IFN642 Applied Cryptography and Network Security

IFN642 Applied Cryptography and Network Security
Lecture 2: Classical Ciphers Semester IFN642

Outline Classical cryptographic schemes
Substitution ciphers: Polyalphabetic Vigenère Beaufort Substitution ciphers: Polygraphic Playfair Hill Cipher Rotor machines: Enigma Semester IFN642

Classical cryptographic schemes
Encryption Operation Decryption Operation Classical cryptographic schemes use simple encryption operations: Transposition and/or Substitution The decryption operation is the inverse of the encryption operation, and the encryption and decryption keys are the same. Referred to as symmetric ciphers plaintext ciphertext Semester IFN642

Simple substitution ciphers: Examples: Caesar, Linear, Random substitution Encryption operation: replace a single plaintext character with a single ciphertext character, under control of key Key: substitution table (or values used to construct it) Example: P/t A B C D E F G H I J K L M C/t X R O V P T N Q S U W Y Z Semester IFN642

Simple substitution ciphers (from Lect 1) are: Monographic: Operate on one character at a time, and Monoalphabetic: Use a single mapping from plaintext to ciphertext, for all characters in the message This results in a weakness: Frequency distribution of plaintext characters not disguised in ciphertext Most frequent plaintext character maps to most frequent ciphertext character, etc Easy to break these using simple frequency analysis Ciphertext only attack is possible -> very insecure ciphers Semester IFN642

Q: Can we make substitution ciphers stronger? Need to alter the frequency distribution of ciphertext characters Break the 1-1 link between plaintext and ciphertext characters that frequency analysis exploited to attack Two approaches – use substitutions that are: Polyalphabetic: Use multiple mappings from plaintext to ciphertext, or Polygraphic: Operate on groups of characters at a time Semester IFN642

Substitution ciphers: Polyalphabetic
Use a set of d monoalphabetic substitutions: d = 1 is simple substitution discussed in Lecture 1. For d >1, have multiple different substitution tables; do not use same substitution table for all characters Secret key determines: Number of substitution tables in set, d, and Order in which they are applied to characters in message Examples: Vigenère Beaufort Semester IFN642

Vigenère Cipher Simplest polyalphabetic substitution cipher
Invented by Blaise de Vigenère Although similar cipher described by Bellaso in 1553 Periodic substitution: Uses multiple different Caesar ciphers Key is sequence of d letters: K = k0 k1 ... Kd-1 Letter indicates required size of shift (A=0, B=1, C=2, …) Use each alphabet in turn Repeat key sequence after processing each d letters of message Example: Message = Unbreakable, Key = BIG d=3, shifts are -> 1, 8, 6 Simply create a set of Caesar cipher alphabets, then use each in turn. Key B I G P/t U N R E A K L C/t V H S M Semester IFN642

Vigenère Cipher Formal description of encryption function:
C= EK(P) = Ek0, k1, k2, ... kd-1(p0 p1 p2 ... pn-1) Specifically, c0 = (p0 + k0) mod 26 c1 = (p1 + k1) mod 26 c2 = (p2 + k2) mod 26 cd-1 = (pd-1 + kd-1) mod 26 cd = (pd + k0) mod 26 General form: ci = (pi + ki mod d ) mod 26 Simply create a set of Caesar cipher alphabets, then use each in turn. Semester IFN642

Vigenère Cipher Formal description of decryption function:
P= DK(C) = Dk0, k1, k2, ... kd-1(c0 c1 c2 ... cn-1) Specifically, p0 = (c0 - k0) mod 26 p1 = (c1 - k1) mod 26 p2 = (c2 - k2) mod 26 pd-1 = (cd-1 - kd-1) mod 26 pd = (cd - k0) mod 26 General form: pi = (ci - ki mod d ) mod 26 Simply create a set of Caesar cipher alphabets, then use each in turn. Semester IFN642

Vigenère Cipher For faster encryption/decryption, shifted alphabets recorded in a table: Vigenère Tableau How to use Vigenère Tableau: For key character k and plaintext character p ciphertext is cell entry in row k, column p Example: Given key letter “B” and plaintext character “U” Look in Vigenère Tableau in cell [B, U] Corresponding ciphertext character is cell entry (V) Semester IFN642

Semester 1 2018 IFN642 A B C D E F G H I J K L M N O P Q R S T U V W X
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Vigenère Cipher Encryption: Example:
Write the plaintext out Write the keyword above it, repeating if necessary Use each key letter as a Caesar cipher key to encrypt corresponding plaintext letter Example: Plaintext: We are discovered Save yourself Keyword: deceptive key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ Semester IFN642

