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Information and Communication Technologies

Arab Open University - AOU

T209Information and Communication

Technologies: People and Interactions

Fourth Session

1 Prepared by: Eng. Ali H. ElayweRevised by: Dr. Hassan SALTI

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Reference Material

This session is based on the following references: Module 5: Security, Book S: Security

Module 5: Security, Book N: Numeracy Skills

Module 5: Security, Book E: Experiments Module 5: Security, (Text Book) Monograph: Security

Techniques in Digital Systems

More references: http://www.cacr.math.uwaterloo.ca/hac/

http://en.wikipedia.org/wiki/Cryptography

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http://www.cacr.math.uwaterloo.ca/hac/http://en.wikipedia.org/wiki/Cryptographyhttp://en.wikipedia.org/wiki/Cryptographyhttp://www.cacr.math.uwaterloo.ca/hac/

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Topics to be covered (1/2)

Part 2 (Encryption) of Book S 1. (S.2.3.1) Encryption using modular arithmetic

Part 3 (Modular arithmetic) of Book N 2. (N.3.1) Introduction

3. (N.3.2) Modular addition

(N.3.2.1) Performing addition in modular arithmetic

(N.3.2.2) The properties of modular addition (N.3.2.3) Summary of section 3.2

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Chapter 2 (Encryption) of Book M(Monograph) 4. (M.2.1) Mathematics

(M.2.1.1) The Caesar code

(M.2.1.2) A numerical version

(M.2.1.3) A mathematicians view

(M.2.1.4) Using a key

(M.2.1.5) Decrypting the Caesar code

(M.2.1.6) The properties of a Group

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Topics to be covered (2/2)

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You should read the following from Books N, E and theMonograph:

1. Book N: Numeracy:

Work through all ofSection 3.1 Introduction and Section 3.2Modular addition in Book N

2. Book E: Experiments:

In Book E work through Part 3 up to the end of Section 3.4.These sections introduce you to some software tools that will

assist your study ofmodular arithmetic. One of these tools (theModular Powers Checker) wont actually be used until later, soskim through it

Topic 1: (S.2.3.1) Encryption usingmodular arithmetic (1/2)

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3. Monograph:

Read from the start ofChapter 2 Encryption, to the end ofSection 2.1 Mathematics. In this reading Monk introduces thepotential ofmodular arithmetic for deriving an algorithm for asimple encryption scheme

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Topic 1: (S.2.3.1) Encryption usingmodular arithmetic (2/2)

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Other names for modular arithmetic:

It is also referred to as modulo arithmetic, clock arithmeticorremainder arithmetic. It usually involves concept of full

rotation (circular) as will be clear later on Set:

A set is a collection of objects

Modulus:

The size (that is, the number of, say, integers a set contains) isknown as the modulus

Topic 2: (N.3.1) Introduction to themodular arithmetic (1/2)

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Modulo n:

Modulus ofn means that the members of the set are the first (n)integers starting from zero {0, 1, 2, 3, , (n-1)}

When we manipulate a set of this kind then we are working inmodulon

Example:

When the set contains the integers 0, 1, 2, 3 and 4, the modulus is5 and we say we are working in modulo 5

In modulus 8, the integers are 0, 1, 2, 3, 4, 5, 6 and 7

The results of operations performed in modular arithmeticdisplay some interesting properties. You will look at operationsusing modular addition,modular multiplicationand modularexponentiation

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Topic 2: (N.3.1) Introduction to themodular arithmetic (2/2)

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Consider a simple addition of two integers: say 5 and 8. Inconventional arithmeticwe could use a number linedemonstrated in Figure 1. We would start at 5, count 8 places

along to the right and read off the result (13)

Topic 3 : (N.3.2) Modular additionSub-Topic 3.1: (N.3.2.1) Performing addition in

modular arithmetic (1/9)

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Figure 1 Simple addition in conventional arithmetic

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Figure 2Addition in modulo 9

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Imagine we are working in modulo 9. Here the highest numberwe can use is an eight, so the conventional method of using anumber line isnt going to work for this calculation. One way ofdealing with this is to repeat the same limited set of integersalong the number line, as shown in Figure 2, and to use the samemethod of starting at 5 and counting forward 8 places

The result of adding 8 to 5 in modulo 9 arithmetic is 4

Sub-Topic 3.1: (N.3.2.1) Performing addition inmodular arithmetic (2/9)

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Circles for modular arithmetic (clock arithmetic):

Since in modular arithmetic the numbers on the numberline are repeated, its easier to just join the ends of the

line together into a circle and count places round thecircle instead

The circles for modulo 4 and modulo 6 are shown inFigure 3.

