I don't know why anyone would be sleep for a lecture on Number Theory! To me, this one of most lovable fields of Mathematics and definitely my favourite. Great lecture, knew most but watched it purely for pleasure. Greetings from Finland :D
I agree. Number Theory is one of the oldest disciplines in the whole of mathematics. It is all around us, as well as being the backbone of the internet and the web (they are not the same!), none of it would work without the mathematics of Number Theory!!
I was tearing my hair out figuratively and literally trying to understand this chapter of my cryptography book. I was so happy to see you have a lecture here thank Dr Paar
I wish I found this series of lectures earlier. I laughed out loud when he said you don’t want to learn the Extended Euclidean Algorithm, because that is what my cryptography professor told our class to do 😆. In fact he made us cover the entire number theory portion of the course on our own.
The ONLY way to truly get comfortable with Euclid Extended is to work on the exercises... and then get a friend to write 100 more for you to solve. Really, do fifty by hand. It helped me to use red ink for the r0 and r1 values. That way I could see them as algebra variables I was trying to isolate. Thank you Prof. Paar !
You may not read this since it's been over 2 years since posting this video, but thanks. I'm currently taking a Cryptography course and missed portions of some lectures and this video was very helpful.
I model and simulate cyber and electronic just recentlyfor employment. Was reading PySDR tutorial by Dr. Marc Lichtman. DPS uses many tangential concepts. Now I will be able to model cyber better.
Remarks gcd(a,b) = Xa+ Yb is called the bezout lemma too. and the extended euclidean algorithm is called eculidean division algorithm. is the algorithm taught to children in the schooll By the way good teacher.my question is if are the proofs in the book ?
Professor Paar. I must simply say excellent lectures. You made be buy your book just because you put those lectures online, Is there a program at Uni Bochum where we can watch also the lecture that you teach on Implementation of Cryptographic Schemes I and II or Symmetric Cryptanalysis?
Introduction to Cryptography by Christof Paar Well then to make it more simple, will that be fine to drive to Bochum during the days you have the Symmetric Cryptanalysis lectures. Will the Uni allow this officially i.e. allowing non official students to stay in during the lectures. I promise not to talk and to go to sleep when you require it :).
Konstantin H Sorry for the late reply. Nobody will mind if you sit in any of the lectures. It might still be a good idea to contact the professor ahead of time. Please note that most lectures are, unfortunately, in German.
Im confused. The GCD of 27 and 27 is 27. Why are we only considering prime factors? Btw I skipped to the euclidean algorithm section in order to get a piece of information. Perhaps it was mentioned earlier that this was a convention being chosen in this particular case???
If we have a very very large number and we don't know if this number is prime or not , how are we going to use the Euler's Phi Function? The computer is going to do the work?
You didn't state Uniqueness (einzigartigkeit)of s,t - not sure, eg gcd(12,4) = 4 right! 4 = s12+ t4 gives (s,t) soln. as { (1,-2), (2,-5) .. so on} so not sure now if t = inv(4 mod 12) is unique? Also not sure if inv(r_1 mod r_0) is unique when gcd(r_0,r_1) is relative prime?
Sir i have a question: in 1:19:00 you say that the complexity of Euler totient function is O( 2^n ) But in my opinion, Euclidian algorithm is O( log(n) ) therefore the complexity of Euler totient function should be O( n.log(n) ) Am i wrong?
Dear Professor, I have a confusion.If you can please see my understanding is right or not. At 1:09:00 you write "The parameter t of the EEA is the inverse of r1 mod r0. So I look back into the example we solved: gcd(973,301) and when we write the following: r4=13*973+(-42)*301. So here my t is (-42) so considering the calculation: ro mod r1=973 mod 301=70 which implies that t (here -42) is the inverse of ro mod r1 (here 70). Also I have another confusion: When the gcd(n,a)=1 then n and a are relatively prime right? so here in the example gcd(973,301)=7 so they are not relatively prime so what is the connection with the example? I hope you reply. Thank you Professor
The only connection is that 973 and 301 are used as initial values to demonstrate the EEA, but when you use those values the EEA will not return the inverse of 201 given modulo 973, since gcd(973,301) is not 1. However, even though gcd(973,301) is not 1, the result of EEA is still useful in finding the inverse of 301/7 given modulo 973/7. The inverse is 973/7-42. We can test this: 301/7 * (973/7-42) = 4171 4171 mod (973/7) = 1 !!!
