Polygraphic system called Hill Cipher

Introduction

We live in a world which is changing rapidly, in the past messages were transmitted via envelopes to ensure the privacy. Now that things have changed messages are now transmitted via the internet, whether its by email or other social networks. The question which is asked the most, is the information transmitted from object A to B safe or secure from third parties. The field that deals with the study of transmitting information in a private manner is known as Cryptography.

According to Anton and Rorres (2013:652) cryptography is the study of writing and solving secret messages.

Even though people want to keep their messages as secret or private, the is always someone else who want to know about the hidden message. The study that deals with breaking the ciphertext is known as cryptanalysis. Even though cryptanalysis can show the vulnerability of a cipher, it can also help to improve the cipher.

This paper discusses a polygraphic system called Hill Cipher.

“A polygraphic system, is a system of cryptography in which the plaintext is divided into sets of n letters, each of which is replaced by a set of n cipher letters” (Anton & Rorres, 2013:653).

Background of Cryptography

The study of cryptography is believed to have existed long time ago, where human beings felt the need to communicate and share information and to communicate selectively.

Literature Review

Methodology

Figure SEQ Figure * ARABIC 1: Cryptanalysis attack

When enciphering or deciphering is performed, the 26 alphabetical letters must be represented by positive integers 0, 1, 2….26, starting with the letter A to Z.

Get quality help now

Writer Lyla

Verified

Proficient in: Communication

5 (876)

“ Have been using her for a while and please believe when I tell you, she never fail. Thanks Writer Lyla you are indeed awesome ”

+84 relevant experts are online

Hire writer

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Table SEQ Table * ARABIC 1: Numerical representation of 26-letter alphabet

Modular Arithmetic

According to Eisenberg (1999:4), we cannot understand Hill ciphers without first understanding modular arithmetic. Therefore, we will take some time and explain what modular arithmetic is and why hill ciphers need it. Since it is also possible to get values greater than 25 after matrix multiplication, whereas our last numerical value is 25. The modular arithmetic comes in handy in those situations, whereby the remainder of the number after dividing by 26 will be taken instead. The following few definitions about modular arithmetic are taken direct from Elementary Algebra book by Howard Anton and Chris Rorres.

Definition 1: “If m is a positive integer a and b are any integers, then we say that a is equivalent to b modulo m, written a = b (mod m) if a – b is an integer multiple of m” (Anton & Rorres:2013:655).

10156055630

Example 2:

38 = 12 (mod 26)

3 = 3 (mod 26)

Anton and Rorres states, “For any modulus m it can be proved that every integer a is equivalent, modulo m, to exactly one of the integers 0, 1, 2, …, m – 1.” This integer is known as the residues of modulo m, and it presented as follows Zm = { 0, 1, 2,…, m – 1} (Anton & Rorres, 2013:655).

References

Eisenberg, M. 1999. Hill Ciphers and Modular Linear Algebra. [06.02.2019].

Anton, H. & Rorres, C. 2013. Elementary Linear Algebra: Applications Version, 11th Edition. Hoboken: Wiley Global Education.