[00:57.720 --> 00:59.780]  Hey, guys, how's it going?
[01:01.260 --> 01:05.740]  Congratulations, you guys are the coolest people here at the conference, you know.
[01:06.800 --> 01:10.340]  Who willingly comes to a math talk in their free time?
[01:10.440 --> 01:15.300]  Like, just as a warning, if you didn't know that's what this was, we are doing a math talk today.
[01:15.900 --> 01:18.420]  But I promise it's not scary.
[01:18.460 --> 01:20.060]  It's not gonna be scary at all.
[01:20.700 --> 01:25.700]  And if there's like any questions at all, like, do not hesitate, right?
[01:25.700 --> 01:31.220]  Because I want everyone to know, math brings up a lot of feelings for people.
[01:31.220 --> 01:34.860]  I get told all the time, they're like, okay, you do math, okay.
[01:35.240 --> 01:41.400]  So, if there's anything at all you don't understand, like, there's no shame.
[01:41.400 --> 01:47.440]  You know, if a lack of comprehension is not a reflection of your abilities, just come up afterwards.
[01:47.760 --> 01:50.680]  You know, math always has more than one way to solve a problem.
[01:50.680 --> 01:52.160]  There's more than one way to learn.
[01:52.200 --> 01:56.700]  So, take a deep breath, you're fine, and we can get going.
[01:56.800 --> 01:57.840]  So, I'm Abby.
[01:57.840 --> 01:59.000]  Thanks for coming.
[01:59.360 --> 02:01.380]  I went and got a math degree.
[02:01.380 --> 02:03.240]  I know, I'm a math nerd.
[02:03.260 --> 02:14.440]  So, I went to Eastern Michigan University, and there I took several classes on cryptography and write the mathematical algorithms used today.
[02:14.620 --> 02:16.660]  So, that's what we'll be going over.
[02:17.300 --> 02:22.300]  Yeah, and then from there, I went on to the PKI world of the cybersecurity.
[02:22.500 --> 02:25.920]  I work for a PKI as a service company.
[02:26.380 --> 02:27.860]  So, that's what I do.
[02:28.060 --> 02:31.280]  So, we will be becoming quantum computers today.
[02:33.480 --> 02:39.480]  We'll begin with a little bit of an introduction to just some basic math concepts.
[02:39.480 --> 02:41.720]  You know, we don't want to dive in to the deep end.
[02:41.720 --> 02:47.060]  We're going to start slow, warm ourselves up, and then we'll go into RSA.
[02:47.060 --> 02:51.340]  We will do an example, and then we'll do elliptic curve, ECC.
[02:51.340 --> 02:54.360]  We'll talk about how those are broken, right?
[02:54.360 --> 03:01.860]  And then we actually will also be doing a post-quantum cryptographic example, but like pretty basic.
[03:01.860 --> 03:07.640]  So, but first, we get to go back to everyone's favorite time, high school.
[03:07.980 --> 03:12.960]  Yeah, because we will not be doing anything higher than high school level math.
[03:13.240 --> 03:15.380]  I don't know if that makes you feel better or not.
[03:15.380 --> 03:20.820]  So, okay, the first concept, exponents, right?
[03:20.820 --> 03:24.440]  Three times three times three is three to the fourth.
[03:24.480 --> 03:25.940]  Which numbers are exponent?
[03:25.960 --> 03:27.300]  Four, right?
[03:28.360 --> 03:29.760]  This make sense so far?
[03:29.760 --> 03:30.780]  We're doing okay?
[03:30.860 --> 03:32.420]  Yeah, okay, good.
[03:33.420 --> 03:35.760]  The next one is we have prime numbers.
[03:36.280 --> 03:38.280]  So, what is a prime number?
[03:38.320 --> 03:41.600]  A prime number has two factors, one and itself.
[03:41.920 --> 03:46.500]  And those numbers are the two numbers that it can be divided by, right, or divisible by.
[03:46.500 --> 03:49.040]  It means when you divide it, there's no decimals.
[03:49.500 --> 03:51.240]  An example is 13, right?
[03:51.240 --> 03:54.080]  Thirteen has two factors, one and thirteen.
[03:54.360 --> 03:57.680]  Another example is 46,853.
[03:59.580 --> 04:05.620]  The reason I have some stake up here is, right, because I like to think of prime numbers like prime rib.
[04:05.780 --> 04:11.240]  You'll share it with, you only want it for yourself, right, but you'll share it with one other person if you have to.
[04:11.520 --> 04:14.100]  So, that's how I kind of remember prime numbers.
[04:14.720 --> 04:18.280]  And obviously, if there's prime numbers, then we have numbers that are not prime.
[04:18.560 --> 04:19.420]  Twenty-four.
[04:19.420 --> 04:21.400]  Twenty-four has way more than two factors.
[04:21.400 --> 04:22.100]  It has eight.
[04:22.100 --> 04:23.460]  I guess it's not that many more.
[04:24.560 --> 04:30.440]  And then, like, 3,652 also has, I think that one has 12 factors.
[04:30.960 --> 04:33.320]  So, okay.
[04:33.660 --> 04:35.400]  We have one more concept.
[04:35.700 --> 04:38.140]  This one you're probably not as familiar with.
[04:38.140 --> 04:41.540]  And if you did talk about it in high school, let me know because that's awesome.
[04:41.760 --> 04:43.760]  So, this is modular arithmetic.
[04:44.340 --> 04:45.540]  Sounds terrifying.
[04:45.540 --> 04:46.500]  I swear it's not.
[04:46.500 --> 04:48.140]  You can do this.
[04:48.460 --> 04:52.640]  So, in modular arithmetic, you only have to know how to subtract and divide.
[04:52.880 --> 04:54.660]  Can you guys subtract and divide?
[04:54.860 --> 04:55.580]  Yeah.
[04:55.860 --> 04:57.120]  That's a maybe.
[04:57.560 --> 04:58.760]  That's all you have to do.
[04:58.760 --> 05:01.280]  You don't even actually have to, like, divide, divide.
[05:01.280 --> 05:04.200]  You can just kind of maybe pretend.
[05:04.560 --> 05:14.800]  So, in modular arithmetic, we say two numbers are congruent modulo m if their difference is divisible by m.
[05:14.900 --> 05:15.340]  Right?
[05:15.340 --> 05:16.200]  What is difference?
[05:16.200 --> 05:17.440]  Subtraction.
[05:17.440 --> 05:18.200]  Divisible?
[05:18.200 --> 05:19.280]  Division.
[05:19.280 --> 05:27.620]  So, two numbers are congruent modulo m if their difference is divisible by m.
[05:27.620 --> 05:29.680]  Let's take a look.
[05:30.080 --> 05:32.900]  I have two numbers, 29 and 14.
[05:33.160 --> 05:35.180]  And I've chosen m to be 5.
[05:35.380 --> 05:36.220]  So, what do I do?
[05:36.220 --> 05:37.740]  First, subtract.
[05:37.780 --> 05:40.560]  What is 29 minus 14 is 15.
[05:41.640 --> 05:42.980]  Pretty straightforward.
[05:42.980 --> 05:46.780]  Next, we ask ourselves, is 15 divisible by 5?
[05:47.320 --> 05:48.780]  And the answer is yes.
[05:48.780 --> 05:49.040]  Right?
[05:49.040 --> 05:51.000]  15 divided by 5 is 3.
[05:51.000 --> 05:52.160]  There are no decimals.
