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The Open Office Impress slides that I've used in the final project of GTR.

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- 1. GTR Final Project [9A(13)]Mathematics Mini-lectureModulus Arithmetic and Field
- 2. But … Whats Modulus Arithmetic?We all know about division...a = bq+r (0≤r<b)...Where q stands for quotient and r for remainderNow, Lets define a mod b as follows...a mod b = r ↔ a = bq+r (0≤r<b)… Which is the action of finding remainderThen we could define congruence...a≡b (mod m) ↔ a mod m = b mod m…Which is equal to say the remainders are the same
- 3. Exercise 1 (10s)1.1) Calculate the following1.1.a) 10 mod 101.1.b) 9 mod 81.1.c) 100 mod 651.2) Find x1.2.a) 100 ≡ x (mod 10), 55<x<651.2.b) 95 ≡ x (mod 87), 180<x<1901.2.c) 54≡ x (mod 8), 15<x<25
- 4. Exercise 1 (Answer)1.1a) 01.1b) 11.1c) 351.2.a) 601.2.b) 1821.2.c) 22
- 5. What about field?To know what is field, we must have to know:set, group, abelian group and ringSet is a group of numbers:e.g. {1,2,3,4,5,...} = N{1,3,5,7,9,...} = set of odd numbers{1,4,9,16,...} = set of square numbersTo confirm that you understand, it is...Exercise...time now!
- 6. Exercise 2 (10s)2.1) Define the following sets.2.1.a) {2,4,6,8,10,...}2.1.b) {2,4,8,16,32,...}2.1.c) {1,2,3,4,5,6,7,8,9,10}2.2) Write down the the following sets.2.2.a) The set of odd numbers between 10 and 202.2.b) The set of triangular numbers2.2.c) The set of multiples of 5 less than 101
- 7. Exercise 2 (Answer)2.1a) Even numbers2.1b) Positive powers of 22.1c) Natural numbers less than 112.2.a) {11,13,15,17,19}2.2.b) {1,3,6,10,15,21,28...}2.2.c) {5,10,15,20,...,90,95,100}
- 8. Then... Whats group?We then have to define operation ■ as...a ■ b is an element of set G...where a and b are elements of G.And we all know the associative law...(a ■ b) ■ c = a ■ (b ■ c)...which means order doesnt affect the resultNext is identity element e ...a ■ e = e ■ a = a...whichs like 0 for addition and 1 for multiplication
- 9. Then... Whats group?At last is inverse element...a ■ b = b ■ a = e...here we say b is the inverse element of aFinally to define group as...1. Related to operation ■ and enclosed2. Associative law is true for all elements3. Identity element exists4. Inverse element exists for all elements...a set that satisfy all these postulates is a groupNote: in (1), a set is related to operation ■ andenclosed means that a ■ b is an element of set G
- 10. Exercise 3 & Answer (10s)Check if the following are groups...a.) N, operation + (addition)b.) N, operation * (multiplication)c.) set of odd numbers, +d.) set of even numbers, +Answers:a.) yesb.) no (no inverse element)c.) no (not closed, odd + odd = even)d.) yes
- 11. After that... Whats abelian group?We all know the commutative law...a ■ b = b ■ a...which is the only difference between a normalgroup and an abelian groupAn abelian group is...a.) a groupb.) commutative law is true for all elements...a set that satisfies both of the above postulates isan abelian group
- 12. Exercise 4 & Answer (10s)Check if the following are abelian groups...a.) N, operation + (addition)b.) set of 2*2 matrix, operation * (multiplication)c.) set of odd numbers, +d.) set of even numbers, +Answers:a.) yesb.) no (not satisfy the commutative law)c.) no (not closed, odd + odd = even)d.) yes
- 13. Next... Whats ring?In a group, we have only one operation......in a ring, we have TWO operations (■ and □)We then have the associative law...(a ■ b) □ c = (a □ c) ■ (b □ c)Now, a ring is...a.) an abelian group with operation ■b.) an abelian group with operation □ but NOTNECESSARY to have inverse elementc.) associative law is true for all elements...a set that satisfies all the above postulates is a ring
- 14. Exercise 5 & Answer (10s)Check if the following are rings...a.) N, operation + & *b.) Q, operation * & +c.) set of odd numbers, + & *d.) set of even numbers, + & *Answers:a.) yesb.) no (no inverse element for 1stoperation)c.) no (not closed, odd + odd = even)d.) yes (even * even = 4n)
- 15. Finally... Whats field?In a ring, we maybe unable to do the reverse of □......in a field, we could do the reverse of □A field is...a.) a ring with operations ■ and □b.) have the reverse element for operation □(EXCEPT 0 when □ is multiplication)...a set that satisfies both of the above postulates is a fieldAnd its EXERCISE TIME again!
