. And that will be a security property we see in public key crypto. Those properties listed here, the totient of one is one. That's the definition. For prime p, the totient of p is p minus one. We saw that. And this property we just saw. The totient of n, when n is made up of multiplying two primes, p and q, is p minus one times q minus one. We'll take advantage of that as well. Next week we'll see some further theorems that take advantage of some of these properties. And then we'll move on to public key cryptography and see an algorithm that uses these properties to make it secure.