02_classicrypt

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Classical Cryptography

Shift cipher

Substitution cipher

Affine cipherVigenre Cipher

Hill Cipher

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References

Douglas Stinson, Cryptography: Theory and

Practice, 3rd, Chapman & Hall CRC, 2006.

Chapter 1

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Secure communication scheme

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The set of integers modulo m:

Remarks:

Congruences

. Examples:

= {0, 1, 2, , 25}.

14 + 20 = 8 in .

5*7 = 9 in .

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We will always consider

(a mod b) >= 0

Lab computing: try

operators + and * modulo

m in C, Java, Maple,

Matlab and Python.


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Lab computing: tryMOD or % operator

in C, Java, Maple, Matlab

and Python.

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The Shift Cipher

For the particular key K = 3, the cryptosystem is often called the

Caesar Cipher, which was purportedly used by Julius Caesar. 6


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The Shift CipherEncryption

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The Shift CipherDecryption

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Caesar cipher (K = 3)

x = NGUYENTHANHNHUT

y = QJXBHQWKDQKQKXW

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owCryptool it!


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Practical cryptosystems1. Each encryption function ekand each decryption

function dkshould be efficiently computable.2. An opponent, upon seeing a ciphertext string y,

should be unable to determine the key k that was

used, or the plaintext string x.

Security

The process of attempting to compute the key k, given

a string of ciphertext y, is called cryptanalysis.

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Security of the Shift cipher The Shift Cipher (modulo 26) is not secure.

It can be cryptanalyzed by the obvious method ofexhaustive key search.

Since there are only 26 possible keys, it is easy to try

every possible decryption rule dk(k = 0, , 25) until a"meaningful" plaintext string is obtained.

On average, a plaintext will be computed using thismethod after trying 26/2 = 13 decryption rules.

a necessary condition for a cryptosystem to besecure is that an exhaustive key search should beinfeasible; i.e., the keyspace should be very large.

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Exercise

Lab computing: build a

ShiftCipher encrypt anddecrypt functions to do this

Take a shortbreak

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The Substitution Cipher

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The Substitution CipherLab

computing:

decrypt thisby define a

SubCipher

decr t

A key for the Substitution Cipher just consists of a

permutation of the 26 alphabetic characters. The number of possible permutations is 26!, which is

more than 4*1026, a very large number.

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?

functionTry Cryptool it!


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Euler phi-function

Lab com utin : define GCD

Ex:

(10) = 4.

(11) = 10.

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and PHI functions in Python tocompute gcd(a,b) and phi(n).

Compare time running with

MAPLE.


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Compute the Euler phi-function

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The multiplicative inverse modulo m

Form = 26:

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Exercise

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The Affine Cipher

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So the three ciphertext characters are 0, 23, and 6, which

corresponds to the alphabetic string AXG.


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Security of the Affine cipher

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1.11

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Take a short

break


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The Vigenre Cipher

Each key k is an alphabetic string of length m, called

a keyword.

The Vigenere Cipher encrypts m alphabetic charactersat a time.

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Try Cryptool it!


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Security of the Vigenere Cipher

The number of possible keywords of length m is 26m,

even for relatively small values of m, an exhaustive keysearch would require a long time.

An alphabetic character can be mapped to one of m

poss e a p a e c c arac ers assum ng a e eywor

contains m distinct characters). Such a cryptosystem is

called apolyalphabetic cryptosystem.

In general, cryptanalysis is more difficult for

polyalphabetic than formonoalphabetic cryptosystems

(Shift Cipher, Substitution Cipher).

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The Hill Cipher

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1.12

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Think about the security of the Hill Cipher!

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The Permutation Cipher

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1.16

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1.17

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The Permutation Cipher is a special case of

the Hill Cipher

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