- 16. Exercise 6 & Answer (10s)Check if the following are fields...a.) Q, operation + & *b.) Z, operation + & *c.) set of odd numbers, + & *d.) R, operation + & *Answers:a.) yesb.) no (no inverse element for 2ndoperation)c.) no (not closed, odd + odd = even)d.) yes
- 17. So... How are they related?In a congruence, +, - & * does not change the congruenceIf a≡b (mod m),a + C ≡ b + C (mod m),a - C ≡ b - C (mod m),a * C ≡ b * C (mod m),But not / …15≡75 (mod 12),(15/5)≡(75/5) (mod 12)(15/3)≢(75/3) (mod 12)Only if Cs coprime with m, the congruence would hold
- 18. So... How are they related?Now define {0,1,2,3,...,m-1} as a group withoperation ,⊠ which = (a / b) mod m, and denote the⊠group Z/mZWe found that only when b is coprime with m, thegroup actually holds.Now define b m as b is coprime with m⊥Define (Z/mZ)xas the group of {x:0≤x<m,x m}with⊥operation . This is the reduced residue class group.⊠
- 19. So... How are they related?Now we could easily prove that (Z/mZ)xis an abeliangroup. We could then define a ring with and ,⊞ ⊠where is (a+b) mod m. The ring is called⊞residue class ring of modulo m.The following is the operation tables of the above ring:⊞ 0 ... m-10 0 ... m-1... ... ... ...m-1 m-1 ... m-2⊠ 0 ... m-10 0 0 0... 0 ... ...m-1 0 ... 1
- 20. So... How are they related?When m is coprime to all elements in Z/mZ, then m isprime. In this case, the residual ring of modulo m wouldbecome a field. This field is called finite field of modulo p,denoted as Fp, where Fp = Z/pZ.Finite field is the final product of modulus arithmetic andgroup theory. This thing is very useful in Maths, particularlyin the proof of Fermats Last theorem, as you may see soon.But before that, lets do some exercise first...
- 21. Exercise 7 & Answer (20s)Write down the two operation tables of F3Answers:⊞ 0 1 2 ⊠ 0 1 20 0 1 2 0 0 0 01 1 2 0 1 0 1 22 2 0 1 2 0 2 1
- 22. How is finite field used?Finite field is used for reduction of the elliptic curve.Substituting the points of the curve with the coordinates ofthe finite fields, the resulting points that still holds is thereduced curve.Finite fields are used because the properties of it. Therational field, Q, has an infinite numbers of elements, buttheres just 1 Q. However, there are infinite numbers of Fp,but each has a finite number of elements. Therefore it isused by mathematicians to explore the Q.
- 23. Why to reduce elliptic curves?Finite field is used for reduction of the elliptic curves,as mentioned above. But why?Actually, this action is an important part in the proof ofthe Fermats Last Theorem by Andrew Wiles. However,the proof is too difficult for talking here and is beyond thescope of this lecture. What you need to know only is that,there is a wide range of uses of finite field in numbertheory.So now, lets say...
- 24. GoodBye!

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