yptosyste ciphertext represent more than one plaintext letter. To give an example of this type of cryptosys- tem, called a polyalphabetic cryptosystem, we will generalize affine codes by using matrices. The idea works roughly the same as before; however, in- stead of encrypting one letter at a time we will encrypt pairs of letters (as before, letters are represented by elements of Z26). We can store a pair of letters ni and ng in a vector Let A be a 2 x 2 invertible matrix with entries in Z26. We can define an encoding function by f(n) = (A © n) ® b, where b is a fixed column vector and matrix operations are performed in Z26- The formula for the decoding function (which is the inverse of the encoding function) is very similar to the decoding function formula that we found for affine encoding: f(m) = (A- © m) e (A-ª © b), where A- is the matriz inverse of A: that is, A-'A = AA¯' = I, where I is the 2 x 2 identity matrix. *Note* that in these formulas, we are using modular matrix multiplication instead of regular matrix multiplication: that is, the regular - and + operations are replaced by © and e: Exercise 9.2.20. Perform the following operations using modular matrix multiplication (mod 26): (e) () 20 (0) ( ; :)(:) ( ")(;) 5 6 12 4 1 (c) 7 8 13 1 (Ъ) 13 13 2 2 13 16 (d) 2 2 13 13
yptosyste ciphertext represent more than one plaintext letter. To give an example of this type of cryptosys- tem, called a polyalphabetic cryptosystem, we will generalize affine codes by using matrices. The idea works roughly the same as before; however, in- stead of encrypting one letter at a time we will encrypt pairs of letters (as before, letters are represented by elements of Z26). We can store a pair of letters ni and ng in a vector Let A be a 2 x 2 invertible matrix with entries in Z26. We can define an encoding function by f(n) = (A © n) ® b, where b is a fixed column vector and matrix operations are performed in Z26- The formula for the decoding function (which is the inverse of the encoding function) is very similar to the decoding function formula that we found for affine encoding: f(m) = (A- © m) e (A-ª © b), where A- is the matriz inverse of A: that is, A-'A = AA¯' = I, where I is the 2 x 2 identity matrix. *Note* that in these formulas, we are using modular matrix multiplication instead of regular matrix multiplication: that is, the regular - and + operations are replaced by © and e: Exercise 9.2.20. Perform the following operations using modular matrix multiplication (mod 26): (e) () 20 (0) ( ; :)(:) ( ")(;) 5 6 12 4 1 (c) 7 8 13 1 (Ъ) 13 13 2 2 13 16 (d) 2 2 13 13
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do part A and C and please show step by step and explain
Transcribed Image Text:9.2.4 Polyalphabetic codes
A cryptosystem would be more secure if a ciphertext letter could represent
more than one plaintext letter. To give an example of this type of cryptosys-
tem, called a polyalphabetic cryptosystem, we will generalize affine codes
by using matrices. The idea works roughly the same as before; however, in-
stead of encrypting one letter at a time we will encrypt pairs of letters (as
before, letters are represented by elements of Z26). We can store a pair of
letters ni and na in a vector
--)
()-
n =
n2
Let A be a 2 x 2 invertible matrix with entries in Z26. We can define an
encoding function by
f(n) = (A © n) e b,
where b is a fixed column vector and matrix operations are performed in
Z26. The formula for the decoding function (which is the inverse of the
encoding function) is very similar to the decoding function formula that we
found for affine encoding:
(m) = (A- ©m) e (A-o b),
where A-1 is the matrix inverse of A: that is, A-1A = AA-1 = I, where
I is the 2 x 2 identity matrix. *Note* that in these formulas, we are using
modular matrix multiplication instead of regular matrix multiplication: that
is, the regular - and + operations are replaced by o and e:
Exercise 9.2.20. Perform the following operations using modular matrix
multiplication (mod 26):
(0 (; :)(:)
:)()
5 6
7 8
12 4
1
(0) (
(c)
13 5
20 20
1.
13
13
2
13
(b) (
(d) (
16
2
13
13
2
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