# William Stallings, Cryptography and Network Security 5/e

Cryptography and Network Security Chapter 2 Fifth Edition by William Stallings Lecture slides by Lawrie Brown Chapter 2 Classical Encryption Techniques "I am fairly familiar with all the forms of secret writings, and am myself the author of a trifling monograph upon the subject, in which I analyze one hundred and sixty separate ciphers," said Holmes.. The Adventure of the Dancing Men, Sir Arthur Conan Doyle Symmetric Encryption or conventional / private-key / single-key sender and recipient share a common key all classical encryption algorithms are

private-key was only type prior to invention of publickey in 1970s and by far most widely used (still) is significantly faster than public-key crypto Some Basic Terminology plaintext - original message ciphertext - coded message cipher - algorithm for transforming plaintext to ciphertext key - info used in cipher known only to sender/receiver encipher (encrypt) - converting plaintext to ciphertext decipher (decrypt) - recovering plaintext from ciphertext cryptography - study of encryption principles/methods cryptanalysis (codebreaking) - study of principles/ methods of deciphering ciphertext without knowing key cryptology - field of both cryptography and cryptanalysis Symmetric Cipher Model Requirements two requirements for secure use of symmetric

encryption: a strong encryption algorithm a secret key known only to sender / receiver mathematically have: Y = E(K, X) = EK(X) = {X}K X = D(K, Y) = DK(Y) assume encryption algorithm is known Kerckhoffs Principle: security in secrecy of key alone, not in obscurity of the encryption algorithm implies a secure channel to distribute key Central problem in symmetric cryptography Cryptography can characterize cryptographic system by: type of encryption operations used substitution transposition product

number of keys used single-key or private two-key or public way in which plaintext is processed block stream Cryptanalysis objective to recover key not just message general approaches: cryptanalytic attack brute-force attack if either succeed all key use compromised Cryptanalytic Attacks ciphertext only only know algorithm & ciphertext, is statistical, can identify plaintext

known plaintext know/suspect plaintext & ciphertext chosen plaintext select plaintext and obtain ciphertext chosen ciphertext select ciphertext and obtain plaintext chosen text select plaintext or ciphertext to en/decrypt Cipher Strength unconditional security no matter how much computer power or time is available, the cipher cannot be broken since the ciphertext provides insufficient information to uniquely determine the corresponding plaintext

computational security given limited computing resources (e.g. time needed for calculations is greater than age of universe), the cipher cannot be broken Encryption Mappings M1 M2 C1 K1 M3 A given key (k) Maps any message Mi to some ciphertext E(k,Mi) Ciphertext image of Mi is unique to Mi under k

Plaintext pre-image of Ci is unique to Ci under k C2 C3 Notation M=set of all plaintexts Cn C=set of all ciphertexts A Mn

A K1 A Ci A Mi K1 key k and Mi in M, ! Cj in C such that E(k,Mi) = Cj key k and ciphertext Ci in C, ! Mj in M such that E(k,Mj) = Ci Ek(.) is one-to-one (injective) If |M|=|C| it is also onto

(surjective), and hence bijective. Encryption Mappings (2) C1 M3 C2 K j K2,K89,... K 3 , K j C3 K m , ...

Mi Mn K1 ,K 7 5 7 Mi is mapped to some ciphertext E(K,Mi) by every key k Different keys may map Mi to the same ciphertext There may be some ciphertexts to which Mi is never mapped by any key Notation key k and Mi in M, ! ciphertext Cj in C such that E(k,Mi) = Cj

It is possible that there are keys k and k such that E(k,Mi) = E(k,Mi) There may be some ciphertext Cj for which key k such that E(k,Mi) = Cj ,... Ci Cn A M2 A given plaintext (Mi) A

M1 Encryption Mappings (3) C1 M2 K17 C2 , . .. K2,K89,... M3 C3 Km

Notation . . . , ... K j 9 4 K , . . .

