1. Number Theory dan Modulo
Multiplicative Inverse

2.  Kriptografi dan Sistem Keamanan Komputer
 Standar Sistem Keamanan
 Enkripsi Kunci Simetrik
 Number Theory
 Enkripsi Kunci Publik
 Autentikasi Pesan dan Fungsi Hash
2011-2012-3 Anung Ariwibowo 2

3.  Himpunan bilangan
 Operasi yang ditentukan terhadap elemen-
elemen himpunan tersebut
 Hasil operasi berada di dalam himpunan
(closure)
 Memenuhi
 associative law: (a.b).c = a.(b.c)
 has identity e: e.a = a.e = a
 has inverses a-1: a.a-1 = e
 Jika memenuhi hukum commutative
 a.b = b.a
 Abelian group

4.  exponentiation
 repeated application of operator
 a3 = a.a.a
 Elemen identitas
 e=a0
 Cyclic group
 Setiap elemen adalah exponent dari sebuah
generator
 b = ak for some a and every b in group
 a = generator
 b = elemen-elemen himpunan selain a

5.  a set of “numbers”
 with two operations (addition and
multiplication) which form:
 an abelian group with addition operation
 and multiplication:
 has closure
 is associative
 distributive over addition: a(b+c) = ab + ac
 if multiplication operation is commutative, it
forms a commutative ring
 if multiplication operation has an identity and
no zero divisors, it forms an integral domain

6.  modulo operator “a mod n” didefinisikan
sebagai sisa bagi jika a dibagi n
 congruence: a ≡ b mod n
 jika dibagi n, a dan b memiliki sisa bagi yang sama
 100 ≡ 34 mod 11
 b disebut residue dari operasi a mod n
 untuk integer: a = qn + b
 biasanya dipilih sisa bagi positif terkecil
 ie. 0 <= b <= n-1
 modulo reduction
 -12 mod 7 = -5 mod 7 = 2 mod 7 = 9 mod 7

7.  a mod n
 a≥0
 a = qn + b
 q≥0
 b≥0
 a<0
 'overshoot'
 a = qn + b
 q<0
 b≥0
2011-2012-3 Anung Ariwibowo 7

8.  a mod n
 0≤a<1
 a = qn + b
 q≥0
 b≥0
 Modular Multiplicative Inverse
 aa-1 = 1
2011-2012-3 Anung Ariwibowo 8

9.  Sebuah bilangan bukan-nol b membagi a jika
untuk sebuah bilangan m didapatkan a=mb
(a,b,m adalah integers)
 d.kl. b membagi a tanpa sisa
 dituliskan: b|a
 dikatakan b adalah pembagi dari a
 Contoh: all of 1,2,3,4,6,8,12,24 divide 24

10.  'clock arithmetic'
 menggunakan sejumlah berhingga bilangan
 loops back from either end
 modular arithmetic is when do addition &
multiplication and modulo reduce answer
 can do reduction at any point
 a+b mod n = [a mod n + b mod n] mod n

11.  bisa diterapkan kepada sembarang group of
integers
 Zn = {0, 1, … , n-1}
 Membentuk commutative ring for addition
dengan multiplicative identity
 note some peculiarities
 if (a+b)=(a+c) mod n
then b=c mod n
 but if (a.b)=(a.c) mod n
then b=c mod n only if a is relatively prime to
n

12. + 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
4 4 5 6 7 0 1 2 3
5 5 6 7 0 1 2 3 4
6 6 7 0 1 2 3 4 5
7 7 0 1 2 3 4 5 6

13.  [0]: {0, 8, 16, 24, . . .}
 [1]: {1, 9, 17, . . .}
 [2]: {2, 10, . . .}
 [3]: {3, 11, . . .}
 [4]: {4, 12, . . .}
 [5]: {5, 13, . . .}
 [6]: {6, 14, . . .}
 [7]: {7, 15, . . .}
2011-2012-3 Anung Ariwibowo 13

14.  a common problem in number theory
 GCD (a,b) of a and b is the largest number that
divides evenly into both a and b
 eg GCD(60,24) = 12
 often want no common factors (except 1) and
hence numbers are relatively prime
 eg GCD(8,15) = 1
 hence 8 & 15 are relatively prime

15.  an efficient way to find the GCD(a,b)
 memanfaatkan teorema:
 GCD(a,b) = GCD(b, a mod b)
 Versi iteratif
EUCLID(a,b)
1. A = a; B = b
2. if B = 0 return A
3. R = A mod B
4. A = B
5. B = R
6. goto 2

16.  Versi rekursif
 Kuis(a,b)
 if b = 0 then return a
 else Kuis(b, a mod b)
2011-2012-3 Anung Ariwibowo 16

17. 1970 = 1 x 1066 + 904 gcd(1066, 904)
1066 = 1 x 904 + 162 gcd(904, 162)
904 = 5 x 162 + 94 gcd(162, 94)
162 = 1 x 94 + 68 gcd(94, 68)
94 = 1 x 68 + 26 gcd(68, 26)
68 = 2 x 26 + 16 gcd(26, 16)
26 = 1 x 16 + 10 gcd(16, 10)
16 = 1 x 10 + 6 gcd(10, 6)
10 = 1 x 6 + 4 gcd(6, 4)
6 = 1 x 4 + 2 gcd(4, 2)
4 = 2 x 2 + 0 gcd(2, 0)

18.  Sebuah integer x yang memenuhi
 a-1 ≡ x mod n
 ax ≡ aa-1 mod n
 ax ≡ 1 mod n
 hanya ada jika a dan n coprime
 Contoh
 3-1 ≡ x mod 11
 3x ≡ 1 mod 11
 x=4
 0 ≤ x < 11
2011-2012-3 Anung Ariwibowo 18

19.  Modular Multiplicative Inverse
http://en.wikipedia.org/wiki/Modular_multiplicative
_inverse
 Stallings, "Cryptography and Network
Security"http://williamstallings.com/Cryptography/
 Schneier, "Applied Cryptography"
http://www.schneier.com/book-applied.html
 Thomas L Noack,
http://ece.uprm.edu/~noack/crypto/
 Slides
tjerdastangkas.blogspot.com/search/label/ikh323
2011-2012-3 Anung Ariwibowo 19