Vigenère Cipher Decryption: Example: using keyword - deceptive
Write the ciphertext out Write the keyword above it, repeating if necessary Use each key letter as a Caesar cipher key to decrypt corresponding ciphertext letter Search row k of VT for ciphertext letter, plaintext is column header Example: using keyword - deceptive key: deceptivedeceptivedeceptive ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ plaintext: wearediscoveredsaveyourself Semester IFN642

Vigenère Cipher Security:
Improved security compared to simple substitution: Multiple ciphertext letters for each plaintext letter, so single letter frequencies obscured Number of keywords of length d = 26d For large d can make brute force attack on key infeasible d 26d 1 26 2 676 3 17,576 4 456,976 5 11,881,376 6 308,915,776 Semester IFN642

Vigenère Cipher Cryptanalysis (breaking Vigenère cipher):
If you know the keyword length: Split ciphertext into outputs of multiple Caesar ciphers, break each one as previously If you don’t know keyword length: Known-plaintext attack makes key recovery easy: You know the plaintext corresponding to some ciphertext, so use Plaintext character = column heading Find ciphertext character in that column Row that ciphertext is in gives key letter Useful if known plaintext is key length or greater Example: ciphertext: ZICVTWQNGRZGVTWAVZHCQYGLMGJ plaintext: wearediscovered key: deceptivedecept Semester IFN642

Vigenère Cipher Cryptanalysis: If you don’t know: keyword length, or
any corresponding plaintext (so, ciphertext alone) Step 1: Figure out the length! Approaches: Try guessing the length: Start small: If d = 1 then ciphertext should have frequency distribution similar to plaintext If d =2 split into two ciphertexts: odd numbered characters and even number characters, then look at frequency distribution of each – is it similar to p/text? If d=3 … Look for clues in ciphertext Other calculations? Semester IFN642

Vigenère Cipher Cryptanalysis: Look for clues in the ciphertext
Kasiski Method: Developed by Freidrich Kasiski in 1863 Found independently by Charles Babbage in 1854 but kept secret To calculate the key length: If a plaintext character group is repeated an exact period apart then same ciphertext character group will repeat Look for repeated letter groups, count distances between them Period will be a divisor of this distance Example: key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ Repeated “VTW” suggests keyword size of 3 or 9 9 Semester IFN642

Vigenère Cipher Cryptanalysis: Other calculations?
Index of Coincidence: Developed by William Friedman in 1920s Measures variation in character frequencies in the ciphertext Defined as the probability that two randomly selected characters in a message will be the same: 𝐼𝐶= 𝑖=0 26 𝑓 𝑖 (𝑓 𝑖 −1) 𝑁(𝑁−1) where 𝑓 𝑖 is frequency of ith character & N is message length Can calculate IC for various plaintext sources: Example: English language text => IC(English) ≈ Letter A B C D E … X Y Z Frequency 0.082 0.015 0.028 0.043 0.127 0.001 0.020 Semester IFN642

Vigenère Cipher Cryptanalysis: Index of Coincidence - expected values
IC values based on letter frequencies, so vary depending on: language used (English, French, German, … ) type of text (novel, recipe book, text book…) Examples for generic texts: Sometimes IC values ‘normalised’ by multiplying by the number of characters in the alphabet: Example: IC(English) ≈ x 26 = 1.734 Language: English French German IC 0.0667 0.0694 0.0734 Semester IFN642

Vigenère Cipher: Cryptanalysis
How to use IC to estimate period: Compute for sequences with various periods Compare results to expected values of IC Example: Source: Ciphertext: RSTCS JLSLR SLFEL GWLFI ISIKR MGL Calculate IC for keyword length 1: Begin with frequency count of characters: 𝐼𝐶= = =0.087 C E F G I J K L M R S T W 1 2 3 6 5 Semester IFN642

Example: using IC … continued Source: Ciphertext: RSTCS JLSLR SLFEL GWLFI ISIKR MGL Calculate IC for keyword length 2: Split into two ciphertext sets: Odds: RTSLL SFLWF IIRG Evens: SCJSR LEGLI SKML Perform frequency counts for each set of characters: Odds: 𝐼𝐶= = =0.0769 After similar calculation, Evens: IC = F G I L R S T W 2 1 3 Semester IFN642