Mn key k and ciphertext Ci in C, ! Mj in M such that E(k,Mj) = Ci There may exist keys k, k and plaintext Mj such that E(k,Mj) = E(k,Mj) = Ci There may exist plaintext Mj such that key k such that E(k,Mj) = Ci 3 K Mi Has a unique plaintext preimage under each k May have two keys that map the same plaintext to it

There may be some plaintext Mj such that no key maps Mj to Ci Ci Cn A K1, A ciphertext (Ci) A M1 Encryption Mappings (4) Under what conditions will there always be some key that maps some plaintext to a

given ciphertext? If for an intercepted ciphertext cj, there is some plaintext mi for which there does not exist any key k that maps mi to cj, then the attacker has learned something If the attacker has ciphertext cj and known plaintext mi, then many keys may be eliminated Brute Force Search always possible to simply try every key most basic attack, exponential in key length assume either know / recognise plaintext Key Size (bits) Number of Alternative Keys Time required at 1 decryption/s

Time required at 106 decryptions/s 32 232 = 4.3 109 231 s = 35.8 minutes 2.15 milliseconds 56 256 = 7.2 1016 255 s = 1142 years

10.01 hours 128 2128 = 3.4 1038 2127 s = 5.4 1024 years 5.4 1018 years 168 2168 = 3.7 1050 2167 s = 5.9 1036 years 5.9 1030 years

26! = 4 1026 2 1026 s = 6.4 1012 years 6.4 106 years 26 characters (permutation) Classical Substitution Ciphers where letters of plaintext are replaced by other letters or by numbers or symbols or if plaintext is viewed as a sequence of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit patterns Caesar Cipher earliest known substitution cipher

by Julius Caesar first attested use in military affairs replaces each letter by 3rd letter on example: meet me after the toga party PHHW PH DIWHU WKH WRJD SDUWB Caesar Cipher can define transformation as: a b c d e f g h i j k l m n o p q r s t u v w x y z = IN D E F G H I J K L M N O P Q R S T U V W X Y Z A B C = OUT mathematically give each letter a number a b c d e f g h i j k l m n o p q r s t u v w x y z 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 then have Caesar (rotation) cipher as: c = E(k, p) = (p + k) mod (26) p = D(k, c) = (c k) mod (26)

Cryptanalysis of Caesar Cipher only have 26 possible ciphers A maps to A,B,..Z could simply try each in turn a brute force search given ciphertext, just try all shifts of letters do need to recognize when have plaintext eg. break ciphertext "GCUA VQ DTGCM" Affine Cipher broaden to include multiplication can define affine transformation as: c = E(k, p) = (ap + b) mod (26) p = D(k, c) = (a-1c b) mod (26) key k=(a,b) a must be relatively prime to 26 so there exists unique inverse a-1

Affine Cipher - Example example k=(17,3): a b z D U M c = L = d e f g h i j k l m n o p q r s t u v w x y IN C T K B S J A R I Z Q H Y P G X O F W N E V OUT example: meet ZTTO

me ZT after DKOTG the OST toga OHBD party YDGOV Now how many keys are there? 12 x 26 = 312 Still can be brute force attacked! Note: Example of product cipher

Monoalphabetic Cipher rather than just shifting the alphabet could shuffle (permute) the letters arbitrarily each plaintext letter maps to a different random ciphertext letter hence key is 26 letters long Plain: abcdefghijklmnopqrstuvwxyz Cipher: DKVQFIBJWPESCXHTMYAUOLRGZN Plaintext: ifwewishtoreplaceletters Ciphertext: WIRFRWAJUHYFTSDVFSFUUFYA Monoalphabetic Cipher Security key size is now 25 characters now have a total of 26! = 4 x 1026 keys with so many keys, might think is secure but would be !!!WRONG!!! problem is language characteristics Language Redundancy and