[05:52.160 --> 05:52.920]  It's a whole number.
[05:52.920 --> 05:55.580]  So, 15 is divisible by 5.
[05:55.720 --> 06:00.040]  So, we have met the two parts of modular arithmetic.
[06:00.040 --> 06:06.140]  So, we can say 29 is congruent to 14 modulo 5.
[06:06.700 --> 06:08.260]  So, that's how we write this.
[06:08.500 --> 06:11.480]  Remember, all we did was subtract and then divide.
[06:11.480 --> 06:16.220]  And then we just write it very complicatedly at the end.
[06:16.600 --> 06:23.820]  So, to say something is congruent, we use an equal sign and then we have the third parallel line on top.
[06:23.820 --> 06:27.480]  You know, it's kind of like an equal sign with like a top hat or, you know, something like that.
[06:27.480 --> 06:31.180]  But those three lines represent congruency.
[06:31.180 --> 06:34.440]  So, it's like they are equal, but they're also not.
[06:35.920 --> 06:41.980]  So, if two numbers can be congruent, that means two numbers can also not be congruent.
[06:42.060 --> 06:43.780]  I mean, look at this example here.
[06:43.900 --> 06:46.680]  Two completely different numbers, 23 and 12.
[06:46.680 --> 06:48.660]  This time my m will be 3.
[06:49.500 --> 06:51.100]  We subtract first, right?
[06:51.100 --> 06:52.960]  23 minus 12 is 11.
[06:52.960 --> 06:55.140]  We do 11 divided by 3.
[06:55.140 --> 06:57.580]  Is 11 divisible by 3?
[06:57.580 --> 06:58.240]  No.
[06:58.240 --> 07:00.500]  There are decimals, there are remainders.
[07:00.820 --> 07:07.860]  So, then we just put a slash through and we say 23 is not congruent to 12 mod 3.
[07:09.260 --> 07:13.160]  And fun fact about mathematicians, we are known to be lazy.
[07:13.160 --> 07:14.660]  It is just a fact.
[07:14.940 --> 07:21.120]  So, essentially, mathematicians find the quickest and most efficient way to do, to make a shortcut.
[07:21.120 --> 07:24.140]  So, right, we don't need to write modulo out every single time.
[07:24.140 --> 07:25.340]  You can just write mod.
[07:26.120 --> 07:28.960]  And that is modular arithmetic.
[07:29.180 --> 07:32.560]  And those are the only concepts you need to know for RSA.
[07:33.240 --> 07:35.200]  So, you guys feel like you can do this?
[07:35.500 --> 07:36.660]  You'll be okay?
[07:36.660 --> 07:37.340]  Okay.
[07:39.880 --> 07:40.580]  Oh, yeah.
[07:40.720 --> 07:42.460]  Congratulations, you repassed high school.
[07:42.460 --> 07:43.620]  Good job, everyone.
[07:43.800 --> 07:45.280]  You can get a second degree.
[07:46.920 --> 07:48.040]  Let's dive right in.
[07:48.040 --> 07:49.560]  RSA, right?
[07:49.560 --> 07:53.460]  RSA is asymmetric cryptography, right?
[07:53.460 --> 07:54.940]  Asymmetric encryption.
[07:55.060 --> 08:01.280]  That means there's two keys, a public key for encryption and a private key for decryption.
[08:02.560 --> 08:10.820]  We, it was created by three men, Rivest, Shamir, and Adelman, RS and A, in, when was it made?
[08:10.880 --> 08:12.020]  1977.
[08:12.020 --> 08:23.940]  So, actually, RSA is almost 50 years old, which, you know, makes sense that we're now coming to get some new algorithms because 50 years old in tech world, oh, that's a long time.
[08:25.120 --> 08:25.500]  Okay.
[08:25.500 --> 08:28.580]  RSA, seven, we have seven components.
[08:28.660 --> 08:32.880]  We start with two positive whole numbers, D and E.
[08:32.880 --> 08:35.980]  D for decryption, it's going to be our private key.
[08:36.140 --> 08:39.120]  E for encryption, it's going to be our public key.
[08:39.980 --> 08:43.560]  Next, we have P and Q, so we're going to have two prime numbers.
[08:43.740 --> 08:44.440]  Good thing, right?
[08:44.440 --> 08:45.420]  We went over that.
[08:45.880 --> 08:48.100]  N is just P times Q.
[08:48.760 --> 08:51.060]  Sorry, I forgot to mention, you have to know how to multiply, too.
[08:51.960 --> 08:56.120]  And then we, right, have M is going to be our plain text, C is our cipher text.
[08:58.830 --> 09:06.150]  And then all seven of these beautiful numbers, we will put together to have three different algorithms for RSA.
[09:08.290 --> 09:10.330]  Here's our first one for encryption.
[09:10.690 --> 09:12.410]  Go ahead and look at it for a second.
[09:12.410 --> 09:16.650]  See if you can, like, read it to yourself the way we were reading modular arithmetic.
[09:23.220 --> 09:23.660]  Okay.
[09:23.660 --> 09:24.380]  Did you read it?
[09:24.380 --> 09:28.860]  C is congruent to M to the E mod N, maybe.
[09:29.500 --> 09:29.820]  Right?
[09:29.820 --> 09:31.640]  How is this different than what we did before?
[09:31.740 --> 09:33.260]  All I did was add an exponent.
[09:33.260 --> 09:34.120]  That's it.
[09:34.660 --> 09:35.180]  Right?
[09:35.180 --> 09:41.340]  So we have our basic, our two numbers, our C and our M, but this time now, right, M has an exponent.
[09:41.780 --> 09:43.860]  So what does this mean, exactly?
[09:43.860 --> 09:53.200]  It means our cipher text is congruent to our message raised to the public key, you know, raised to the encryption, mod N.
[09:53.620 --> 09:57.900]  And if you forget how we find that, that's what the bottom just says.
[09:57.900 --> 10:01.780]  It says C minus M to the E is divisible by N.
[10:02.100 --> 10:05.340]  That's all we're doing, is we're subtracting and then we divide.
[10:07.700 --> 10:08.500]  Take a second.
[10:08.500 --> 10:09.720]  This is the decryption.
[10:09.900 --> 10:10.860]  Is it different?
[10:10.860 --> 10:11.440]  Maybe?
[10:15.950 --> 10:16.550]  Okay.
[10:16.550 --> 10:29.650]  So in decryption, we switch, right, the message and the cipher text, because now we need the cipher text to decrypt, which is now, instead of our exponent being E with the public key, now we have our exponent being D with the private key.
[10:30.930 --> 10:34.890]  And I forgot to mention this, but all of these algorithms are framed.
[10:34.890 --> 10:36.090]  They are works of art.
[10:36.090 --> 10:37.470]  They belong in a museum.
[10:37.470 --> 10:38.890]  They are beautiful.
[10:39.670 --> 10:40.570]  But, yeah.
[10:41.050 --> 10:45.710]  And then, right, so our message minus C to the D is divisible by N.
[10:47.610 --> 10:49.190]  I know what you're all thinking now.
[10:49.250 --> 10:51.670]  We have our encryption and our decryption.
[10:51.970 --> 10:53.730]  Why, what is the third algorithm?
[10:53.730 --> 10:55.130]  Why do we need a third one?
[10:55.930 --> 11:00.030]  So this third one is what makes RSA secure.
[11:00.030 --> 11:01.530]  It's not like a name for it.
[11:01.530 --> 11:05.870]  I just call it our make secure algorithm, to be straight forward.