I've tried to calculate the inverse of a. For example gcd(89,37) = 1 Extended euclidian algorithm gives me: 5*89 - 12*37 = 1 Then I thought 12*37 (mod 89) = 1 but it's 88. Where did I go wrong? Thanks, Lars.
Here is the mistake: It holds that -12 * 37 = 1 mod 89 i.e., the inverse of 37 mod 89 is equal to -12 which, in turn, is equal to 77 mod 89. Thus, 77 * 37 = 1 mod 89 regards, christof
Dear Professor, I know the computations: r1=37 and r2=15=89+(-2)*37 and r3=7=(-2)*89+5*37 and r4=1=5*89+(-12)*37 ............ My question is what you did after it like from where 77 came :( I mean 89-12=77 ???but can u please explain a bit i want the feeling of step by step you usually follow. What I did is I follow all steps you mention at 1:09:32 and gcd(n,a)=1=s*n+t*a so gcd(89,37)=1=5*89+(-12)*37. so here -12 is t or we call is a inverse (sorry for the notations). then I also get you say about division by mod n (n=89) so when i divide 5*89+(-12)*37 i come up with 0 for first term(5*89) and then i am left with (-12)*37/mod 89. Then I couldnt follow :( I hope you reply.
1:07:50 S.n+t.a=1 Then we take mod n. (S.n+t.a) mod n = 1 mod n S.n mod n=0 So t.a mod n =1 mod n But you are saying t.a=1 mod n How t.a mon n = t.a? Can you please clarify professor?
Look at the definition of inverse again. a x a-1 == 1 mod n t . a-1 == 1 mod n Substituting a-1 for ta makes ta the inverse. Gcd(n, a) the n goes to zero(n divided by n leaves zero remainder) that leaves ta
sir your book 6.3.4 Fermat's Little Theorem and Euler's Theorem def. First line last word (crpytography) word wrong . Right word (cryptography). Please sir don't mind .Please correct this incorrect word .Because your book very popular . thank you sir.
@@TheGenerationGapPodcast extendedEu :: Integer -> Integer -> (Integer, Integer) extendedEu a 0 = (1, 0) extendedEu a b = (t, s - q * t) where (q, r) = quotRem a b (s, t) = extendedEu b r
@@balthazarbeutelwolf9097 as someone that's learning this course, if he were to explain it lime that it would be much more difficult to understand. I believe his approach is to make this more approachable without knowledge from outside the course
Bonsoir pour le couple des entiers (973 ; 301) on aura pat le SCHEMA d'OURAGH ......973.......301.......70.......21........7 ........0............-3........-4........-3 .......13.........-42........13.......-3........1 et donc 973(13)+301(-42)=7 très simplement . Cordialement.
2:40 Euclidean Algorithm
23:50 extended Euclidean Alg.
1:12:20 Some theorems
Thanks
I don't know why anyone would be sleep for a lecture on Number Theory! To me, this one of most lovable fields of Mathematics and definitely my favourite. Great lecture, knew most but watched it purely for pleasure. Greetings from Finland :D
they are sleep because his like cliffhangers. they end at the exciting part. only in his case the are no episodes. you are just hung.
I agree. Number Theory is one of the oldest disciplines in the whole of mathematics. It is all around us, as well as being the backbone of the internet and the web (they are not the same!), none of it would work without the mathematics of Number Theory!!
"You can start playing cards again". Don't think I've ever heard a prof say that before.
Professor Paar, Thank you very much for your methods of teaching. it makes everything so clear. Big applause.
I had such a horrible number theory professor. I would beg to have a professor like this guy.
Shouldve went to UCF for good math profs
The best lecture about Number Theory for PKC out there! Thank You Prof. Paar!
I was tearing my hair out figuratively and literally trying to understand this chapter of my cryptography book. I was so happy to see you have a lecture here thank Dr Paar
I wish I found this series of lectures earlier. I laughed out loud when he said you don’t want to learn the Extended Euclidean Algorithm, because that is what my cryptography professor told our class to do 😆. In fact he made us cover the entire number theory portion of the course on our own.
Hello from Kent State University. I am using your material for individual study and the lectures help me TREMENDOUSLY! Thanks!
The ONLY way to truly get comfortable with Euclid Extended is to work on the exercises... and then get a friend to write 100 more for you to solve. Really, do fifty by hand. It helped me to use red ink for the r0 and r1 values. That way I could see them as algebra variables I was trying to isolate. Thank you Prof. Paar !
