ikh323-06

Enkripsi Kunci Simetrik

 Matrix arithmetic modulo 26  Menyamarkan distribusi frekuensi (diffusion)  Substitusi n simbol  Perkalian matriks n × n  Encipher menggunakan matriks K  Decipher menggunakan matriks K-12011-2012-3 Anung Ariwibowo 2

 Plaintext ACT CAT  [0 2 19 2 0 19]  Key GYBNQKURP  [ [6 24 1] [13 16 10] [20 17 15] ]  Ciphertext  POH FIN  [15 14 17 5 8 13]2011-2012-3 Anung Ariwibowo 3

 Matrix arithmetic modulo 26  Determinan matriks  Adjoin matriks  Invers matriks2011-2012-3 Anung Ariwibowo 4

 A = [ [5 8] [17 3] ]  inv(A) mod 26 = [ [9 2] [1 15] ]  A × inv(A) = [ [53 130] [156 79] ] mod 26  = [ [1 0] [0 1] ]2011-2012-3 Anung Ariwibowo 5

 Matriks 2 × 2  A = [ [a b] [c d] ]  |A| = ad – bc  Matrix 3 × 3  A = [ [a11 a12 a13] [a21 a22 a23] [a31 a32 a33] ]  |A| = a11 × C11 + a12 × C12 + a13 × C132011-2012-3 Anung Ariwibowo 6

 Cij  Cofactor matriks A dengan menghapus baris i dan kolom j  Untuk matriks bujursangkar n  A = [ [a11 a12 . . a1n] [a21 a22 . . a2n] . . [an1 an2 . . ann] ]  |A| = Σ a1j × C1j, 1 ≤ j ≤ n2011-2012-3 Anung Ariwibowo 7

 |A| = Σ a1j × C1j, 1 ≤ j ≤ n  Indeks i = 1 dapat diganti dengan indeks baris- baris yang lain  1≤i≤n  Dapat dibuktikan hasil determinannya sama  Strategi: Cari baris yang paling banyak mengandung nilai nol2011-2012-3 Anung Ariwibowo 8

 Gaussian Elimination  Matriks yang diperluas (augmented matriks)  Operasi Elementer  Matriks Cofactor dan aturan Cramer  Minor matriks  Cofactor matriks  Determinan matriks2011-2012-3 Anung Ariwibowo 9

 K = [ [17 17 5] [21 18 21] [2 2 19] ]  Minor matriks Mij  Submatriks yang didapat dengan menghapus baris i dan kolom j  Hitung determinan dari submatriks tersebut  Baris 1  M11 = 18 × 19 – 21 × 2  M12 = 21 × 19 – 21 × 2  M13 = 21 × 2 – 18 × 22011-2012-3 Anung Ariwibowo 10

 Baris 2  M21 = 17 × 19 – 5 × 2  M22 = 17 × 19 – 5 × 2  M23 = 17 × 2 – 17 × 2  Baris 3  M31 = 17 × 21 – 5 × 18  M32 = 17 × 21 – 5 × 21  M33 = 17 × 18 – 17 × 212011-2012-3 Anung Ariwibowo 11

 Cofactor Cij  (-1)(i+j) Mij  Baris 1  C11 = (-1)(1+1) × 18 × 19 – 21 × 2  C12 = (-1)(1+2) × 21 × 19 – 21 × 2  C13 = (-1)(1+3) × 21 × 2 – 18 × 2  Baris 2  C21 = (-1)(2+1) × 17 × 19 – 5 × 2  C22 = (-1)(2+2) × 17 × 19 – 5 × 2  C23 = (-1)(2+3) × 17 × 2 – 17 × 22011-2012-3 Anung Ariwibowo 12

 Baris 3  C31 = (-1)(3+1) × 17 × 21 – 5 × 18  C32 = (-1)(3+2) × 17 × 21 – 5 × 21  C33 = (-1)(3+3) × 17 × 18 – 17 × 212011-2012-3 Anung Ariwibowo 13

 Matriks yang elemen-elemennya adalah cofactor dari matriks asal  C = [ [C11 C12 C13] [C21 C22 C23] [C31 C32 C33] ]2011-2012-3 Anung Ariwibowo 14

 |A| = Σ a1j × C1j, 1 ≤ j ≤ n  Ekspansi cofactor sepanjang baris 1  |A| = Σ ai2 × Ci2, 1 ≤ i ≤ n  Ekspansi cofactor sepanjang kolom 2  Hasilnya pasti sama  Untuk mencari determinan, gunakan ekspansi cofactor pada kolom/baris yang paling banyak mengandung nilai nol2011-2012-3 Anung Ariwibowo 15

 Invers sebuah matriks didapat dengan mengalikan invers determinan dengan transpos matriks Cofactor  A-1 = (1/|A|) × CT2011-2012-3 Anung Ariwibowo 16

 Tugas Mandiri tentang  Number Theory  Matriks  Primality testing  UAS  Alat hitung  Substitution, Transposition, Number Theory, Public key2011-2012-3 Anung Ariwibowo 17

 http://en.wikipedia.org/wiki/Hill_cipher  http://en.wikipedia.org/wiki/Modular_multiplicative_inverse  http://en.wikipedia.org/wiki/Cofactor_(linear_algebra)  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 18

ikh323-06

by Anung Ariwibowo on Feb 13, 2012

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