[11:06.010 --> 11:13.370]  So this describes the relationship between the private key and the public key, our D and our E.
[11:13.430 --> 11:21.550]  So here we can see D times E is congruent to 1 mod P minus 1 times Q minus 1.
[11:22.050 --> 11:25.870]  And if you remember before on the previous ones, we had mod N.
[11:26.370 --> 11:27.670]  And what was N?
[11:27.670 --> 11:29.530]  N was P times Q.
[11:29.770 --> 11:31.850]  So we no longer have P times Q anymore.
[11:31.850 --> 11:33.430]  We have something a little different.
[11:33.950 --> 11:35.350]  Don't worry, you don't have to memorize it.
[11:35.350 --> 11:36.390]  It's on Google.
[11:39.110 --> 11:39.930]  Okay.
[11:40.030 --> 11:40.950]  Let's do it.
[11:40.950 --> 11:41.830]  You guys, are you ready?
[11:42.090 --> 11:43.550]  Let's do some RSA.
[11:44.770 --> 11:49.290]  So in any good crypto system, we have our two main characters, Alice and Bob.
[11:49.290 --> 11:51.090]  So we're going to start with Alice.
[11:51.130 --> 11:59.590]  Alice is going to create the RSA crypto system and Bob is going to use it to encrypt a message and then Alice will decode.
[12:00.230 --> 12:03.230]  So we begin with our private key, our D.
[12:03.230 --> 12:04.570]  We have 37.
[12:04.810 --> 12:09.550]  We have two prime numbers, P is 71, Q, 97.
[12:09.690 --> 12:13.690]  Multiply them together and you get 6,887.
[12:16.510 --> 12:17.810]  Let's see.
[12:19.650 --> 12:20.450]  Okay.
[12:21.190 --> 12:24.230]  Now, we have our private key.
[12:24.470 --> 12:27.190]  We can calculate our public key.
[12:27.410 --> 12:29.790]  So I have our make secure algorithm, right?
[12:29.790 --> 12:35.130]  D times E is congruent to 1 mod P minus 1 times Q minus 1.
[12:35.290 --> 12:37.230]  All I have to do is plug in what we know.
[12:37.530 --> 12:41.250]  We know 37 and we know P and Q.
[12:41.670 --> 12:43.250]  I hope you all can subtract 1.
[12:43.250 --> 12:45.250]  So then we get 70 times 96.
[12:45.650 --> 12:51.090]  And we get 37E is congruent to 1 mod 6720.
[12:52.930 --> 12:59.250]  And then, obviously, you can see the answer is E is 1,453, which is our public key.
[12:59.550 --> 13:02.810]  And you all know exactly how I got from line 3 to line 4, right?
[13:03.790 --> 13:05.490]  Makes complete sense, right?
[13:05.490 --> 13:06.990]  I subtracted and then divided.
[13:07.130 --> 13:08.930]  And you're like, but there's a variable.
[13:09.010 --> 13:10.570]  How do I know what?
[13:10.830 --> 13:11.270]  Okay.
[13:11.270 --> 13:13.830]  So fun fact, I did not solve this.
[13:13.830 --> 13:16.890]  I had an online calculator solve this.
[13:17.250 --> 13:20.230]  So my favorite is Wolfram Alpha, right?
[13:20.230 --> 13:21.870]  You guys don't, we don't need to do this by hand.
[13:21.870 --> 13:27.850]  It's really, that would take like 10 more slides and nobody's got time for that.
[13:28.130 --> 13:32.010]  So, I mean, because modern computers can solve RSA problems.
[13:32.010 --> 13:33.490]  Like, that's not the issue.
[13:33.670 --> 13:37.710]  Well, they can solve small RSA problems, right?
[13:38.150 --> 13:48.310]  The reason we use large numbers for real-time RSA is because computers take thousands and thousands of years of computation to solve them.
[13:48.310 --> 13:50.790]  It's not that they're unsolvable, unbreakable.
[13:50.950 --> 13:53.990]  It's that it's just, they're not breakable in our lifetime.
[13:54.270 --> 13:59.190]  So if you use numbers like 37, a computer can't solve that.
[13:59.190 --> 14:01.970]  And then I just prefer Wolfram Alpha.
[14:02.890 --> 14:06.970]  But now that we have that, we can plug it back into our table.
[14:07.370 --> 14:09.310]  And Alice's part is done.
[14:09.310 --> 14:11.170]  So let's move on to Bob.
[14:12.490 --> 14:13.530]  Okay.
[14:13.530 --> 14:15.970]  Bob wants to say hi to Alice.
[14:16.250 --> 14:23.010]  So we turn his alpha characters into numeric characters.
[14:23.370 --> 14:26.610]  So that means our message is 0809.
[14:26.650 --> 14:27.910]  How did I get that?
[14:27.910 --> 14:30.450]  Well, H is the eighth letter of the alphabet.
[14:31.030 --> 14:32.210]  Nine, right?
[14:32.330 --> 14:34.590]  Or I is the ninth letter of the alphabet.
[14:36.830 --> 14:37.750]  Actually, that's it.
[14:37.750 --> 14:39.130]  That's all Bob needs to do.
[14:40.090 --> 14:40.790]  Okay.
[14:40.870 --> 14:43.250]  Now we're going to use our encryption algorithm.
[14:44.610 --> 14:51.030]  Remember, we have our ciphertext is congruent to our message raised to the public key, mod N.
[14:51.670 --> 14:53.030]  Plug in what we know.
[14:53.030 --> 14:56.210]  We know our message, 0809.
[14:56.550 --> 15:00.770]  We're going to raise it to the 1,453rd power.
[15:01.330 --> 15:07.850]  And then we're going to, essentially we call it simplifying mod N or mod 6,887.
[15:08.150 --> 15:13.910]  Once again, if you're crazy like me, feel free to do this by hand, but that's a very large exponent.
[15:13.910 --> 15:15.850]  You don't need to do that.
[15:16.010 --> 15:27.030]  So plug it in online, you know, write software to do it for you, and you will find out that your ciphertext is going to be congruent to 5,593.
[15:27.530 --> 15:33.090]  So now we have a complete RSA system.
[15:33.890 --> 15:34.930]  That's it.
[15:34.930 --> 15:36.950]  You guys are still alive.
[15:37.210 --> 15:37.870]  Yeah.
[15:38.590 --> 15:42.970]  But we would not be good mathematicians if we didn't double check our work, right?
[15:43.750 --> 15:44.790]  So let's do that.
[15:44.790 --> 15:45.990]  Let's be Alice.
[15:46.090 --> 15:51.090]  So Alice is given the ciphertext, and she is the only one who has the private key.
[15:51.430 --> 15:53.230]  So let's plug in what we know.
[15:53.490 --> 15:57.210]  So our ciphertext, right, was 5,593.
[15:57.310 --> 15:59.790]  Our private key, if we remember, was 37.
[16:00.130 --> 16:04.730]  So we're going to raise 5,593 to the 37th power.
[16:04.930 --> 16:14.150]  We will simplify it, mod N, and then we can get M is congruent to 809, which was our original message.
[16:15.730 --> 16:30.270]  And then, thing to note, right, because in encryption a lot, we have to repad zeros to the ends of things, or in this case, in the beginning, which just means you have to know the alpha-to-numeric system you're using.
[16:30.270 --> 16:40.370]  And, right, in this case, I know that there are not 80 letters in the English alphabet, so I'm not going to use... the zero does not attach to the eight on the other side.