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Figure 3 Circles for modular arithmetic

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We can use these to work out addition in the same way as weused the number line, but we must always move in a

clockwise direction(I expect now you can see why modular

arithmetic is also referred to asclock arithmetic !!)

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Activity 19 (self-assessment) Using the clock faces in Figure 3 calculate the following:

(a) 3 + 9 in modulo 4

0 modulo 4 (b) 5 + 8 in modulo 6

1 modulo 6

(c) 4 + 11 in modulo 6

3 modulo 6

(d) 2 + 6 in modulo 40 modulo 4

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Residue Method (remainder arithmetic): The above could be a tedious method of calculation

especially when you need to go round the circle more than

once!

Another way of performing the same arithmetic is to add the

two integers together in the conventional way, divide the

result by the modulus and express the answer as the remainder,or residueas it is often known

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Example of Residue Method of calculation: Calculate 3 + 5 in modulo 6

Solution: First add the two integers together in the conventional way:

3 + 5 = 8 Then divide the result by 6:

8 6 = 1 remainder 2

Express the answer as the remainder (or residue)

The answer is 2

The conventional way of writing this calculation is: 3 + 5 2 mod 6

Note the use of the symbol which is read as is congruent

to rather than the symbol = which is read as is equal to

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I f i d C i i T h l i

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Congruence:

Two integers a and b are said to be congruentmodulo n if

(a mod n) = (b mod n)

If this is the case then the remainder is identical

when both a and b are divided by n. We write

this as:

a b mod n

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Activity 20 (self-assessment)

State which if any of the following pairs are congruentmodulo 7:

(a) 10, 3

10 7 = 1 remainder 3, 3 7 = 0 remainder 3so 10 and 3 are congruent modulo 7

(b) 12, 512 7 = 1 remainder 5, 5 7 = 0 remainder 5

so 12 and 5 are congruent modulo 7

(c) 14, 6

14 7 = 2 remainder 0, 6 7 = 0 remainder 6so 14 and 6 are not congruent modulo 7

(d) 26, 1226 7 = 3 remainder 5, 12 7 = 1 remainder 5

so 26 and 12 are congruent modulo 7

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Addition Tables: Figure 4shows a set of addition tables for modulus 4, 5, 6 and 7.

These tables give the result of adding together any two

numbers less than the modulus To use the tables, find the first number in the top horizontal

number line, and the second number in the left vertical number

line. The result of modular addition is in the matrix where the

column and row of the numbers intersect

Sub-Topic 3.2: (N.3.2.2) The properties ofmodular addition (1/16)

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Figure 4Addition tables for modulo 4, 5, 6 and 7

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Activity 21 (self-Assessment) Using the modular addition tables in Figure 4, find the results

of the following:

(a) 5 + 3 mod 75 + 3 mod 7 1 mod 7

(b) 1 + 3 mod 4

1 + 3 mod 4 0 mod 4

(c) 3 + 2 mod 6

3 + 2 mod 6 5 mod 6

(d) 4 + 4 mod 5

4 + 4 mod 5 3 mod 5

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I f ti d C i ti T h l gi

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1- Concept of Group (Group Theory): A set and a particular operation are together called a group if they

possess certain properties

One such property is that ofclosure; another is the property of

identity

2- Closure Property: In modular arithmetic the result of any operation involving

members of a group must give a result that is a member of thesame set (the result should not be some number outside the set).

