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

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

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)

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

Vigenere Cipher

The same letter need not be enciphered as the same letterKey: vector=(21,4,2,19,14,17) h e 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)

Sherlock Holmes

A visual substitution (use a visual pattern to replace each English letter)

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

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

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]

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

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 : ; ¡ = ¿ ? @

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

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

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

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.

• (plaintext) 1011001110001111

• (key) + 0100001001011001

• (ciphertext) 1111000111010110

Cryptanalysis

• Suppose Xn+2=C0Xn+C1Xn+1

0

1

11

10

1

0

C

C

Cryptanalysis

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

)2(mod 0

.......

......

...

...

det)det(

64

43

132

21

XX

XX

XXX

XXX

M m

m

Irreducible Polynomial mod 2

• xn+m=c0xn+c1xn+1+····+cm-1xn+m-1 (mod 2)

• f(T)=Tm –cm-1 Tm-1 - - c‥‥ 1T1 – 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

Enigma

• A mechanical encryption device used by the Germans in World War II.

• A rotor machine

Enigma

Schematic diagram of Enigma

Schematic diagram of Enigma

Schematic diagram of Enigma

• K:keyboard

• R:revering drum

• S:plugboard

• L,M,M:rotors

Single Rotor

• 26 substitution cipher

• A1,A27,A53….

• A2,A28,A54….

• Frequency analysis

Three Rotors

• 26*26*26*6=105456 possibilities.

• 100391791500 ways of interchanging six pairs of letters on the plugboard.

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.

To Attack Enigma

• Message key:a sequence of three letters,for example,r,f,u.

• rfurfu

• Daily setting

• Encrypting the message key

• Reset

To Attack Enigma

• dmqvbn

• vonpuy

• pucfmq

• AD=(dvpf..)…

• AD=(dvpfkxgzyo)(eijmunqlht)(bc)(rw)(a)(s)

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

Bletchley Park

Exercises

• Problems from 2.13 Exercises

on p.55~59

• Problems from 2.14 Exercises

• on p.59~62