[16:40.370 --> 16:42.250]  That's how I know it attaches to the nine.
[16:43.050 --> 16:48.990]  That's just like, mathematicians can get picky, you know, they're like, what are you doing over there?
[16:49.030 --> 16:50.750]  So that's how we do that.
[16:51.330 --> 16:56.270]  Right, and we have our keys on here because we're unlocking the ciphertext, essentially.
[17:00.400 --> 17:10.220]  Oh, I did also want to mention that the numbers chosen for this RSA cryptosystem are technically just arbitrarily, randomly chosen.
[17:10.220 --> 17:14.420]  You're not using that in like an actual public key infrastructure.
[17:14.420 --> 17:21.280]  There are rules surrounding how you choose these values, but, right, for the purpose of an example, it doesn't matter.
[17:22.840 --> 17:25.460]  That's actually it for RSA.
[17:25.880 --> 17:29.340]  So, now you guys know how to encrypt all of your data.
[17:30.500 --> 17:33.780]  Or, I guess, decrypt it, if you really want to.
[17:34.620 --> 17:35.900]  But let's move on.
[17:35.900 --> 17:37.100]  Now we have elliptic curves.
[17:37.100 --> 17:38.200]  We have ECC.
[17:39.220 --> 17:40.500]  Completely different.
[17:40.700 --> 17:44.060]  We are actually, we're not really going to use any of the math we just did.
[17:44.120 --> 17:46.940]  So, sorry if you're like, I finally got it.
[17:47.120 --> 17:49.360]  We're completely switching gears here.
[17:49.760 --> 17:54.600]  So, ECC is going to be using more graphing.
[17:54.600 --> 18:03.480]  If you guys remember, like, X, Y coordinates, you know, your X axis, your Y axis, coordinate plane, that's what we're doing here.
[18:04.420 --> 18:06.300]  And then just a bit of history.
[18:06.460 --> 18:11.840]  ECC is, what was it, eight years after RSA.
[18:11.840 --> 18:14.200]  So, it still has been around for a long time.
[18:14.420 --> 18:20.360]  I think RSA is the more common one, right, but ECC is definitely picking up traction.
[18:21.080 --> 18:28.260]  These keys tend to be smaller than RSA, right, because remember how I mentioned you have to use extremely large numbers for RSA.
[18:28.700 --> 18:31.240]  So, that's why ECC tends to be a little more popular.
[18:33.990 --> 18:36.610]  Just like before, here are our basic components.
[18:36.910 --> 18:38.230]  We always start with E.
[18:38.230 --> 18:41.630]  E is going to be our equation for the elliptic curve graph.
[18:42.530 --> 18:44.510]  P is not a prime number.
[18:44.630 --> 18:48.150]  So, it's the same variable but different concept.
[18:48.410 --> 18:52.070]  So, P is going to be a point on our graph, a point on the curve.
[18:52.070 --> 18:54.930]  And this point is going to be our public key.
[18:55.910 --> 18:58.430]  Now, we have two positive whole numbers.
[18:58.470 --> 19:00.470]  We have N A and N B.
[19:01.070 --> 19:07.510]  Q A will be N A times P, right, and then Q B will be N B times P.
[19:08.430 --> 19:16.730]  And then to get the private key, which will be a second point on the curve, we do N A times Q B, right.
[19:16.730 --> 19:19.650]  I have an A and a B in this scenario.
[19:20.150 --> 19:22.210]  So, these are the components.
[19:23.230 --> 19:24.910]  Let's put it all together.
[19:27.090 --> 19:28.670]  Alice and Bob are back.
[19:28.670 --> 19:30.430]  We always miss Alice and Bob.
[19:30.570 --> 19:35.910]  So, we start with our equation E and our public key, our point P.
[19:37.170 --> 19:42.790]  Alice chooses her secret value, which is N A, right, A for Alice.
[19:42.790 --> 19:50.530]  So, Alice chooses a secret value, multiplies it by the public key, and gets her Q A value.
[19:51.170 --> 19:54.450]  Bob is actually going to do the exact same thing.
[19:54.450 --> 20:04.930]  Bob's also choosing a secret value, his N B, multiplying it by P, right, the public key, and getting his own Q B value.
[20:05.230 --> 20:07.410]  But why do they both do this?
[20:07.410 --> 20:12.010]  It's actually because Alice and Bob are both going to have access to the private key.
[20:12.010 --> 20:14.250]  They're both calculating the private key.
[20:14.410 --> 20:16.710]  How is that different, right, than RSA?
[20:16.730 --> 20:20.270]  In RSA, only Alice had the private key.
[20:20.990 --> 20:23.670]  So, how do they both calculate the private key?
[20:23.670 --> 20:27.470]  Well, what they're going to do is then swap Q's with each other.
[20:28.230 --> 20:32.890]  So, when Alice calculates her Q value, she gives it to Bob.
[20:32.910 --> 20:37.030]  And then when Bob calculates his Q value, he sends it to Alice.
[20:37.230 --> 20:40.310]  But remember, they both have their secret value still.
[20:40.310 --> 20:41.690]  So, it's still secure.
[20:41.690 --> 20:45.170]  There's still information that is not becoming public.
[20:46.070 --> 20:50.410]  So, then once we swap Q's, we can calculate the private key, right?
[20:50.410 --> 20:57.010]  We have Q B times N A, and then that will equal Q A times N B, right?
[20:57.010 --> 21:00.510]  A private key is always going to have an A and a B or a B and an A.
[21:00.510 --> 21:03.590]  We're never calculating A A or B B.
[21:03.770 --> 21:07.790]  So, that's kind of how to help you remember because there's so many N's and so many Q's, right?
[21:07.790 --> 21:10.930]  So, we need an A and a B to calculate our private keys.
[21:13.580 --> 21:21.560]  The first algorithm for our elliptic curve here, this is how we graph our elliptic curves.
[21:21.900 --> 21:26.800]  We have Y squared is equal to X cubed plus A X plus B.
[21:27.380 --> 21:34.960]  What's different about this than what you're probably normally used to in like a high school algebra class is that our Y is squared, right?
[21:35.100 --> 21:37.960]  Because normally you're like, okay, yeah, I've seen this before.
[21:37.960 --> 21:38.920]  This is not that bad.
[21:39.100 --> 21:43.880]  But now we just square the Y and that makes like a ton of a difference.
[21:44.120 --> 21:54.080]  Because what it's going to do, so I have an example over here, is it turns it sideways and makes it look like a weird little octopus squid thing.
[21:54.200 --> 21:58.060]  So, this is what an elliptic curve looks like.
[21:58.800 --> 22:04.180]  Here, I graph the example Y squared equals X cubed minus 3 X plus 3.
[22:04.180 --> 22:10.380]  And then we get this fun, I don't know, horn trumpet graph.
[22:10.380 --> 22:13.200]  Whatever your imagination makes it look like.
[22:13.360 --> 22:15.620]  This is just a random example.
[22:15.620 --> 22:22.560]  I know there's a Bitcoin example that's Y squared equals X cubed plus 7.
[22:23.340 --> 22:30.600]  So, there are other, like the A and your B values are dependent on what you're trying to do.
[22:34.270 --> 22:35.610]  Here's our second algorithm.
[22:35.610 --> 22:39.070]  ECC is only going to have these two, so we're not going to have a third one.
[22:39.370 --> 22:42.130]  So, this is how we calculate the Qs.