This is known as closure

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Activity 22 (exploratory) Look again at the tables in Figure 4 to satisfy yourself that

they do indeed demonstrate the property of closure

The members of each set modulo n contain all the integers from 0

to (n 1). You will see from the tables in Figure 4 that combiningany members of the same set by modular arithmetic produces a

result that is itself also a member of the set. The tables therefore

indicate that modular addition has this property ofclosure

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Figure 4


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3- Identity element:

Where there is one element (which we will call e) of a setwhich, when combined mathematically with any other

element (which we shall call a), returns a result ofa,then the element e is known as the identity of the group

This can be expressed generally as:

a ea mod n or e aa mod n

where the symbol denotes any mathematical operation

In the case of is the modular addition, the identityelement will always be e=0

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4- The additive inverse property: For any element (which we will call a) in the group, there is

another element (which we will call ) in the group which,when combined by addition, returns a result equal to theidentity of the group

This can be expressed mathematically as: a + e or + a e

is the additive inverseof a

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Activity 23 (exploratory) Look again at the modulo 7 table in Figure 4. Are there any

two members in this set which, when combined underaddition, result in the identity (0) for modular arithmetic?

Yes. In fact, as well as the obvious 0 + 0, there are three pairs ofnumbers that would result in the identity. These are: 1 + 6; 2 +5; and 3 + 4

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Figure 4



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Activity 24 (self-assessment) What pairs of numbers (other than 0 + 0) form additive

inverses with each other:

(a) in modulo 4 arithmetic?

1 and 3; 2 and 2 (b) in modulo 5 arithmetic?

1 and 4; 2 and 3

(c) in modulo 6 arithmetic?

1 and 5; 2 and 4; 3 and 3

(d) in modulo 10 arithmetic?1 and 9; 2 and 8; 3 and 7; 4 and 6; 5 and 5

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Note that any two members of a set, which together add up to themodulus, form an additive inverse pair!


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Finding solutions: In a generalized form we say that find x given a and b such that:

a + x b mod n

where a and b can be any element within the group

Single solution (In modular addition): In fact, any equation of the form a + xb mod n produces a

single solution for x

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Activity 25 (self-assessment) Using the modular addition tables in Figure 4, find solutions for

x for the following:

(a) 2 + x 4 mod 5

x = 2

(b) 3 + x 5 mod 6

x = 2

(c) 6 + x 4 mod 7

x = 5

(d) 1 + x 3 mod 4 x = 2

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Figure 4


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5- The associative property:

If, when two or moreoperations are carried out in modulararithmetic, the order in which the operations are performed

does not affect the result, this is known as the associativeproperty


(a b) c mod n a (b c) mod n where the symbol denotes any mathematical operation

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Example: Imagine we want to perform the calculation 2 + 3 + 4 in modulo 5: it

doesnt matter in which order I add the elements, the result willalways be the same

First well try (2 + 3) + 4 mod 5: 2 + 3 mod 5 0 mod 5

0 + 4 mod 5 4 mod 5

Next well try 2 + (3 + 4) mod 5:

3 + 4 mod 5 2 mod 5

2 + 2 mod 5 4 mod 5

The result of the calculation is independent of the order inwhich the operations are performed

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6- The commutative property:

If, when an operation is carried out in modular arithmetic, the

order in which the integers are placed does not affect theresult, this is known as the commutative property


(a b) mod n (b a) mod n

where the symbol denotes any mathematical operation

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Conventional Maths: Addition is commutative but subtraction is not

4 3 3 - 4

Multiplication is commutative, so

43 = 34

but division is not commutative, so

4 3 3 4

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Note: (Activity 26) When working with a modulus ofn you should discover that

taking a number y and adding it to another number a gives youa result that is y + a mod n. Later you can add , the additive

inverse ofa, and you could predict the result would be y + a + y + 0 y mod n

However, with these three numbers it does not matter in whatorder you perform the addition: you always get the result y.For example, given a+y+ y+a+ mod n, the property of

commutativity allows us to swap a and y without changing theresult, so

a + y + y + a + mod n

which we know gives the result y

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Definition of congruence:

Two integers a and b are said to be congruentmodulo n if (amod n) = (b mod n). This is written as a b mod n

Working in modular addition:

Any operation carried out on the group results in closure

The group has one element, the identitye, such that a + e a,or e + a a. For modular addition the identity e is 0.

