SYMMETRIC CRYPTOSYSTEMS Symmetric Cryptosystems 20/10/2015 | pag. 2.

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Presentation transcript:

SYMMETRIC CRYPTOSYSTEMS Symmetric Cryptosystems 20/10/2015 | pag. 2

Block Ciphers: Classical examples Symmetric Cryptosystems 20/10/2015 | 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 20/10/2015 | pag. 4

Remark Cryptography 20/10/2015 | pag. 5 Implementation of a (non-linear!) substitution often occurs through a look-up table, called S-box.

Block Ciphers: Advanced examples Symmetric Cryptosystems 20/10/2015 | pag. 6 DES – Feistel Cipher AES – Rijndael

DES: Feistel Cipher Cryptography 20/10/2015 | 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 Cryptography 20/10/2015 | pag. 8

DES: Feistel Cipher Cryptography 20/10/2015 | pag. 9

DES: Algorithm Cryptography 20/10/2015 | pag. 10

DES: Algorithm Cryptography 20/10/2015 | pag. 11

DES: Algorithm Cryptography 20/10/2015 | pag. 12

DES: Algorithm Cryptography 20/10/2015 | pag. 13

DES: Algorithm Cryptography 20/10/2015 | pag. 14

DES: Algorithm Cryptography 20/10/2015 | pag. 15

DES: Algorithm Cryptography 20/10/2015 | pag. 16

DES: Algorithm Cryptography 20/10/2015 | pag. 17

DES: Algorithm Cryptography 20/10/2015 | pag. 18

DES: S-Boxes Cryptography 20/10/2015 | pag. 19

DES: Algorithm Cryptography 20/10/2015 | pag. 20

DES: Algorithm Cryptography 20/10/2015 | pag. 21

DES: Algorithm Cryptography 20/10/2015 | pag. 22

DES: Algorithm Cryptography 20/10/2015 | pag. 23 Round number Number of left rotations

DES: Algorithm Cryptography 20/10/2015 | pag. 24

DES: Algorithm Cryptography 20/10/2015 | pag. 25

DES: Algorithm Cryptography 20/10/2015 | pag. 26

AES: Rijndael Cipher Cryptography 20/10/2015 | pag. 27 We again need some algebra first!

Intermezzo: Polynomials over Rings Cryptography 20/10/2015 | pag. 28

Example: Polynomials over Rings Cryptography 20/10/2015 | pag. 29

Intermezzo: Polynomials over Rings Cryptography 20/10/2015 | pag. 30

Example: Polynomials over Rings Cryptography 20/10/2015 | pag. 31

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 32

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 33

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 34

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 35

Example: Polynomials over Fields Cryptography 20/10/2015 | pag. 36

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 37

Intermezzo: Polynomials over Fields Cryptography 20/10/2015 | pag. 38

Example: Polynomials over Fields Cryptography 20/10/2015 | pag. 39

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|=p n, with n a strictly positive integer. Cryptography 20/10/2015 | pag. 40

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 41 Hence we can also denote it by GF(p). Note that char(GF(p))=p.

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 42

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 43 2

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 44

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 45 For every prime p and positive integer n there is an irreducible polynomial of degree n in Z p [x] !

Intermezzo: Construction of Finite Fields Theorem Let p be a prime and f(x) an irreducible polynomial of degree n in Z p [x]. Then Z p [x]/ (or Z p [x] mod f(x) ) is a field with p n elements. Proof As we can choose as coset representatives polynomials of the form a 0 + a 1 x + a 2 x a n-1 x n-1, we get a ring of order p n. As in Z n 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 20/10/2015 | pag. 46

Example: Construction of Finite Fields Cryptography 20/10/2015 | pag. 47

Example: Construction of Finite Fields Cryptography 20/10/2015 | pag. 48

Intermezzo: Construction of Finite Fields Cryptography 20/10/2015 | pag. 49 Conclusion: For every prime p and positive integer n the field GF(p n ) exists!