[22:42.630 --> 22:46.910]  Not how we calculate the, well, kind of also how we calculate the private keys.
[22:47.490 --> 22:57.070]  So, when we look at this here, we see Q is equal to P plus P plus, right, plus P, or just N times P.
[22:57.510 --> 23:00.250]  I'm sure, right, this is just multiplication, right?
[23:00.250 --> 23:03.670]  If I have 5 apples, 5 times, like 25 apples.
[23:04.990 --> 23:09.090]  Unfortunately, guys, I hate to break it to you, this is not that simple.
[23:09.230 --> 23:15.450]  There is an extra step that we have to do to do this multiplication.
[23:16.510 --> 23:17.770]  Let's do it together.
[23:18.470 --> 23:19.950]  Okay, we start with point P.
[23:19.950 --> 23:21.410]  Point P is on the left.
[23:22.030 --> 23:23.470]  I want to get to 2P.
[23:23.470 --> 23:25.730]  All I want to do is just P plus P.
[23:26.190 --> 23:28.250]  And you can see it's on the bottom right.
[23:28.450 --> 23:29.490]  You're like, what?
[23:29.490 --> 23:31.650]  That is not multiplication.
[23:32.330 --> 23:40.730]  So, what we did is I start with point P and I draw a line through it and then I stop once I hit the graph again.
[23:41.310 --> 23:43.110]  That becomes point R.
[23:43.110 --> 23:44.510]  Point R is irrelevant.
[23:44.510 --> 23:45.890]  It's just like a middleman.
[23:45.890 --> 23:47.610]  We're not really going to use it.
[23:47.730 --> 23:56.210]  But from point R, then we go straight down and that new point is our final 2P or Q.
[23:56.670 --> 24:06.370]  So, essentially, when we say we're multiplying by the points, right, we're multiplying by, like, the public key, which is a point on the curve, we're doing point translation.
[24:06.370 --> 24:11.550]  We're going from point to point to point using straight lines.
[24:14.340 --> 24:15.740]  Oh, did you have a question?
[24:15.740 --> 24:16.580]  Yeah.
[24:19.600 --> 24:22.700]  Oh, that's a good question.
[24:23.960 --> 24:29.380]  I think it's the tangent line.
[24:29.480 --> 24:31.280]  So, it's where the derivative is zero.
[24:31.980 --> 24:33.580]  I believe that's what we choose.
[24:34.660 --> 24:37.820]  So, at P.
[24:38.020 --> 24:39.020]  Yes, at P.
[24:39.220 --> 24:39.780]  Yeah.
[24:40.080 --> 24:43.650]  Yeah.
[24:45.430 --> 24:46.310]  Okay.
[24:47.130 --> 24:48.470]  Let's put it all together now.
[24:48.470 --> 24:50.990]  Let's put ECC and let's walk through this.
[24:51.430 --> 24:53.150]  So, here's, I start with my equation.
[24:53.150 --> 25:00.410]  I have Y squared equals X cubed plus 324X plus 1,287.
[25:00.910 --> 25:04.090]  My public key, right, is a point on the curve.
[25:04.090 --> 25:06.570]  So, my X coordinate is 920.
[25:06.570 --> 25:09.450]  My Y coordinate is 303.
[25:09.790 --> 25:12.350]  Now I have my public key.
[25:13.130 --> 25:20.630]  And not to confuse you guys, but if you plug in 920 for X, you will not get 303.
[25:21.510 --> 25:22.970]  I apologize.
[25:22.970 --> 25:32.750]  But the reasoning for that is because essentially we also apply modular arithmetic to elliptic curve equations.
[25:32.750 --> 25:34.710]  We're not going to do that here.
[25:34.790 --> 25:36.750]  It's just, we went over it before.
[25:36.750 --> 25:39.390]  Just going to add, you know, like 30 more slides.
[25:39.710 --> 25:41.910]  So, we also apply modular arithmetic.
[25:42.070 --> 25:48.310]  We also apply your elliptic curve graph over a finite field.
[25:48.550 --> 25:54.290]  And since I promised we were only going to do high school math, we will not be talking about finite fields today.
[25:54.770 --> 26:04.550]  But just a basic example is essentially a finite field, has a bunch of points in it, and you remove some of them because they don't fit the criteria.
[26:05.410 --> 26:08.050]  That's like a basic example of a finite field.
[26:08.050 --> 26:12.810]  So, essentially, if you plug in 920, you're not going to get 303.
[26:13.490 --> 26:16.630]  I mean, if you were, if you want to, like, and see what the actual answer is.
[26:16.630 --> 26:19.490]  If you want to do math in your free time, like, I'm here for it.
[26:20.110 --> 26:22.990]  So, okay.
[26:22.990 --> 26:27.150]  Handy-dandy table because there's a lot of values and numbers and I get confused.
[26:27.150 --> 26:28.530]  So, we have our equation.
[26:28.530 --> 26:30.150]  We have our public key.
[26:30.430 --> 26:32.830]  Alice has chosen her secret value, right?
[26:32.830 --> 26:34.390]  Because remember, we need our NA.
[26:34.590 --> 26:38.030]  She chooses 1,194.
[26:38.710 --> 26:41.670]  Now, she does her multiplication, right?
[26:41.670 --> 26:45.110]  Her NA times P to get her Q value.
[26:45.630 --> 26:48.410]  Remember, it was our point translation.
[26:48.410 --> 27:00.290]  So, essentially, she will start at the public key and do the point translation 1,194 times over and over and over again until she gets the brand-new point.
[27:00.370 --> 27:08.050]  So, she gets the point 2067 for X and 2178 for Y, and that is her Q value.
[27:09.630 --> 27:12.190]  And with that, Alice is done.
[27:12.490 --> 27:13.770]  Say goodbye, Alice.
[27:13.970 --> 27:14.610]  Okay, Bob.
[27:14.610 --> 27:15.770]  Bob's back.
[27:16.110 --> 27:19.970]  Bob has his secret value, 1,759.
[27:20.490 --> 27:22.050]  He does the same thing, right?
[27:22.050 --> 27:24.030]  Because he needs a Q value as well.
[27:24.030 --> 27:38.010]  So, he starts with the public key, does his point translation 1,759 times to get his brand-new point, 3684 for X, 3125 for Y.
[27:40.070 --> 27:43.470]  And then we get to everyone's favorite part.
[27:43.770 --> 27:46.850]  Alice and Bob now get to swap Qs with each other.
[27:46.850 --> 27:49.810]  Because what else did we learn in kindergarten, guys?
[27:49.810 --> 27:51.330]  Sharing is caring.
[27:51.330 --> 27:54.050]  So, unless it's encrypted data.
[27:54.050 --> 27:58.430]  So, Bob sends his QV value to Alice.
[27:58.430 --> 28:01.270]  Alice sends her QA value to Bob.
[28:02.610 --> 28:04.110]  We'll start with Alice.
[28:04.150 --> 28:06.310]  Because, right, what do they both do?
[28:06.390 --> 28:10.670]  They are both computating the private key.
[28:10.970 --> 28:15.470]  So, we will calculate Alice and we will calculate Bob and we will make sure they get the same thing.
[28:16.670 --> 28:17.670]  What do we know?
[28:17.670 --> 28:21.990]  We know Alice's secret value, NA, is 1,194.
[28:22.510 --> 28:28.970]  We know Bob's Q value, QV, is the point 3684, 3125.