I paid for this on Coursera and it was awful. Prof Christof Paar, you are the sweetest and the brightest!
You may not read this since it's been over 2 years since posting this video, but thanks. I'm currently taking a Cryptography course and missed portions of some lectures and this video was very helpful.
Euclidean Algorithm 2:40
extended Euclidean Alg. 24:00
Some theorems 1:12:30
Hello from Vietnam. Thanks Profession Paar so much :)
wooow this Professor is insane at explaining Concepts🤩🤩🤩
58:58 yes, also in latin the plural nominative case of "formula" is "formulae". Thanks so much, Prof. Paar, super great lecture!
Well thanks for uploading it!
The moment, 1:03:00 when Professor says you can sleep now, you can start playing your cards now :D
This was amazing!
Thank you so much for recording these lectures! :)
Excellent lecture, this was very helpful.
Thank you!
thanks for making this video!
Dear Professor Paar. Superb lecture I ever watched!
I model and simulate cyber and electronic just recentlyfor employment. Was reading PySDR tutorial by Dr. Marc Lichtman. DPS uses many tangential concepts. Now I will be able to model cyber better.
Remarks
gcd(a,b) = Xa+ Yb is called the bezout lemma too.
and the extended euclidean algorithm is called eculidean division algorithm. is the algorithm taught to children in the schooll
By the way good teacher.my question is if are the proofs in the book ?
Thanks, Mr. Paar. It is the really nice lecture.
Professor Paar. I must simply say excellent lectures. You made be buy your book just because you put those lectures online, Is there a program at Uni Bochum where we can watch also the lecture that you teach on Implementation of Cryptographic Schemes I and II or Symmetric Cryptanalysis?
No, unfortunately not (yet). I plan to do this in 1-2 years. Coming up with "polished" lectures takes a lot of time :)
Introduction to Cryptography by Christof Paar Well then to make it more simple, will that be fine to drive to Bochum during the days you have the Symmetric Cryptanalysis lectures. Will the Uni allow this officially i.e. allowing non official students to stay in during the lectures. I promise not to talk and to go to sleep when you require it :).
Konstantin H Sorry for the late reply. Nobody will mind if you sit in any of the lectures. It might still be a good idea to contact the professor ahead of time. Please note that most lectures are, unfortunately, in German.
You are great. Thank you.
Loved the lecture! Need a different camera operator and reduce motion by about 50 percent. :-)
Thank you :))
Great Professor
Im confused. The GCD of 27 and 27 is 27. Why are we only considering prime factors? Btw I skipped to the euclidean algorithm section in order to get a piece of information. Perhaps it was mentioned earlier that this was a convention being chosen in this particular case???
Let p1
Very difficult problem
When explaining extended gcd, better start with a concrete example first.
Thanks for upload!!!
You can do it in oneline using Xor swaps and modulo
Vielen vielen Dank, Sir
If we have a very very large number and we don't know if this number is prime or not , how are we going to use the Euler's Phi Function? The computer is going to do the work?
You didn't state Uniqueness (einzigartigkeit)of s,t - not sure, eg gcd(12,4) = 4 right!
4 = s12+ t4 gives (s,t) soln. as { (1,-2), (2,-5) .. so on} so not sure now if t = inv(4 mod 12) is unique? Also not sure if inv(r_1 mod r_0) is unique when gcd(r_0,r_1) is relative prime?
sir very nice good excellent lectures
Sir i have a question:
in 1:19:00 you say that the complexity of Euler totient function is O( 2^n )
But in my opinion, Euclidian algorithm is O( log(n) )
therefore the complexity of Euler totient function should be O( n.log(n) )
Am i wrong?
How cute is the brush at 46:34
nice lecture
1:45:00 best part 😌
24:07 Extended Euclidean Algorithm; you can also to to goo.gl/hD8kmg (the box on the right is the running tutorial.)
Dear Professor, I have a confusion.If you can please see my understanding is right or not. At 1:09:00 you write "The parameter t of the EEA is the inverse of r1 mod r0. So I look back into the example we solved: gcd(973,301) and when we write the following: r4=13*973+(-42)*301. So here my t is (-42) so considering the calculation: ro mod r1=973 mod 301=70 which implies that t (here -42) is the inverse of ro mod r1 (here 70). Also I have another confusion: When the gcd(n,a)=1 then n and a are relatively prime right? so here in the example gcd(973,301)=7 so they are not relatively prime so what is the connection with the example? I hope you reply. Thank you Professor
Multiplicative inverses only exist when the gcd is 1.