Example: index of coincidence: Source: Ciphertext: RSTCS JLSLR SLFEL GWLFI ISIKR MGL Calculated IC for a range of keyword lengths: Keyword length likely to be length with highest average IC: 3 Divide up ciphertext assuming this length: Attack as three separate Caesar ciphers Length Indices of Coincidence Average IC 1 0.0760 2 0.0769, 0.0714 3 0.1111, , 0.1574 4 0.0476, , , 0.0476 Semester IFN642

Beaufort cipher Simple polyalphabetic substitution cipher
Created by Sir Francis Beaufort Similar to Vigenère, but involves subtraction Periodic substitution: Key is sequence of d characters: K = k0 k1 ... Kd-1 Use each alphabet in turn Repeat from start after processing d letters of message Encryption: C= Ek(P) = Ek0, k1, k2, ... kd-1(p0 p1 p2 ... pn-1) c0 = (k0 - p0) mod 26 c1 = (k1 - p1) mod 26 cd-1 = (kd-1 - pd-1) mod 26 cd = (k0 - pd) mod 26 ci = (ki mod d - pi) mod 26 Decryption: similar but pi = (ki mod d - ci) mod 26 Note: This is reciprocal cipher: encryption & decryption are same Semester IFN642

Polygraphic substitution
Operate on multiple characters at a time: Divide plaintext/ciphertext into blocks of d characters, Perform encryption/decryption of each block Simple substitution, but much larger alphabet as each item is composed of multiple characters Examples: Playfair cipher Hill cipher Semester IFN642

Playfair Cipher 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
Invented by Charles Wheatstone in 1854 named after his friend, Baron Playfair Polygraphic cipher with block size 2: Operates on pairs of characters (called digraphs) Key is a 5X5 matrix of letters based on a keyword (25 letters so set I=J) To construct key matrix: Write keyword removing duplicate letters Example: LETTERS -> LETRS Working from top left of the matrix, (L to R across row then down to next row) Write down letters of keyword Fill rest of matrix with remaining letters Example: matrix 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 Fill in the 5x5 matrix, L to R, top to bottom, first with keyword after duplicate letters have been removed, and then with the remaining letters, with I/J used as a single letter. This example comes from Dorothy Sayer's book "Have His Carcase", in which Lord Peter Wimsey solves this, and describes the use of a probabl word attack. Semester IFN642

Playfair Cipher Encryption
Divide plaintext into digrams: If a pair is a repeated letter, insert a filler like 'X’ Example: BALLOON is recorded as BA LX LO ON Use matrix to encrypt each two letter block: Find both letters in the matrix If both letters are: in the same row replace each with letter to right (wrapping back to start from end) Example: ar encrypts to RM (aR, rM) in the same column replace each with the letter below it (wrapping back to top from bottom), Example: mu encrypts to CM (mC, uM) 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 Note the various rules, and how you wrap from right side back to left, or from bottom back to top. Decrypting of course works exactly in reverse. Can see this by working the example pairs shown, backwards. Semester IFN642

Playfair Cipher Encryption
Continuing from previous slide: Use matrix to encrypt each two letter block: Find both letters in the matrix If both letters are: In the same row (previous slide) In same column(previous slide) Otherwise: each letter is replaced by the letter in its row, but in the column of the other letter of the pair Examples: hs encrypts to BP ea encrypts to IM or JM (as desired) Encryption Exercise: Encrypt your name (Use pencil and paper) 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 Semester IFN642

Playfair Cipher Decryption
Divide ciphertext into digraphs Use matrix to decrypt each digraph: Find both letters in the matrix If both letters are: In the same row replace each with letter to left (wrap if needed) Example: RM decrypts to ar In same column replace each with letter above (wrap if needed) Example: CM decrypts to mu Otherwise: each letter replaced by the letter in its row, but in the column of the other letter of pair Example: BP decrypts to hs Remove padding X and adjust i/j if needed 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 Semester IFN642

Playfair Cipher Security
Improved security compared to monoalphabetic: Substitution cipher But substituting digraphs rather than single characters 26 x 26 = 676 digrams instead of 26 letters (AA, AB, AC, …ZZ) Widely used for many years: US & British military used Playfair cipher in WW1 However: Can use frequency analysis to break this: Need a 676 entry frequency table to analyse, and correspondingly more ciphertext Can be broken given a few hundred letters of ciphertext it still has much of the plaintext structure (think digraph frequency) Semester IFN642

Playfair Cipher Cryptanalysis
Same plaintext digram always encrypts to same ciphertext digram Use this to form an attack: Compute digram frequencies for ciphertext Compare to known digram frequencies for plaintext language Match most frequent digrams May be able to reconstruct matrix Identify letters in same row, column, etc Semester IFN642