[28:30.050 --> 28:43.850]  We do our point translation over the N and the Q and we get the brand-new point, which is going to be our private key, for 3347 for X, 1242 for Y.
[28:47.860 --> 28:49.180]  Was I going to mention you guys?
[28:49.180 --> 28:50.740]  It must not have been that important.
[28:53.420 --> 28:53.960]  Okay.
[28:54.580 --> 28:56.940]  So, now let's do Bob.
[28:56.940 --> 28:58.760]  Let's make sure Bob gets the same thing.
[28:59.100 --> 29:01.320]  We start with his secret value, right?
[29:01.320 --> 29:04.420]  Because Bob is the only one who knows his secret value.
[29:04.460 --> 29:07.180]  Alice is the only one who knows her secret value.
[29:07.700 --> 29:11.200]  So, that's our NB, 1,759.
[29:11.960 --> 29:15.320]  We know Alice's Q value because they swapped them with each other.
[29:15.500 --> 29:23.620]  We do our point translation and we get the brand-new point, 3347 for X, 1242 for Y.
[29:24.800 --> 29:31.060]  And as you can see, Bob and Alice did get the exact same point, so they both now have the private key.
[29:32.340 --> 29:33.820]  Oh, that's what I was going to mention.
[29:33.980 --> 29:38.900]  Just like with RSA, how do you do this point translation if not by hand?
[29:39.720 --> 29:41.660]  Type it into an online calculator.
[29:41.860 --> 29:43.540]  There are ones that exist.
[29:43.560 --> 29:45.120]  They can do this for you.
[29:45.840 --> 29:49.420]  Or, you know, write something if you really want the challenge.
[29:49.840 --> 29:52.480]  But, yeah, you do not, I did not do any of this by hand.
[29:52.480 --> 29:53.500]  There's no need.
[29:53.740 --> 29:56.400]  So, another fun fact.
[29:56.400 --> 29:57.660]  There's so many fun facts.
[29:58.060 --> 30:03.520]  When we have our private key, finally, we actually only take the X coordinate.
[30:04.060 --> 30:07.800]  Y, sorry, Y, W, H, Y.
[30:08.240 --> 30:10.860]  Because our elliptic curves are symmetrical, right?
[30:10.860 --> 30:13.100]  Whatever's on the top is on the bottom.
[30:13.360 --> 30:16.760]  So, our Y value actually could be positive or negative.
[30:17.300 --> 30:19.060]  And that doesn't mean it's wrong, right?
[30:19.060 --> 30:22.000]  That doesn't mean they got the wrong private key.
[30:22.400 --> 30:29.840]  It just means, sorry, the modular arithmetic they were doing brought them on top or brought them on bottom.
[30:29.840 --> 30:32.880]  So, that's why we just get rid of the Y value.
[30:32.880 --> 30:34.440]  We're like, we don't need the added negatives.
[30:34.960 --> 30:36.820]  So, we just keep the X value.
[30:40.130 --> 30:42.410]  We're going to review all of this again.
[30:42.630 --> 30:49.530]  Because elliptic curve seems like it would be fine, but it's suddenly way more complicated than RSA.
[30:50.430 --> 30:55.450]  So, we always start with our elliptic curve equation, our E.
[30:55.710 --> 30:58.390]  And we have a public key, our point P.
[30:58.950 --> 31:04.570]  Alice and Bob both choose a secret value, N A and N B.
[31:04.610 --> 31:07.010]  They both have access to the public key, right?
[31:07.010 --> 31:08.510]  The point P.
[31:08.830 --> 31:14.370]  So, they both multiply it and get their own Q values.
[31:14.370 --> 31:17.050]  Alice has Q A, A for Alice.
[31:17.050 --> 31:19.510]  Bob has Q B, B for Bob.
[31:20.230 --> 31:26.690]  They swap Q values with each other because they both need to calculate the private key.
[31:27.770 --> 31:34.590]  So, that means Alice calculates the private key with her secret value and Bob's Q value.
[31:34.590 --> 31:40.490]  Bob does the same thing, but he uses his secret value and Alice's Q value.
[31:41.090 --> 31:42.370]  How do we remember that?
[31:42.370 --> 31:44.210]  Right, we need an A and a B.
[31:44.210 --> 31:47.010]  We need an A and a B or a B and an A.
[31:47.290 --> 31:51.510]  We do not calculate private keys with an A and an A or a B and a B.
[31:54.140 --> 31:56.340]  It's kind of a lot, I guess.
[31:57.260 --> 32:00.020]  Okay, are we still alive?
[32:00.080 --> 32:03.380]  Doing a bunch of math on a Thursday?
[32:03.380 --> 32:04.180]  Okay.
[32:05.180 --> 32:07.400]  But now, let's talk about the fun stuff.
[32:07.400 --> 32:09.680]  How do we break these algorithms?
[32:10.180 --> 32:19.540]  So, these mathematical algorithms are used for encryption and cryptography because going one direction is extremely easy.
[32:19.680 --> 32:21.680]  Well, extremely is relative.
[32:21.680 --> 32:25.500]  And then, going backwards is extremely difficult.
[32:26.380 --> 32:29.060]  I like to use the metaphor of like quicksand, right?
[32:29.060 --> 32:31.340]  You step into quicksand, that's so easy.
[32:31.340 --> 32:34.460]  You know, you can just take a step forward and all of a sudden, you're in the quicksand.
[32:34.520 --> 32:38.300]  But then, you try to get out and all of a sudden, you're stuck and you can't.
[32:38.540 --> 32:40.520]  That's what these algorithms are like.
[32:40.940 --> 32:46.240]  So, for RSA, the quicksand is going to be factorization.
[32:46.900 --> 32:50.900]  So, essentially, if we remember, we had an N value.
[32:50.900 --> 32:52.080]  And what was N?
[32:52.140 --> 32:54.060]  N was P times Q.
[32:55.400 --> 33:01.160]  So, N is really easy to get if I gave you P and Q, right?
[33:01.160 --> 33:02.600]  You guys know how to multiply.
[33:02.600 --> 33:05.620]  All you have to do is multiply P and Q together and you get N.
[33:06.160 --> 33:09.740]  But then, if I'm like, okay, here's N, now factor it.
[33:09.740 --> 33:15.000]  And not only factor it, find the two prime numbers that multiply together and get it.
[33:15.000 --> 33:16.200]  And you're like, what?
[33:16.880 --> 33:20.800]  And like, sure, like 6,887, that was our N.
[33:20.800 --> 33:22.560]  Like, you could do that.
[33:22.560 --> 33:24.440]  You could probably even do that by hand.
[33:25.760 --> 33:28.920]  But, I mean, computers also can do that if you don't want to.
[33:29.300 --> 33:34.280]  But all of a sudden, like, right, those aren't the level of numbers we're using in a real cryptosystem.
[33:34.360 --> 33:37.680]  We're using our 2048 bits, our 4096 bits.
[33:37.680 --> 33:47.880]  All of a sudden, if I hand you that number and I say factor it, and not only factor it, factor it using prime numbers, you're like, um, what?
[33:48.460 --> 33:52.900]  So, that's what makes it easy one direction, hard the other.
[33:53.800 --> 33:55.640]  Elliptic curves, right?
[33:55.640 --> 34:01.700]  So, for elliptic curves, we call the quicksand the discrete log problem.
[34:02.060 --> 34:07.460]  But it's essentially doing that point translation backwards, right?
[34:07.460 --> 34:13.140]  So, when you start with a point, it's really easy to find two lines to find the next one.