The only connection is that 973 and 301 are used as initial values to demonstrate the EEA, but when you use those values the EEA will not return the inverse of 201 given modulo 973, since gcd(973,301) is not 1. However, even though gcd(973,301) is not 1, the result of EEA is still useful in finding the inverse of 301/7 given modulo 973/7. The inverse is 973/7-42. We can test this:
301/7 * (973/7-42) = 4171
4171 mod (973/7) = 1 !!!
merci !
for C: int E(int r,int s){return r%s>0?E(s,r%s):s;}
I've tried to calculate the inverse of a. For example gcd(89,37) = 1
Extended euclidian algorithm gives me: 5*89 - 12*37 = 1
Then I thought 12*37 (mod 89) = 1 but it's 88.
Where did I go wrong? Thanks, Lars.
Here is the mistake: It holds that
-12 * 37 = 1 mod 89
i.e., the inverse of 37 mod 89 is equal to -12 which, in turn, is equal to 77 mod 89. Thus,
77 * 37 = 1 mod 89
regards, christof
Dear Professor, I know the computations:
r1=37 and
r2=15=89+(-2)*37 and
r3=7=(-2)*89+5*37 and
r4=1=5*89+(-12)*37 ............
My question is what you did after it like from where 77 came :( I mean 89-12=77 ???but can u please explain a bit i want the feeling of step by step you usually follow.
What I did is I follow all steps you mention at 1:09:32 and gcd(n,a)=1=s*n+t*a so gcd(89,37)=1=5*89+(-12)*37. so here -12 is t or we call is a inverse (sorry for the notations). then I also get you say about division by mod n (n=89) so when i divide 5*89+(-12)*37 i come up with 0 for first term(5*89) and then i am left with (-12)*37/mod 89.
Then I couldnt follow :(
I hope you reply.
Hi Professor i got it after solving Problems in book :)
1:07:50
S.n+t.a=1
Then we take mod n.
(S.n+t.a) mod n = 1 mod n
S.n mod n=0
So t.a mod n =1 mod n
But you are saying t.a=1 mod n
How t.a mon n = t.a?
Can you please clarify professor?
Look at the definition of inverse again. a x a-1 == 1 mod n
t . a-1 == 1 mod n
Substituting a-1 for ta makes ta the inverse. Gcd(n, a) the n goes to zero(n divided by n leaves zero remainder) that leaves ta
sir your book 6.3.4 Fermat's Little Theorem and Euler's Theorem def.
First line last word (crpytography) word wrong . Right word (cryptography). Please sir don't mind .Please correct this incorrect word .Because your book very popular . thank you sir.
Karim Khan Thanks for catching this. We will correct it in the 2nd edition. Thanks, christof
ok sir Thank you.
Please explain about mod arithmetic, sir.
Please have a look at Lecture 2 of this series. regards, christof
Euclidean bytes = 3.05 =Phi = Pi
God the lecture was golden but sadly it seems like the students in the class were taking this too much for granted
doing this as an iterative algorithm is a pain - why not use a functional language and have this done and over with in 2 minutes?
Explain. We are all hears and eyes. Enlighten us
@@TheGenerationGapPodcast extendedEu :: Integer -> Integer -> (Integer, Integer)
extendedEu a 0 = (1, 0)
extendedEu a b = (t, s - q * t)
where (q, r) = quotRem a b
(s, t) = extendedEu b r
@@balthazarbeutelwolf9097 as someone that's learning this course, if he were to explain it lime that it would be much more difficult to understand.
I believe his approach is to make this more approachable without knowledge from outside the course
Rule number one for a teacher: Never assume that everyone is on the same page!
1:02:57 savage asf
function gcdEA(n,m) { if (n mod m == 0) return n; else gcdEA(m, n mod m); }
Bonsoir
pour le couple des entiers (973 ; 301) on aura pat le SCHEMA d'OURAGH
......973.......301.......70.......21........7
........0............-3........-4........-3
.......13.........-42........13.......-3........1
et donc 973(13)+301(-42)=7
très simplement .
Cordialement.
Euler = Euclidean 0 = 5/01/02/03/04/05/06/07/08/09/01
this guy wastes a lot of time....he kibbitzes and repeats himself and kibbitzes....9 minutes in and still basically nothing
what's "kibbitzes"?