Systems of Equations as Matrices and Hill Cipher.

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Presentation transcript:

Systems of Equations as Matrices and Hill Cipher. Annela Kelly Bridgewater State University

Matrix Algebra Algebra ax=b 5x=3 x= 3 5 = 5 −1 3 Ax=b 5 0 2 1 𝑥 1 𝑥 2 = 3 4 𝑥 1 𝑥 2 = 5 0 2 1 −1 3 4 What is 5 0 2 1 −1 ? Matrix multiplication review applet at: https://www.khanacademy.org/math/algebra2/alg2-matrices/matrix-multiplication-alg2/e/multiplying_a_matrix_by_a_matrix or http://www.mathsisfun.com/algebra/matrix-multiplying.html

Matrix inverse formula 𝐴 𝐴 −1 = 1 0 0 1 Matrix inverse for 2× 2matrix: EXAMPLE: To get more details and in-depth discussion about inverses: http://www.mathsisfun.com/algebra/matrix-inverse.html

Cryptology Caesar Cipher (100 BC)

Hill cipher As time progressed, the study of cryptography began to involve higher level mathematics. With this more advanced math came more advanced ciphers based on the idea of encryption and decryption keys. Encryption keys are a special value or set of values used in an encryption algorithm to convert a plaintext into a cipher text. A decryption key is the opposite. One encryption scheme that utilizes more advanced mathematics, as well as encryption and decryption keys is a cipher from 1929 called the Hill cipher. The Hill cipher is based on matrix multiplication and is a lot more secure than the Caesar cipher that was previously discussed.

Numbers into letters Example: BED 1 4 3

Modular Calculations What if a number is bigger than 26 or smaller than 0? Use “clock arithmetic”: 12 ≡ 12 27 ≡ 1 -1 ≡ 25 53 ≡ 1 Worksheet on clock arithmetic!

(Matrix) inverses formula modulo 26 Algebra 5 ∙6=30 5 ∙21=105 5∙ 1 5 =1 i.e. 5 −1 = 1 5 Modulo 26 Algebra 5 ∙6 ≡ 4 5 ∙21 ≡ 1 5 −1 ≡ 21 Worksheet on inverses mod 26!

Encoding in Hill Cipher Convert letters into numbers Write message into blocks (matrices) of two Multiply decoding matrix A with the vectors Convert numbers into letters

Decoding in Hill Cipher Convert numbers into letters: Multiply decoding matrix 𝐴 −1 with the vectors: Convert numbers into letters Worksheet on encoding and decoding!

Exchanging secrets MESSAGE: CALCULUS CODE: EGUPDAWC -1 2 −1 3 4 CODE: EGUPDAWC -1 DECODED MESSAGE: CALCULUS More info on Hill Ciphers at: http://www.unc.edu/~marzuola/Math547_S13/Math547_S13_Projects/R_Doyle_Section001_Cryptography.pdf