symmetric cryptosystems symmetric cryptosystems 20/10/2015 | pag. 2

Symmetric Cryptosystems

Symmetric Cryptosystems21/04/23 | pag. 2

Block Ciphers:Classical examples

Symmetric Cryptosystems21/04/23 | pag. 3

• Affine Cipher

• Affine Linear and Linear Cipher

• Vigenère

• Hill

Block Ciphers:Remark

Secure block ciphers must not be (affine) linear or easy to approximate by linear functions!!!

Cryptography 21/04/23 | pag. 4

Remark

Cryptography 21/04/23 | pag. 5

Implementation of a (non-linear!) substitution often occurs through a look-up table, called S-box.

Block Ciphers:Advanced examples

Symmetric Cryptosystems21/04/23 | pag. 6

• DES – Feistel Cipher

• AES – Rijndael

DES:Feistel Cipher

Cryptography 21/04/23 | pag. 7

An iterated block cipher is a block cipher involving the sequential repetition of an internal function called rounds.

an iterated block cipher

DES:Feistel Cipher

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DES:Feistel Cipher

Cryptography 21/04/23 | pag. 9

DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:S-Boxes

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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DES:Algorithm

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Roundnumber

Number ofleft

rotations

1 1

2 1

3 2

4 2

5 2

6 2

7 2

8 2

9 1

10 2

11 2

12 2

13 2

14 2

15 2

16 1

DES:Algorithm

Cryptography 21/04/23 | pag. 24

DES:Algorithm

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DES:Algorithm

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AES:Rijndael Cipher

Cryptography 21/04/23 | pag. 27

We again need some algebra first!

Intermezzo:Polynomials over Rings

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Example:Polynomials over Rings

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Intermezzo:Polynomials over Rings

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Example:Polynomials over Rings

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Intermezzo:Polynomials over Fields

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Intermezzo:Polynomials over Fields

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Intermezzo:Polynomials over Fields

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Intermezzo:Polynomials over Fields

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Example:Polynomials over Fields

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Intermezzo:Polynomials over Fields

Cryptography 21/04/23 | pag. 37

Intermezzo:Polynomials over Fields

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Example:Polynomials over Fields

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Intermezzo:Finite Fields

• Let R be a ring. If there is a least positive integer n such that nr=0 for all r in R, then we say that R has characteristic n and write char(R)=n. When no such integer exists, we set char(R)=0.

• Let F be a field with char(F)>0, then char(F) is prime.

• Any finite field F has char(F)=p, where p is prime.

• Let F be a finite field, where char(F)=p, then |F|=pn , with n a strictly positive integer.

Cryptography 21/04/23 | pag. 40

Intermezzo:Construction of Finite Fields

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Hence we can also denote it by GF(p). Note that char(GF(p))=p.

Intermezzo:Construction of Finite Fields

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Intermezzo:Construction of Finite Fields

Cryptography 21/04/23 | pag. 43

2

Intermezzo:Construction of Finite Fields

Cryptography 21/04/23 | pag. 44

Intermezzo:Construction of Finite Fields

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For every prime p and positive integer n there is an irreducible polynomial of degree n in Zp[x] !

Intermezzo:Construction of Finite Fields

Theorem

Let p be a prime and f(x) an irreducible polynomial of degree n in Zp[x]. Then

Zp[x]/ < f(x) > (or Zp[x] mod f(x) ) is a field with pn elements.

ProofAs we can choose as coset representatives polynomials of the form a0 + a1x + a2x2 + ... + an-1xn-1 , we get a ring of order pn. As in Zn we use the analogue of the Extended Euclidean algorithm to find the inverse of an element.Let g(x) be a coset representative of a non-zero element of the ring. Since f(x) is irreducible it is not divisible by any lower degree polynomial and so the gcd(g(x), f(x)) = 1. Then by the analogue of the Extended Euclidean algorithm 1 = a(x)g(x) + b(x)f(x) for some polynomials a(x), b(x). Then a(x) is a coset representative for the inverse of g(x).

Cryptography 21/04/23 | pag. 46

Example:Construction of Finite Fields

Cryptography 21/04/23 | pag. 47

Example:Construction of Finite Fields

Cryptography 21/04/23 | pag. 48

Intermezzo:Construction of Finite Fields

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Conclusion: For every prime p and positive integer n the field GF(pn) exists!