1 / 32

Classical Cryptosystems

Classical Cryptosystems. Shift Ciphers (Caesar) y= x+k (mod 26) Affine Ciphers y=ax+b (mod 26) Vigenere Ciphers codes=(02,14,03,04,18) Substitution Ciphers (26! Permutations) Sherlock Holmes P27 (Visual Substitution) The Playfair and ADFG[V]X Ciphers Block Ciphers

lazaro
Download Presentation

Classical Cryptosystems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Classical Cryptosystems • Shift Ciphers (Caesar) y= x+k (mod 26) • Affine Ciphers y=ax+b (mod 26) • Vigenere Ciphers codes=(02,14,03,04,18) • Substitution Ciphers (26! Permutations) • Sherlock Holmes P27 (Visual Substitution) • The Playfair and ADFG[V]X Ciphers • Block Ciphers • PseudoRandom Number Generators

  2. Shift Cipher y=x+k (mod 26) • attack  XQQXZH (k=23 mod 26) • great  ITGCV (k=2 mod 26) a b c d e f g h i j 00 01 02 03 04 05 06 07 08 09 k l m n o p q r s t 10 11 12 13 14 15 16 17 18 19 u v w x y z a b c d 20 21 22 23 24 25 00 01 02 03

  3. Affine Cipher y=ax+b mod 26 • how are you  QZNHOBXZD, (a,b)=(5,7) • wo??er?u?  NZUWBOGDK, (a,b)=(5,7) gcd(a,26)=1 is required Table for ax=1 mod 26 1(1) 7(15) 15( 7) 21(5) 3(9) 9( 3) 17(23) 23(17) 5(21) 11(19) 19(11) 25(25)

  4. Frequencies of Letters in English a b c d e f g h i .082 .015 .028 .043 .127 .022 .020 .061 .070 j k l m n o p q r .002 .008 .040 .024 .067 .075 .019 .001 .060 s t u v w x y z .063 .091 .028 .010 .023 .001 .020 .001

  5. Vigenere Cipher The same letter need not be enciphered as the same letter Key: vector=(21,4,2,19,14,17) he r e i s h o w i t w o r k s v e c t o r v e c t o r v e c t C I T X W J C S Y B H N J V M L Attacks according to the following information (1) The frequencies of letters in English A0=[.082, .015, .028, …,.020,.001] <A0,A0> is larger than <A0,Aj>, j=1,2,...,25 (2) Key length (3) Digrams (e.g., WX) or trigrams (e.g., FHQ)

  6. Sherlock Holmes A visual substitution (use a visual pattern to replace each English letter)

  7. The Playfair p l a y f i r b c d e g h k m n o q s t u v w x z meet at the schoolhouse  me et at th es ch ox ol ho us ex EG MN FQ QM KN BK SV VR GQ XN KU

  8. ADFGX Cipher A D F G X A p g c e n D b q o z r F s l a f t G m d v i w X k u y x h Kaiser Wilhelm  XA FF GG FA AG DX GX GG FD XX AG FD GA

  9. Block Ciphers • Hill cipher, DES, AES, RSA, Electronic Codebook, Elliptic Curve cryptosystems • Find the inverse of A and B (mod 26) • A=3 4 inv-A=7 22 5 7 21 3 M = [1 2 3; 4 5 6; 11 9 8] inv-M=[22 5 1; 6 17 24; 15 13 1]

  10. Hill Cipher • blockcipherx (1 11 14) (2 10 2) (8 15 7) (4 17 23) (1 11 14)M=(17 1 25) (mod 26) = RBZ (2 10 2)M =(12 20 4) (mod 26) = MUE blockcipher  RBZMUEPYONOM

  11. Binary Numbers and ASCII • ASCII – American Standard Code for Information Interchange • A=65=01000001 ~ Z=90=01011010 • a=97=01100001 ~ z=122=01111010 • [33~47] ! “ # $ % & ' ( ) * + , - . / • [48~64] 0 1 2 3 4 5 6 7 8 9 : ; ¡ = ¿ ? @