Hill Cipher Invented by Lester S. Hill in 1929 Polygraphic cipher
Character block size m: Operates on m-tuples of characters Encryption/decryption are mathematical operations: Linear combinations of plaintext/ciphertext and key (modulo 26) Key is mxm matrix (integers but may be obtained from a keyword) 𝐾= 𝑘 1,1 ⋯ 𝑘 1,𝑚 ⋮ ⋱ ⋮ 𝑘 𝑚,1 ⋯ 𝑘 𝑚,𝑚 Semester IFN642

Hill cipher Two representations used in literature to describe Hill cipher encryption, both use matrices: Encryption sometimes written as: 𝑐 1 ⋮ 𝑐 𝑚 = 𝑝 1 … 𝑝 𝑚 𝑘 1,1 ⋯ 𝑘 1,𝑚 ⋮ ⋱ ⋮ 𝑘 𝑚,1 ⋯ 𝑘 𝑚,𝑚 Or alternatively as: 𝑐 1 ⋮ 𝑐 𝑚 = 𝑘 1,1 ⋯ 𝑘 1,𝑚 ⋮ ⋱ ⋮ 𝑘 𝑚,1 ⋯ 𝑘 𝑚,𝑚 𝑝 1 ⋮ 𝑝 𝑚 CrypTool uses this second representation - we will too Semester IFN642

Hill cipher The corresponding decryption operations make use of the inverse of matrix K, denoted K-1: First representation: If 𝑐 1 ⋮ 𝑐 𝑚 = 𝑝 1 … 𝑝 𝑚 𝐾 then 𝑝 1 ⋮ 𝑝 𝑚 = 𝑐 1 … 𝑐 𝑚 𝐾 −1 Second representation (the one CrypTool uses): If 𝑐 1 ⋮ 𝑐 𝑚 =𝐾 𝑝 1 ⋮ 𝑝 𝑚 then 𝑝 1 ⋮ 𝑝 𝑚 = 𝐾 −1 𝑐 1 ⋮ 𝑐 𝑚 Semester IFN642

Hill Cipher Encryption example: Steps to follow:
Encrypt plaintext: HILL using Key K = Steps to follow: Note block size: m = 2 (since K is 2x2 matrix) Divide plaintext into m-character blocks: Block 1: (H,I) Block 2: (L,L) Now encode as integers: (H,I) -> (?,?) (L,L) -> (?,?) 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 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 Semester IFN642

Supporting Mathematics: Modular arithmetic
Recall from last week: Let a, b and n be integers with n  1: Modular addition: (a  b) mod n  ((a mod n)  (b mod n)) mod n Modular multiplication: (a x b) mod n  ((a mod n) x (b mod n)) mod n Semester IFN642

Hill Cipher Encryption example continued:
Use numerical coding of characters as input to encryption equation with 2x2 matrix K (calculate mod 26) 𝑐 1 𝑐 2 = ≡ 𝑐 3 𝑐 4 = ≡ ? ? Output is encoded digrams, convert to ciphertext characters: (11,25) -> (L,Z) (?,?) -> (?,?) Semester IFN642

Supporting Mathematics: Matrices
Recall multiplicative inverse for integers: The integer b is the multiplicative inverse of a mod n if (a x b) mod n  1 mod n Notation: a-1 denotes the multiplicative inverse of a It is similar for matrices: The identity matrix is I = 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 1 I has all diagonal entries =1 and all other entries = 0 The matrix K-1 is the inverse of matrix K if K.K-1 = I Hill cipher decryption needs inverse of the encryption matrix K Semester IFN642

Example: Our encryption matrix is: K = Multiply by the 2x2 identity matrix: K I = = =𝐾 Multiply by another matrix to obtain the identity: So this matrix is the inverse for K = ≡ Semester IFN642

Finding inverse of matrix K: Can do this mathematically (below) or use online tools All operations are modulo 26 Adjoint All operations are in modulo 26 Semester IFN642

Hill Cipher Decryption example: Use the inverse of matrix K:
Since K = and ≡ 𝐾 −1 = Use 𝐾 −1 to decrypt first ciphertext pair: (L,Z) 𝑝 1 𝑝 2 = = = ≡ 7 8 Then decode to obtain plaintext (H,I) Now use 𝐾 −1 to decrypt next ciphertext pair: (B,G) 𝑝 3 𝑝 4 = = = ≡ Semester IFN642