Cryptanalysis human languages are redundant e.g., "th lrd s m shphrd shll nt wnt" letters are not equally commonly used in English E is by far the most common letter followed by T,R,N,I,O,A,S other letters like Z,J,K,Q,X are fairly rare have tables of single, double & triple letter frequencies for various languages English Letter Frequencies English Letter Frequencies Sorted Frequencies of Letters in English 14 12 10 8 6 4

2 0 e t a o i n s h r d l c u m w f g y p b v k

j x q z What kind of cipher is this? 14 12 10 8 6 4 2 0 0 1 2 3

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Frequency of Characters in English 14 12 10 8 6

4 2 0 a b c d e f g h i

j k l m n o p q r s t

u v w x y z What kind of cipher is this? Cipher-3 14 12 10 8 6 4

2 0 a b c d e f g h i j

k l m n o p q r s t

u v w x y z r s t u v

w x y z Frequency of Characters in English 14 12 10 8 6 4 2 0 a b

c d e f g h i j k l

m n o p q Sorted Frequencies of Letters in Cipher-3 14 12 10 8 6 4 2 0 e

t a o i n s h r d l c

u m w f g y p b v k

j x q z Sorted Frequencies of Letters in English 14 12 10 8 6 4 2 0 e t

a o i n s h r d l c u

m w f g y p b v k j

x q z Use in Cryptanalysis key concept - monoalphabetic substitution ciphers do not change relative letter frequencies discovered by Arabian scientists in 9th century calculate letter frequencies for ciphertext compare counts/plots against known values if caesar cipher look for common peaks/troughs peaks at: A-E-I triple, N-O pair, R-S-T triple troughs at: J-K, U-V-W-X-Y-Z for monoalphabetic must identify each letter tables of common double/triple letters help (digrams and trigrams) amount of ciphertext is important statistics!

Example Cryptanalysis given ciphertext: UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ count relative letter frequencies (see text) Example Cryptanalysis given ciphertext: UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ guess P & Z are e and t guess ZW is th and hence ZWP is the proceeding with trial and error finally get: it was disclosed yesterday that several informal but direct contacts have been made with political

representatives of the viet cong in moscow Playfair Cipher not even the large number of keys in a monoalphabetic cipher provides security one approach to improving security was to encrypt multiple letters the Playfair Cipher is an example invented by Charles Wheatstone in 1854, but named after his friend Baron Playfair Playfair Key Matrix a 5X5 matrix of letters based on a keyword fill in letters of keyword (sans duplicates) fill rest of matrix with other letters eg. using the keyword MONARCHY M O N

A R C H Y B D E F G

I/J K L P Q S T U V W X

Z Encrypting and Decrypting plaintext is encrypted two letters at a time 1. if a pair is a repeated letter, insert filler like 'X 2. if both letters fall in the same row, replace each with letter to right (wrapping back to start from end) 3. if both letters fall in the same column, replace each with the letter below it (wrapping to top from bottom) 4. otherwise each letter is replaced by the letter in the same row and in the column of the other letter of the pair Playfair Example Message = Move forward Plaintext = mo ve fo rw ar dx Here x is just a filler, message is padded and segmented

mo -> ON; ve -> UF; fo -> PH, etc. Ciphertext = ON UF PH NZ RM BZ M O N A R C H Y B D

E F G I/J K L P Q S T

U V W X Z Security of Playfair Cipher security much improved over monoalphabetic since have 26 x 26 = 676 digrams would need a 676 entry frequency table to analyse (versus 26 for a monoalphabetic) and correspondingly more ciphertext was widely used for many years eg. by US & British military in WW1 it can be broken, given a few hundred letters since still has much of plaintext structure