  12. One-time Pads • By Gilbert Vernam and Joseph Mauborgne around 1918 • The key is a random sequence of 0’s and 1’s of the same length as the message. Once a key is used, it is discarded and never used again. • 00101001⊕10101100=10000101 • Used in “hot line” between USSR and US

  13. Pseudo-random Bit Generation • Rand() based on a linear congruential generator xn=axn-1 + b (mod m) with gcd(a,m)=1, m=231-1=2147483647 • x0=seed, a=16847, b=314759 • Blum-Blum-Shub (BBS) bit generator • Select n=pq, the product of two primes • x0=seed=x2 (mod n), where gcd(x,n)=1 • xj=(xj-1)2 (mod n) and bj= xj ^ 1

  14. Linear Feedback Shift Register (LFSR) Sequences (mod 2) • Plaintext • 01000010 01011001 11110001 10111010 • (x1, x2, x3, x4 , x5)=(0,1,0,0,0) • Xn+5 =Xn + X n+2 (mod 2) • Xn+m=c0xn+c1xn+1+····+cm-1xn+m-1 (mod 2) • Xn+31 =Xn + X n+3 (mod 2) has period 231-1

  15. Proposition • Let M be a matrix (mod 2) {x1 x2 x3··· xm x2 x3 x4··· xm+1 ︰ ︰ xm xm+1 xm+2··· x2m-1}={xj} If the sequence {xj} satisfies a linear recurrence of length less than m, then det(M)=0. Cinversely, if the sequence satisfies a linear recurrence of length m and det(M)=0, then the sequence also satisfies a linear recurrence of length than m.

  16. (plaintext) 1011001110001111 • (key) + 0100001001011001 • (ciphertext) 1111000111010110

  17. Cryptanalysis • Suppose Xn+2=C0Xn+C1Xn+1

  18. Cryptanalysis • If the linear recurrence of length is less than m,then

  19. Irreducible Polynomial mod 2 • xn+m=c0xn+c1xn+1+····+cm-1xn+m-1 (mod 2) • f(T)=Tm –cm-1 Tm-1 - ‥‥ - c1T1 – c0 • If f(T) is irreducible, then its period divides 2m – 1, an interesting case is when 2m – 1 is a prime (Mersenne primes) • 231 – 1 =2147483647 is a prime number • Further discuss this topic later

  20. Enigma • A mechanical encryption device used by the Germans in World War II. • A rotor machine

  21. Enigma

  22. Schematic diagram of Enigma

  23. Schematic diagram of Enigma

  24. Schematic diagram of Enigma • K:keyboard • R:revering drum • S:plugboard • L,M,M:rotors

  25. Single Rotor • 26 substitution cipher • A1,A27,A53…. • A2,A28,A54…. • Frequency analysis

  26. Three Rotors • 26*26*26*6=105456 possibilities. • 100391791500 ways of interchanging six pairs of letters on the plugboard.

  27. To Attack Enigma • A codebook containing the daily settings. • During a given day,every first letters in plaintexts is encrypted in the same substitution cipher.

  28. To Attack Enigma • Message key:a sequence of three letters,for example,r,f,u. • rfurfu • Daily setting • Encrypting the message key • Reset

  29. To Attack Enigma • dmqvbn • vonpuy • pucfmq • AD=(dvpf..)… • AD=(dvpfkxgzyo)(eijmunqlht)(bc)(rw)(a)(s)

  30. The Effect of the Plugboard • AD has cycles of length 10,10,2,2,1,1. • SADS-1 has cycles of length 10,10,2,2,1,1. • The cycle lengths remain unchanged. • Substitution cipher

  31. Bletchley Park

  32. Exercises • Problems from 2.13 Exercises on p.55~59 • Problems from 2.14 Exercises • on p.59~62

More Related