Every element a in the group has an additive inverse such that

a + e mod n = 0 mod n Modular addition is associative: (a + b) + c mod n a + (b + c)

mod n

Modular addition is commutative: a + b mod n b + a mod n

Sub-Topic 3.3: (N.3.2.3) Summary ofSection (N.3.2)

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Encryption methods are thoroughly analyzed by long-established mathematical processes

Mathematicians not only produce results that areuseful for code makers they also have ways of

identifying mathematical problems that are veryhard to do

The task of coding data can then be reformulated tobe a way of processing the data so that a potential

cracker is faced with a mathematical problem that isvery hard to solve

Topic 4: (M.2.1) Mathematics

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In Table 1 the alphabet in the bottom row has been movedup three places; the alphabet in the lower row thereforestarts with the letter D and continues in alphabetical orderuntil the letter Z is written under the letter W

Sub-Topic 4.1: (M.2.1.1) The Caesar code(1/3)

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Table 1 An encryption table for the Caesar code

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

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


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o at o a d Co u cat o ec olog es

Table 2 An encryption table for the Caesar code highlighting the

encrypted letters C, A and T

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Example:To encrypt a message like the sequence of lettersCAT, each letter of the message is found in the upper row andthen the corresponding letter of the lower row is written down.The C in CAT, for instance, corresponds to the letter F in thelower row as highlighted in the copy of the encryption table inTable 2. CAT is hence encrypted as FDW.



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


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g

Table 3 A decryption table for the Caesar code of Table 2. Theletters F, D and W have been picked out (CAT is decrypted from

FDW)

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

Example:To decrypt a message, a decryption table should bedrawn (Table 3). Each letter of the encrypted message is found inthe upper row and then the corresponding decrypted letter of thelower row is written down


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g

The message CAT would then be coded as thenumerical sequence 2 0 19 simply by substituting thenumbers that correspond to the letters in Table 4. Thisoriginal message in its alphabetic or numeric form is

often called the plaintext Table 5 provides an operation that adds an element of

confusion; it is a version ofTable 2 but with the lettersre-coded as numbers

Table 5 is chosen to generate an unconventional re-coding of the alphabet. It is a re-coding designed toconfuse, and is referred to as encryption (orenciphering)

Sub-Topic 4.2: (M.2.1.2) A numericalversion (1/3)

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Table 4 A possible numerical coding scheme for the alphabet

Table 5 An encryption scheme for numerical codes

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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 Z0 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

0 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

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2

Encryption example:The plaintext word CAT represented by the

numerical sequence 2 0 19 is encrypted using Table 5 first by taking thenumber 2 in the numerical sequence of the message and noting that a 2 in theupper row corresponds to the number 5 in the lower row of the table and so on.The whole encrypted sequence is 5 3 22


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Using a decryption table (Table 6) in exactly the sameway as Table 5 was used, the numerical version of theciphertext 5 3 22 translates to the sequence 2 0

19, which is the numerical version of the originalplaintext CAT (using Table 4)

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0 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

23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Table 6 A decryption table for the encryption operation shown

in Table 5


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g

Group Theory: It works with a fixed number of things, like the letters in the

alphabet or a limited collection of numbers

A collection of whole numbers, the integers, that we areworking with would be denoted by Z and to indicate that weare dealing with twenty six numbers from 0 to 25 we can usethe suffix 26 and write Z26

A Group, in Group Theory, has a number of distinctiveproperties

One of the properties requires that the results of operationsthat can be performed on a Group always end up being in theGroup. These kinds of operations, it is said, guarantee closure

Sub-Topic 4.3: (M.2.1.3) A mathematiciansview (1/3)

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The operation described by Table 5, for instance, guaranteesclosure since it can only be applied to integers taken from Z26 andthe results are all in Z26

To satisfy the closure property of a Group, mathematicians adopta special way of performing addition called modulo 26 addition(modulo n in general)

For the Z26 that we are dealing with, numbers are added in theusual way and:

1. If the result is between 0 and 25 then the result stands

2. If the result is 26 or over then 26 is subtracted

3. If the result is still 26 or more then 26 is subtracted again and thisis repeated until the result is in Z26, in other words the result is anumber from 0 to 25

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The Caesar code using Modulo n Maths (instead oftables and graphs etc): The encryption operation in Table 5 can now be summed up in a

formula:

c p + 3 mod 26 where p is a numerically encoded plaintext letter and c is the

corresponding numerically encoded ciphertext

The 3 in the above formula represents the shift of 3associated with Caesar code

The encrypted version of the plaintext word CAT, which isnumerically coded as 2 0 19, is:

2 + 3 5 mod 26, 0 + 3 3 mod 26, 19 + 3 22 mod 26

giving the sequence5 3 22, which can be interpreted usingTable 4 as the cryptogramFDW

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The secrecy in the Caesar code is linked to thealphabet shift i.e. 3and it is generally called a Key

Specifying the encryption operation in terms of

modulo 26 arithmetic makes it unnecessary toconstruct the encryption table

Using modulo 26 arithmetic encryption involvesperforming the calculation implied by the formula:

c p + Kmod 26

Where, in the examples so far, we have replaced Kby3

Sub-Topic 4.4: (M.2.1.4) Using a key(1/3)

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General value for K: Changing the value that we substitute for K is equivalent to

shifting the alphabet up by different amounts. Thus shifting thealphabet by nineteen places implies setting Kto 19 so

encryption operation becomes: c p + 19 mod 26

The plaintext word CAT is still represented by the sequence2 0 19 but its encrypted version is different. Using the rulesof modulo 26 addition the encrypted result is given by:

2 + 19 21 mod 26, 0 + 19 19 mod 26,

19 + 19 38 38 26 12 mod 26

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The outcome is the sequence 211912, which when translatedback into letters using Table 4gives the ciphertext VTM

Clearly the value ofK affects the outcome of the encryptionprocess and provides the key to decryption

Without knowing the value of K, the key, eavesdroppers might findit difficult to work out what the ciphertext said

People interested in obfuscating their messages might use thisversion of the Caesar code but they would need to keep K secretto make it more difficult for others to interpret their ciphertext

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Sub-Topic 4.5: (M.2.1.5) Decrypting theCaesar code (1/8)

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Summary: Encrypting by using the Caesar code and shifting the alphabet

up by 3 is equivalent to using the formula:

c p + 3 mod 26

Decryptingmessages that have been encrypted using theCaesar code and shifting the alphabet three places isequivalent to using the formula:

p c + 23mod 26

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Example: The word CAT that was encrypted using a key ofK = 3 produced

the ciphertext5 3 22 and using the decryption key of 23 isdecrypted as:

5 + 23 28 2mod 26, 3 + 23 26 0 mod 26,22 + 23 45 19mod 26

Which gives the result 2 0 19 to spell the original plaintext word CAT

Decryption of the Caesar code, therefore, can be carried out usingthe same operation as encryption but using a different key

The decryption key however must be chosen so that it

complements the encryption key. In the case of the Caesar code,someone who is expected to decrypt the message either needs toknow the decryption key or to know how to perform the decryptionoperation given the encryption key

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Symmetric and Asymmetric Cipher:( Important )1- Schemes where the sender and the receiver both work fromtheir knowledge of the encryption key are called symmetric

2- Schemes where the sender knows an encryption key and thereceiver knows the complementary decryption key are said tobe asymmetric

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There are five principal characteristics of a Group:

Firstly, a Group consists of a number ofelements and anoperation that can be performed on those elements ( e.g.,

modulo addition) Secondly, the closure property

Thirdly, the identity property

Fourthly, the existence of the inverse operation

Fifthly, the associative property

End of section 2.1 from the Monograph studymaterial

Sub-Topic 4.6: (M.2.1.6) The properties of aGroup

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Continue reading about Module 5:

1- Part 2 (Encryption) of Book S

(S.2.3.2) Breaking a code

2- Chapter 2 (Encryption) of Book M (Monograph)

(M.2.2) Working with codes

The due date ofTMA01 is 10 May

Topic 5: Preparation for next session

Documents

T209B_S4 lecture