Number Theory dan Modulo  Multiplicative Inverse
     Kriptografi dan Sistem Keamanan Komputer      Standar Sistem Keamanan      Enkripsi Kunci Simetrik      Number Th...
   Himpunan bilangan   Operasi yang ditentukan terhadap elemen-    elemen himpunan tersebut   Hasil operasi berada di d...
   exponentiation       repeated application of operator       a3 = a.a.a   Elemen identitas       e=a0   Cyclic gro...
   a set of “numbers”   with two operations (addition and    multiplication) which form:   an abelian group with additi...
   modulo operator “a mod n” didefinisikan    sebagai sisa bagi jika a dibagi n   congruence: a ≡ b mod n       jika di...
     a mod n             a≥0               a = qn + b               q≥0               b≥0             a<0           ...
     a mod n             0≤a<1               a = qn + b               q≥0               b≥0      Modular Multiplicat...
   Sebuah bilangan bukan-nol b membagi a jika    untuk sebuah bilangan m didapatkan a=mb    (a,b,m adalah integers)   d....
   clock arithmetic   menggunakan sejumlah berhingga bilangan   loops back from either end   modular arithmetic is whe...
   bisa diterapkan kepada sembarang group of    integers       Zn = {0, 1, … , n-1}   Membentuk commutative ring for ad...
+ 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 4 5 6 7 02 2 3 4 5 6 7 0 13 3 4 5 6 7 0 1 24 4 5 6 7 0 1 2 35 5 6 7 0 1 2 3 46 6...
     [0]: {0, 8, 16, 24, . . .}      [1]: {1, 9, 17, . . .}      [2]: {2, 10, . . .}      [3]: {3, 11, . . .}      [4...
   a common problem in number theory   GCD (a,b) of a and b is the largest number that    divides evenly into both a and...
   an efficient way to find the GCD(a,b)   memanfaatkan teorema:       GCD(a,b) = GCD(b, a mod b)   Versi iteratif    ...
     Versi rekursif         Kuis(a,b)         if b = 0 then return a         else Kuis(b, a mod b)2011-2012-3         ...
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 9...
     Sebuah integer x yang memenuhi         a-1 ≡ x mod n         ax ≡ aa-1 mod n         ax ≡ 1 mod n      hanya ada...
     Modular Multiplicative Inverse       http://en.wikipedia.org/wiki/Modular_multiplicative       _inverse      Stalli...
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ikh323-05

  1. 1. Number Theory dan Modulo Multiplicative Inverse
  2. 2.  Kriptografi dan Sistem Keamanan Komputer  Standar Sistem Keamanan  Enkripsi Kunci Simetrik  Number Theory  Enkripsi Kunci Publik  Autentikasi Pesan dan Fungsi Hash2011-2012-3 Anung Ariwibowo 2
  3. 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. 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. 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. 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. 7.  a mod n  a≥0  a = qn + b  q≥0  b≥0  a<0  overshoot  a = qn + b  q<0  b≥02011-2012-3 Anung Ariwibowo 7
  8. 8.  a mod n  0≤a<1  a = qn + b  q≥0  b≥0  Modular Multiplicative Inverse  aa-1 = 12011-2012-3 Anung Ariwibowo 8
  9. 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. 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. 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. 12. + 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 4 5 6 7 02 2 3 4 5 6 7 0 13 3 4 5 6 7 0 1 24 4 5 6 7 0 1 2 35 5 6 7 0 1 2 3 46 6 7 0 1 2 3 4 57 7 0 1 2 3 4 5 6
  13. 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. 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. 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. 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. 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. 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 < 112011-2012-3 Anung Ariwibowo 18
  19. 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/ikh3232011-2012-3 Anung Ariwibowo 19

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