Discrete Maths 2003 Lecture 36 3 Slides Pp

8/13/2019 Discrete Maths 2003 Lecture 36 3 Slides Pp

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Lecture 36, 17-October-200Discrete Mathematics 2003

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Introduction to Cryptography This work is based loosely on Chapter 12

(Number Theory) of the text

Security is very important in IT, and a key aspect

of this isprivacy (or confidentiality) e.g. incredit card transactions on the internet

We want secure communication, where only thesender & receiver of a message can understand it

This is achieved by encrypting (or enciphering)the message at the sender site, & decrypting (ordeciphering) it at the receiver site

Hopefully, an intruder who obtains the encryptedmessage is unable to understand its contents

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Caesars Cipher A very early example of encryption is due to

Julius Caesar (10044 BC)

We call data that is not encryptedplaintext(orcleartext), while encrypted data is ciphertext

Caesar encrypted his messages using the table

Plaintext 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 zCiphertext 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

Thus discrete is encrypted as GLVFUHWH, andSULYDFB is decrypted asprivacy

This cipher is termed a monoalphabetic cipher,

since each plaintext letter is always encrypted bythe same ciphertext letter

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Caesars Cipher (continued)

This example of Caesars cipher involves a shiftof each letter of the alphabet by 3 positions

There is nothing particularly significant about theuse of 3 we could have shifted each letter by 7

positions, or 19 positions, etc Altogether, there are 26 different ciphers that can

be constructed using Caesars approach(including the trivial cipher)

They can be described in the following way

Suppose our cipher involves a shift of each letterby spositions


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Caesars Cipher (continued) Identify a with 1, b with 2, c with 3, , x with

24, y with 25, and z with 0

To encipher a plaintext letter:

1. Encode the letter as its corresponding number

2. Add s to the number

3. If the result is 25, translate it back to a letter

4. If the result is 26, divide it by 26, andtranslate the remainderback to a letter

Examples: Encipher the letters b and x usingCaesars cipher with a shift of 6 positions

Because of Step 2 above, the Caesar ciphers areknown as additive ciphers

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Multiplicative Ciphers A cipher can also be obtained by multiplication

by a number t, as follows

To encipher a plaintext letter:

1. Encode the letter as its corresponding number

2. Multiply the number by t

3. Use the remainder when this number isdivided by 26, and translate it back to a letter

If t= 2, we obtain

Plaintext 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 zCiphertext B D F H J L N P R T V X Z B D F H J L N P R T V X Z

Note this is nota useful cipher, as the messageLRN could mean fig,fit,sit, etc

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Multiplicative Ciphers (cont)

The multiplicative cipher with t= 2 isnt usefulbecause we need to be able to reconstruct theplaintext uniquely from the ciphertext

However, if t= 3, we do obtain a useful cipher

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

It can be checked that valid multiplicative ciphersare obtained if t= 1 (the trivial cipher), 3, 5, 7, 9,11, 15, 17, 19, 21, 23 and 25 so there are 12multiplicative ciphers


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Diagram of Secret Key Encryption

(Diagram from Foundations of Computer Scienceby

Behrouz A. Forouzan, Brooks/Cole, 2003, p. 308)

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Data Encryption Standard (DES)

Secret key encryption has been used for morethan 2000 years

At first the algorithms were simple, and the keyswere easy to guess

Today, the algorithms are very sophisticated

The most common method nowadays is theDataEncryption Standard(DES), developed in the1970s (essentially by IBM)

The method, in which data is scrambled in a very

complicated fashion, is used widely, particularlyin banking

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DES (continued)

In DES, the data is transformed into a string of

bits (e.g. use ASCII code), which is broken into

segments of 64 bits

Each segment is then encrypted in a many-stageprocess that uses a 56-bit key

The DES method is a monoalphabetic cipher,

since, for a given key, each particular 64-bit

plaintext segment is always encrypted as the

same 64-bit ciphertext string


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Secret Key Algorithms

Secret key algorithms are usually efficient theytake less time to implement than the alternativepublic key algorithms well discuss shortly

So secret key encryption is especially useful forlong messages

However, there are 2 disadvantages

Firstly, each pair of users must have a secret key so, if npeople are to use secret key encryption,there must be n (n 1)/2 secret keys

Thus, for 1 million people to communicate, thereneeds to be about half-a-trillion secret keys!

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Disadvantages of Secret Key

Encryption

So the first disadvantage of secret key encryption

relates to the managementof the secret keys.

That is, how can so many keys be kept secret?

The second disadvantage is associated with the

distribution of the keys i.e. how does the sender

securely inform the receiver what the key is?

These 2 disadvantages of secret key encryption

are addressed in public key encryption, which

well introduce in the next lecture

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Discrete Maths 2003 Lecture 36 3 Slides Pp