[34:13.140 --> 34:20.560]  But if you don't know where you're going, it's really difficult, and we just call that the discrete log problem.
[34:21.740 --> 34:26.600]  There's also, um, what was it?
[34:26.720 --> 34:28.000]  Eh, that's not going to be important.
[34:28.000 --> 34:28.680]  Okay.
[34:31.370 --> 34:39.890]  So, essentially, right, the struggle is real, as the millennials say, with going backwards for RSA and ECC.
[34:40.230 --> 34:44.950]  But the struggle is not real if you're a quantum computer.
[34:45.210 --> 34:51.750]  So, essentially, right, quantum computers run on qubits instead of our regular bits.
[34:52.630 --> 35:01.530]  And they, using that, can do these, you know, going backwards, they can do the factorization, they can do the discrete log problem in less than a day.
[35:01.870 --> 35:04.970]  Right, what did we say before with modern computers?
[35:04.970 --> 35:08.810]  For modern computers, it takes thousands of years to go backwards.
[35:09.090 --> 35:13.830]  And, you know, all of a sudden, quantum computers show up and they're like, eh, I got it, like, today.
[35:13.830 --> 35:16.570]  And actually, one has.
[35:16.570 --> 35:27.410]  There is, um, I don't remember where, but a quantum computer using a quantum tunneling process actually did break an RSA 2048.
[35:27.490 --> 35:33.690]  So, I don't remember how long it took, but it did break it.
[35:34.210 --> 35:36.330]  But that's just one example.
[35:36.330 --> 35:43.040]  Quantum computers are also typically, they could use, like, Shor's algorithm, if you guys have heard of that.
[35:43.040 --> 35:52.760]  So, a Shor's algorithm is going to be using probability to factor and probability to find the discrete log.
[35:53.300 --> 36:00.920]  And when you use probability, essentially what that means is you do not have to check every single number.
[36:01.540 --> 36:06.380]  And so because they don't have to check every single number now, it cuts their time, right?
[36:06.380 --> 36:09.580]  And as you can tell, it cuts the time drastically.
[36:13.810 --> 36:14.470]  Okay.
[36:14.770 --> 36:21.190]  So that's essentially, without going too, like, my brain hurts, into quantum computers.
[36:22.030 --> 36:27.290]  They do also have post-quantum cartography, right?
[36:27.290 --> 36:34.470]  So if we're like, okay, well, if this quantum computer is coming and it's about to break my RSA and my ECC, right, what do we do?
[36:34.470 --> 36:38.150]  We use post-quantum, which is my favorite.
[36:38.150 --> 36:39.430]  It's just more math.
[36:40.730 --> 36:43.610]  You guys want to see some post-quantum cartography?
[36:44.030 --> 36:45.110]  Yeah.
[36:45.910 --> 36:46.990]  Okay.
[36:47.270 --> 36:48.110]  Here you go.
[36:48.110 --> 36:49.070]  Here it is.
[36:50.410 --> 36:52.970]  So, post-quantum cartography.
[36:53.030 --> 36:58.330]  NIST just standardized an algorithm, MLDSA.
[36:58.330 --> 36:59.610]  Feel free to look it up.
[36:59.610 --> 37:01.610]  There's three strengths to it.
[37:03.030 --> 37:05.530]  MLDSA is, like, the government name for it.
[37:05.530 --> 37:10.790]  It's way less boring than the original name, which is Dilithium, if that sounds familiar.
[37:11.190 --> 37:16.750]  So Dilithium essentially uses lattices and learning with errors.
[37:16.790 --> 37:18.390]  So here's a lattice.
[37:18.390 --> 37:19.790]  It's a bunch of dots.
[37:20.270 --> 37:25.650]  And then my encrypted message is the red dot that I have just over on the left.
[37:26.250 --> 37:37.410]  So when you, in a post-quantum example, I have my encrypted message and I say, to break it, you need to find the closest dot on the lattice that it is to.
[37:38.490 --> 37:41.310]  Do you guys know which dot the red one is closest to?
[37:41.430 --> 37:43.270]  The bottom left, right?
[37:43.810 --> 37:44.430]  Yeah.
[37:44.430 --> 37:46.790]  And it's really easy just by looking at it.
[37:47.170 --> 37:54.350]  Now, what if you had a 3D lattice, a 4D lattice, you know, a 20th dimension lattice?
[37:54.490 --> 37:56.730]  It's not as easy anymore.
[37:58.150 --> 38:03.350]  And so that, right, because this is just a pretty basic visual example.
[38:05.070 --> 38:12.490]  But if you don't like the graph aspect, this will explain the learning with errors a little bit more.
[38:12.810 --> 38:16.650]  So here's my lattice, but instead of having a graph, now I have it algebraically.
[38:17.130 --> 38:20.290]  If you remember, this is what we call a systems of equations.
[38:20.750 --> 38:25.730]  I'm sorry if I just, like, tweaked something in your brain you hadn't thought about in a long time.
[38:26.590 --> 38:32.150]  So systems of equations, I have six equations, I have six variables, and I have six answers.
[38:32.370 --> 38:34.290]  That means I could solve this.
[38:34.290 --> 38:36.350]  I could solve this by hand if I wanted to.
[38:38.330 --> 38:40.150]  So I have, well, don't actually solve this.
[38:40.150 --> 38:41.330]  I just made this one up.
[38:41.330 --> 38:43.290]  So this is not going to work.
[38:44.190 --> 38:46.470]  So here's my lattice, right?
[38:47.070 --> 38:49.530]  Now pay attention as I go to the next slide.
[38:50.650 --> 38:52.730]  What's different about it?
[38:52.930 --> 38:55.270]  All the answers are different.
[38:55.690 --> 39:03.330]  So here I can see the answers 13, 790, 31, 414, 66, 35.
[39:03.630 --> 39:06.170]  Let's go back, right?
[39:06.170 --> 39:14.950]  But the actual answers are 12, 789, 30, 413, 65, and 34.
[39:14.990 --> 39:16.690]  So what did I do?
[39:16.690 --> 39:19.730]  I gave my answers an error.
[39:20.110 --> 39:22.590]  It's learning with errors.
[39:22.590 --> 39:29.330]  So essentially what this means is the public key is the error matrix, right?
[39:29.330 --> 39:30.590]  The error lattice.
[39:30.590 --> 39:32.390]  That's the public knowledge.
[39:32.390 --> 39:37.430]  Because you have no idea how much I shifted my answers by, right?
[39:37.430 --> 39:42.250]  Unless I told you the original answers, you have no idea I shifted everything by one, right?
[39:42.250 --> 39:45.870]  I could shift it either direction as much as I wanted.
[39:46.530 --> 39:52.650]  So that's where that error comes in and why it makes it extremely difficult to solve these problems.
[39:53.210 --> 40:00.210]  Because unless you start like just going through and figuring, you know, like quantum computers even can't solve this.
[40:00.210 --> 40:07.870]  So that's why we have this learning with errors and lattices as our post-quantum cryptography.
[40:09.250 --> 40:12.730]  And with that, we have successfully made it through.
[40:12.810 --> 40:14.530]  We have done our RSA, right?
[40:14.530 --> 40:15.770]  What was RSA?
[40:15.850 --> 40:21.230]  We have prime numbers, modular arithmetic, and then we have our elliptic curves, right?
[40:21.230 --> 40:24.050]  We have our little squid octopus graph thing.
