Modular Arithmetic (Part 1)

Network Security: Modular Arithmetic (Part 1)
Topics discussed:
1) Introduction to modular arithmetic with a real-time example.
2) Clock analogy for understanding the concept of modulus.
3) Usage of congruence in cryptography.
4) Explanation on the usage of congruence instead of equality in cryptography.
5) Identification of a few valid and invalid congruence with suitable explanations.

Music:
Axol x Alex Skrindo - You [NCS Release]

#NetworkSecurityByNeso #Cryptography #NetworkSecurity #ModularArithmetic

[anmol]

Thanks for these videos. My discrete prof teaches like he's explaining the concepts to himself rather than first time learners, these videos really help.

[rajupujari]

yes, both are valid,
first -> -8 = 7(mod 5) formula here is (a = b(mod n)) => (a-b)/n => -8-7/5 => remainder is 0 => first one is valid
second -> -3 = -8(mod 5) => -3-5 = -8 => second on is valid

[sakriyapyakurel]

Both are valid
For first :
-8-7 = -15 (divisible by 5)
For second:
-3-(-8) = 5(divisible by 5)

[akankshasonkar]

Thanks for this i tried your concepts for 5 hrs and now i mastered 🙏

[mahdi-hasan]

Note: remember, a = b (mod m) is same as b = a (mod m).

1) -8 = 7 (mod 5)
=> 0 = 15 (mod 5) [Adding 8 on both sides]
=> true. [15 mod 5 = 0]

2) -3 = -8 (mod 5)
=> 5 = 0 (mod 5) [Adding 8 on both sides]
=> true. [5 mod 5 = 0]

[sravyasirikonda]

How to find if a modular arithmetic is valid/ invalid?
Simple.
a=b (mod n) only if (a-b)/n is divisible.

[bhavyas]

Both are valid first one using the property n|(a-b) and the second one using the property a cong b(mod n) => b cong a(mod n)

[alyalyalyalyalyalya]

THANK YOU I COULDN’T UNDERSTAND THIS FOR THREE DAYS

[PrithaMukherjee-cl]

Sir,
Both are valid.
First one,
-8 divided by 5, if the quotient used is -3 then the remainder is coming 7 so the expression is coming valid.
For the second,
-3 divided by 5, the quotient is 0 then the remainder is -3 which can also be written as -8 (also be represented as -8 + 5=-3). So the second expression is also coming valid.
Regards.

[darshanshah]

5*(-3) + 7 = 8
5*(1)-8 = -3

hence both are Valid Congruent .

[nayang]

I never put comments because this account is from my mom, but OH MY GOD, this video is crazy, my teacher always says things he believed everyone knows but it was my first time to see those modular congruence for AMC 10 competitions for a freshman, Thank you!

[davidbarker]

Exceptional video. This is how teaching is done. Thank you!

[RitikTUF]

a congruence b (mod n) represents n divides (a-b)
like
i) 15 cong 3 (mod 12) ==>> (15-3)/12 =1 it is divisible
similarly
-8 cong 7 (mod 5) ==>> (-8-7)/5 = -15/5 = -3 it is also true
similarly -3 - (-8) = 5 / 5 = 1 also true. hence both last questions are valid.

[bossthekhani]

I really understand valid and invalid congruence
1) -8==7(mode 5) wrong
2) -3==-8(mode 5) right

[carlosgarcia]

Both are valid. -n mod m = m - ( n mod m) or n|(a-b), a mod m == b is a== b mod m, a/m gets a remainder = rem when b/m.

[rafastyles]

Damn I never expected that a clock was the perfect example of modular operations.
Like 22 == 10 (mod 12)

Thank you for explaining. I had seen this before in training for math olympiads and never understood

[adityaprasadsahoo]

The congruence -8 ≡ 7 (mod 5) is true, as both -8 and 7 leave the same remainder when divided by 5.

[vasulibhai]

1-) -8%5 --> -8+5 = -3 --> -3+5 = 2
and
7%5 => 2. Hence correct
2-) -3%5 --> -3+5 = 2
and
-8%5 --> -8+5 = -3 --> -3+5 = 2.
Hence correct.

[KenderGuy]

I saw a math puzzle using modular arithmetic and couldn't solve it, so I looked up some videos to brush up
was skipping through most of the video to see where my knowledge ends, I hope part 2 has new information for me
I was checking and got all the answers right that you showed, but you said to put my answer to the last two in the comments
so I decided to humor it

the last two are both valid because
5 * (-2) + 2 = -8, therefore -8 mod 5 is congruent with 2
7 is also a multiply of 5, plus 2, so it is also congruent mod 5 with 2
same with the last one, because -5 + 2 = -3 and -10 + 2 = -8

[akshaygamez]

a=km+b it is same as euclid's division lemma (a=bq+r)