This document discusses symmetric key encryption techniques using matrix arithmetic modulo 26. It explains how a plaintext is encrypted using a key matrix through matrix multiplication and diffusion techniques. It then discusses how to calculate the inverse of a matrix using determinants, minors, cofactors and Cramer's rule in order to decrypt the ciphertext. Examples are provided to illustrate matrix multiplication, inverses, determinants and the encryption/decryption process.

1. Enkripsi Kunci Simetrik

2.  Matrix arithmetic modulo 26
 Menyamarkan distribusi frekuensi (diffusion)
 Substitusi n simbol
 Perkalian matriks n × n
 Encipher menggunakan matriks K
 Decipher menggunakan matriks K-1
2011-2012-3 Anung Ariwibowo 2

3.  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

4.  Matrix arithmetic modulo 26
 Determinan matriks
 Adjoin matriks
 Invers matriks
2011-2012-3 Anung Ariwibowo 4

5.  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

6.  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 × C13
2011-2012-3 Anung Ariwibowo 6

7.  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 ≤ n
2011-2012-3 Anung Ariwibowo 7

8.  |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 nol
2011-2012-3 Anung Ariwibowo 8

9.  Gaussian Elimination
 Matriks yang diperluas (augmented matriks)
 Operasi Elementer
 Matriks Cofactor dan aturan Cramer
 Minor matriks
 Cofactor matriks
 Determinan matriks
2011-2012-3 Anung Ariwibowo 9

10.  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 × 2
2011-2012-3 Anung Ariwibowo 10

11.  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 × 21
2011-2012-3 Anung Ariwibowo 11

12.  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 × 2
2011-2012-3 Anung Ariwibowo 12

13.  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 × 21
2011-2012-3 Anung Ariwibowo 13

14.  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

15.  |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 nol
2011-2012-3 Anung Ariwibowo 15

16.  Invers sebuah matriks didapat dengan
mengalikan invers determinan dengan
transpos matriks Cofactor
 A-1 = (1/|A|) × CT
2011-2012-3 Anung Ariwibowo 16

17.  Tugas Mandiri tentang
 Number Theory
 Matriks
 Primality testing
 UAS
 Alat hitung
 Substitution, Transposition, Number Theory, Public
key
2011-2012-3 Anung Ariwibowo 17

18.  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/ikh323
2011-2012-3 Anung Ariwibowo 18