Polyalphabetic Ciphers polyalphabetic substitution ciphers improve security using multiple cipher alphabets make cryptanalysis harder with more alphabets to guess and flatter frequency distribution use a key to select which alphabet is used for each letter of the message use each alphabet in turn repeat from start after end of key is reached Vigenre Cipher simplest polyalphabetic substitution cipher effectively multiple caesar ciphers key is multiple letters long K = k1 k2 ... kd ith letter specifies ith alphabet to use use each alphabet in turn repeat from start after d letters in message decryption simply works in reverse Example of Vigenre Cipher

write the plaintext out write the keyword repeated above it use each key letter as a caesar cipher key encrypt the corresponding plaintext letter eg using keyword deceptive key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ Aids simple aids can assist with en/decryption a Saint-Cyr Slide is a simple manual aid a slide with repeated alphabet line up plaintext 'A' with key letter, eg 'C' then read off any mapping for key letter can bend round into a cipher disk or expand into a Vigenre Tableau Security of Vigenre Ciphers

have multiple ciphertext letters for each plaintext letter hence letter frequencies are obscured but not totally lost start with letter frequencies see if it looks monoalphabetic or not if not, then need to determine number of alphabets, since then can attack each Frequencies After Polyalphabetic Encryption Letter Relative Frequency sub-title sub-title 14.000 12.000 10.000 8.000 6.000 4.000

Homework 2 Due Fri Question 1: What is the best flattening effect you can achieve by carefully selecting two monoalphabetic substitutions? Explain and give an example. What about three monoalphabetic substitutions? Kasiski Method method developed by Babbage / Kasiski repetitions in ciphertext give clues to period so find same plaintext a multiple of key length apart which results in the same ciphertext of course, could also be random fluke e.g. repeated VTW in previous example distance of 9 suggests key size of 3 or 9 then attack each monoalphabetic cipher individually using same techniques as before

Example of Kasiski Attack Find repeated ciphertext trigrams (e.g., VTW) May be result of same key sequence and same plaintext sequence (or not) Find distance(s) Common factors are likely key lengths key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ Autokey Cipher ideally want a key as long as the message Vigenre proposed the autokey cipher with keyword is prefixed to message as key knowing keyword can recover the first few letters use these in turn on the rest of the message but still have frequency characteristics to attack eg. given key deceptive key:

deceptivewearediscoveredsav plaintext: wearediscoveredsaveyourself ciphertext:ZICVTWQNGKZEIIGASXSTSLVVWLA Homophone Cipher rather than combine multiple monoalphabetic ciphers, can assign multiple ciphertext characters to same plaintext character assign number of homophones according to frequency of plaintext character Gauss believed he made unbreakable cipher using homophones but still have digram/trigram frequency characteristics to attack e.g., have 58 ciphertext characters, with each plaintext character assigned to ceil(freq/2) ciphertext characters so e has 7 homophones, t has 5, a has 4, j has 1, q has 1, etc. Vernam Cipher ultimate defense is to use a key as long as the

plaintext with no statistical relationship to it invented by AT&T engineer Gilbert Vernam in 1918 specified in U.S. Patent 1,310,719, issued July 22, 1919 originally proposed using a very long but eventually repeating key used electromechanical relays One-Time Pad if a truly random key as long as the message is used, the cipher will be secure called a One-Time pad (OTP) is unbreakable since ciphertext bears no statistical relationship to the plaintext since for any plaintext & any ciphertext there exists a key mapping one to other can only use the key once though problems in generation & safe distribution of key

Transposition Ciphers now consider classical transposition or permutation ciphers these hide the message by rearranging the letter order without altering the actual letters used can recognise these since have the same frequency distribution as the original text Rail Fence cipher write message letters out diagonally over a number of rows use a W pattern (not column-major) then read off cipher row by row eg. write message out as: m e m a t r h t g p r y e t e f e t e o a a t giving ciphertext MEMATRHTGPRYETEFETEOAAT