[40:24.050 --> 40:26.050]  We have our point-to-point translation.
[40:26.490 --> 40:30.850]  We talked about breaking them using factorization and discrete logs.
[40:31.750 --> 40:34.510]  And then, right, our bonus post-quantum example.
[40:35.490 --> 40:44.150]  So, if you guys are still alive after doing just a bunch of math on a Thursday afternoon, give yourselves a little pat on the back.
[40:45.490 --> 40:48.390]  But we now have... I don't know what time it is.
[40:48.730 --> 40:49.610]  Is there time?
[40:50.950 --> 40:51.630]  Oh, yeah.
[40:51.630 --> 40:53.710]  We have some time for questions if there are any.
[40:54.010 --> 40:57.810]  If there's not, you can also just come up after if you don't want to ask it out loud.
[40:57.810 --> 40:58.590]  But thank you.
[41:05.470 --> 41:06.830]  Oh, yes?
[41:13.330 --> 41:18.850]  Well, I mean, like, in other nation-states, we don't really know unless they, like, tell us, right?
[41:19.010 --> 41:37.850]  I think... I mean, I'm kind of leaning... like, IBM has done a lot, you know, and I really like... so, with IBM, you can actually go online and get in line to use a quantum computer, which I think is awesome.
[41:37.990 --> 41:45.810]  I don't know how long that line is, but they also have, like, fake quantum software out there as well.
[41:45.810 --> 41:59.050]  If you want to, like, practice coding on a quantum computer, you can go on their website, and I think that's awesome, especially since, like, so just giving that access and that knowledge to the public, I think what makes... sets IBM ahead.
[41:59.170 --> 42:00.790]  You know, they're, like, not fooling around.
[42:00.790 --> 42:02.310]  They're actually doing the work.
[42:03.350 --> 42:04.410]  I mean, what's the other one?
[42:04.410 --> 42:04.930]  Google.
[42:04.930 --> 42:11.330]  Google has a really big one, but I'd probably lean a little more towards IBM, but that's just my opinion.
[42:17.020 --> 42:18.000]  Oh, yeah.
[42:42.550 --> 42:44.670]  What, 2035 is what, 10 years?
[42:45.910 --> 42:59.710]  I mean, I'm not a betting person, so I prefer to be, right, you know , 10 years is just unrealistic.
[43:00.430 --> 43:02.170]  So I think 10 years is fine.
[43:02.290 --> 43:04.590]  I mean, at the rate...
[43:04.590 --> 43:17.790]  first of all, right, when quantum computers come out, and then for them to be actually, like, worrisome for the average consumer, they have to be plentiful, right?
[43:17.790 --> 43:23.250]  If just IBM has a quantum computer, then is that really... how dangerous is that, right?
[43:23.250 --> 43:25.390]  So I guess that's kind of what you want to ask yourself.
[43:25.750 --> 43:34.630]  So if IBM gets one working in 10 years, how many years after that will then other people who are more dangerous get access to one?
[43:34.690 --> 43:36.630]  So I think 10 years is fine.
[43:36.630 --> 43:43.630]  Also, like, because we don't even have access to that many post quantum keys yet, and those things are large.
[43:43.630 --> 43:51.910]  If you haven't looked at a post quantum key, it is very large, and unfortunately, right, a lot of our machines aren't going to be able to handle those yet.
[43:52.090 --> 43:57.810]  So, I mean, and just getting that tech up, I don't... yeah, I think 10 years is fine.
[43:58.070 --> 44:03.230]  I mean, I even think 15, but more people... but that's me.
[44:03.230 --> 44:07.850]  Other people like to be a little, yeah, a little more on top of it.
[44:08.930 --> 44:10.310]  Did that answer?
[44:10.490 --> 44:11.830]  Okay, yeah.
[44:13.730 --> 44:14.210]  Oh
[44:33.910 --> 44:35.730]  , yeah, yeah.
[44:35.730 --> 44:37.770]  Like how you get a 20th dimension?
[44:49.030 --> 44:54.370]  Yeah, so, yeah.
[44:54.370 --> 45:01.650]  So when we say 20th dimension, like, I guess I don't mean it in, like, our 3D world.
[45:01.650 --> 45:06.230]  Like here, right, I have six equations with six variables.
[45:06.230 --> 45:09.270]  Each variable just gives me another dimension, right?
[45:09.270 --> 45:09.610]  Yeah.
[45:09.610 --> 45:12.330]  So a 20th dimension just has 20 variables.
[45:12.330 --> 45:12.970]  Yeah.
[45:13.090 --> 45:14.670]  Does that... yeah, okay.
[45:17.860 --> 45:19.020]  Oh, yeah?
[45:32.900 --> 45:34.180]  Yeah.
[45:34.340 --> 45:46.220]  So the symmetric... to repeat the question, why are, like, symmetric like AES and hashing less susceptible to quantum computing?
[45:47.140 --> 45:49.980]  It's essentially like...
[45:54.420 --> 45:56.480]  how do I explain this?
[46:00.730 --> 46:04.390]  I mean, it's not that they're, like, not, right?
[46:04.390 --> 46:09.690]  But they use completely different algorithms.
[46:09.930 --> 46:17.590]  So, like, what we went over today is, like, that's not even at all what symmetric encryption uses, right?
[46:18.050 --> 46:20.630]  And AES, there's, like, what, 256?
[46:21.230 --> 46:27.150]  I'm not as familiar with them, but essentially, right, because they're on... what's the word?
[46:27.150 --> 46:29.730]  You know, they use more the blockchain, right?
[46:29.730 --> 46:33.390]  So, like, they're just... yeah, yeah.
[46:33.390 --> 46:39.030]  So with that concept, it's just going to... quantum computers are going to attack it differently.
[46:39.030 --> 46:44.010]  I guess we just haven't reached that point where they can attack it as efficiently.
[46:44.070 --> 46:44.710]  Yeah.
[47:05.470 --> 47:07.550]  There are several others.
[47:08.330 --> 47:11.910]  You mean, like, other post-quantum algorithms?
[47:12.090 --> 47:12.930]  Is that what you're talking about?
[47:12.930 --> 47:13.590]  Yeah.
[47:14.750 --> 47:15.710]  Yeah.
[47:18.870 --> 47:19.590]  So I've...
[47:19.590 --> 47:29.410]  well, for encryption, there's a homogenous one, but it was proven to be broken.
[47:29.730 --> 47:33.190]  So that's why NIST didn't move forward with standardizing it.
[47:33.670 --> 47:37.090]  There's also, like, Sphinx Plus for digital signing.
[47:37.090 --> 47:39.430]  There's several others out there for digital signing.
[47:40.070 --> 47:45.170]  And those are going to be using... what is it?
[47:46.530 --> 47:48.870]  Those actually fall back onto the hashing.
[47:48.870 --> 47:55.550]  They're going to use things like Merkle trees and hashing for a lot of their digital signing.
[47:56.270 --> 47:57.490]  So that exists out there.
[47:57.490 --> 48:05.730]  But as for, like, just an actual encryption system, they're just not up to standards.
[48:06.010 --> 48:09.790]  So there are... they do exist, right?
[48:09.790 --> 48:15.990]  And you can, like, look up what are some other ones, but they just haven't been able to keep up to par.
[48:16.570 --> 48:17.170]  Yeah.
[48:24.250 --> 48:25.070]  Okay.
[48:25.070 --> 48:25.430]  Sweet.
[48:25.430 --> 48:26.670]  Thanks, everyone.