Row Transposition Ciphers is a more complex transposition write letters of message out in rows over a specified number of columns then reorder the columns according to some key before reading off the rows Key: 4312567 Column Out 4 3 1 2 5 6 7 Plaintext: a t t a c k p o s t p o n e d u n t i l t w o a m x y z Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ Block Transposition Ciphers arbitrary block transposition may be used specify permutation on block repeat for each block of plaintext Key: 4931285607 Plaintext: attackpost poneduntil twoamxyzab Ciphertext: CTATTSKPAO DLEONIDUPT MBAWOAXYTZ

Homework 2 Due Fri Question 2: Mathematically specify an arbitrary block transposition cipher with block length B and permutation :[0..B-1] [0..B-1] for plaintext P=p0p1p2p3pN-1, where N is a multiple of B. Product Ciphers ciphers using substitutions or transpositions are not secure because of language characteristics hence consider using several ciphers in succession to make harder, but: two substitutions make a more complex substitution two transpositions make more complex transposition but a substitution followed by a transposition makes a new much harder cipher this is bridge from classical to modern ciphers

Homework 2 Due Fri Question 3: a. What is the result of the product of two rotational substitutions? b. What is the result of the product of two affine substitutions? c. What is the result of the product of two block transpositions? Rotor Machines before modern ciphers, rotor machines were most common complex ciphers in use widely used in WW2 German Enigma, Allied Hagelin, Japanese Purple implemented a very complex, varying substitution cipher used a series of cylinders, each giving one substitution, which rotated and changed after

each letter was encrypted with 3 cylinders have 263=17576 alphabets Hagelin Rotor Machine Rotor Machine Principles Rotor Ciphers Each rotor implements some permutation between its input and output contacts Rotors turn like an odometer on each key stroke (rotating input and output contacts) Encryption key is the sequence of rotors and their initial positions Note: Enigma also had steckerboard permutation, which was part of the key Enigma Usage Keys were changed daily (at least) How many keys? Pressing a key on the keyboard completed

a circuit that lit up the light by a character on the output To encrypt, type plaintext on the keyboard and copy the lit up characters of ciphertext Army and air force allowed local key choice, Navy used book of random keys Rotor Ciphers View more explanations and simulation: http://enigma.louisedade.co.uk/howitworks.html Note that the rotor machine has the amazing and handy property that, if the rotors and plugboard are initialized the same (i.e., the same key is used), then typing ciphertext in yields plaintext out! Decryption was the same as encryption! Rotor Ciphers Decryption was the same as encryption! Really?? Prove it!!

Must mathematically model permutations: Plugboard Rotor disks Rotations of disks Reflector Note that the Reflector has to be an exchange that is never the identity Modeling Rotor Ciphers Decryption was the same as encryption! Plugboard permutation: P Rotor disk permutations: D1, D2, etc. Reflector: R, where R-1 = R Rotations of disks: ri D ri-1 where ri is a rotation of i positions ri(x) = x+i % N Rather than writing as f(x), or the usual

right to left composition, we'll write in order Modeling Rotor Ciphers Encryption applies: P ri1 D1 ri1-1 ri2 D2 ri2-1 ri3 D3 ri3-1 R then back through the disks and the plugboard in the reverse direction This is for a single plaintext character after each character is entered, the rotations change odometer-style HW2 Bonus For Enigma-style rotor cipher (see above) a) What is period of rotor machine? b) Show that if c = E(K,p) is ciphertext character with K = , then p = E(K, c) c) Give formula for the jth ciphertext symbol, including the odometer type rotations of of the disks using the

notation above (assume D1 is fastest) Steganography an alternative to encryption hides existence of message using only a subset of letters/words in a longer message marked in some way using invisible ink hiding in LSB in graphic image or sound file hide in noise has drawbacks high overhead to hide relatively few info bits advantage is can obscure encryption use Steganography Cover bmp with embedded data (2lsb)

Embedded data Recovered from cover Summary have considered: classical cipher techniques and terminology monoalphabetic substitution ciphers cryptanalysis using letter frequencies Playfair cipher polyalphabetic ciphers transposition ciphers product ciphers and rotor machines steganography

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