Hill Cipher Cryptanalysis: Example for 2x2:
Weakness: encryption operations are all linear So known plaintext attack easy given m ciphertext blocks: Set up matrix with variables for entries, Form linear equations and solve Example for 2x2: [ 𝑐1 𝑐2 ]= 𝑎 𝑏 𝑐 𝑑 𝑝 1 𝑝 2 Insert known values for pairs: Need 4 linear equations to solve to recover the 4 variables a, b, c, d. Two equations from (H,I) -> (L,Z) [ ]= 𝑎 𝑏 𝑐 𝑑 Two equations from (L,L) -> (B,G) [ 1 6 ]= 𝑎 𝑏 𝑐 𝑑 Then solve simultaneous linear equations and recover variables Semester IFN642

Hill Cipher Cryptanalysis: Known plaintext attack:
Need m known plaintext blocks to solve Hill cipher Sometimes in practice might need a little more to determine uniquely Once K is determined, can find K-1 and decrypt the remaining ciphertext Ciphertext only attack: Perform frequency analysis of non-overlapping ciphertext m-tuples Try to match with common plaintext m-tuples Then compute matrix, as on previous slide Semester IFN642

Rotor machines Improving cipher security:
Ciphers using substitution or transposition alone are not secure: Characteristics of plaintext language used as basis for attack Caesar Linear Playfair Hill Increase security by performing several operations in succession: Example: A substitution followed by a transposition Improved security because complicates frequency analysis Requires more effort to encrypt/decrypt Motivation to create mechanical encryption devices Semester IFN642

Rotor Machines Timing for development of cryptographic machines linked to major world events: wars Prior to WW1: Encryption, decryption and cryptanalysis usually performed manually Disadvantages: Time consuming, prone to error After WW1: Development of mechanical devices to automate encryption/decryption Many used rotors: cylinder or disc with characters on rim Code operator used a keyboard to enter message Device encrypted/decrypted automatically Output was displayed and code operator recorded it Examples: Hagelin, Enigma, Lacida, Green Machine, Purple Machine Semester IFN642

Rotor Machines Mechanical cryptographic devices on display in museums around the world, particularly defence related museums: Australian War memorial in Canberra has an Enigma machine on display in the WWII gallery  Crypto Museum is an excellent virtual museum for information on mechanical cryptographic devices: Semester IFN642

Enigma: The most famous rotor-based cryptographic device
German military used Enigma machine during World War II 3-5 rotors were commonly used Picture: A 3-rotor Enigma machine Semester IFN642

Rotors - the heart of the German Enigma machines.
Had to be set correctly before messages could be processed. Semester IFN642

Each cylinder represent a monoalphabetic substitution
Each cylinder has 26 input and 26 output pins Internal wiring connects each input pin to a unique output pin For every complete rotation of inner cylinder, middle cylinder rotates one. For every complete rotation of middle cylinder, outer cylinder rotates one. Semester IFN642

Enigma Operating Enigma machine:
Prior to use: operator had to insert rotors and set key Military use: had a new key each day May also have had plugboard connections to set Encryption: Operator presses plaintext character key on keyboard Electrical current ran through wires in rotors Bulb glows under letter on lamp panel Operator notes which character is illuminated: this is corresponding ciphertext character Semester IFN642

Enigma Operating Enigma machine: Symmetric and reciprocal cipher:
same key and same process used for decryption Decryption: Operator sets up rotors using key Operator presses ciphertext character key on keyboard Electrical current ran through wires in rotors Bulb glows under letter on lamp panel Operator notes which character is illuminated: this is corresponding plaintext character Semester IFN642

Enigma Security of Enigma machine: Each rotor: Combining these:
Has 26 electrical contacts on each face (input & output) Internal wiring from contact on one side to other Implements a monoalphabetic substitution cipher Combining these: Output from one rotor is input to next For 3 rotors, 263 = 17,576 different substitutions possible Choice or ordering of rotors results in different subsequent substitutions Rotor movement - for a given choice of 3 rotors: Move through 263 = 17,576 substitutions before we are back to original setting Large period defeats frequency analysis Enigma was broken by Polish and British during WW2 Semester IFN642

Enigma CrypTool2 Task: Look in Classical cryptography templates for Enigma, Open template and start the cipher machine Expand the Enigma component to full screen, and watch the animation Semester IFN642

Summary Evolution of ciphers: Simple monoalphabetic substitution
Easily defeated by frequency analysis Polyalphabetic substitution ciphers: Examples: Vigenère, Beaufort Alter single character frequency distribution Cryptanalysis possible - need to determine period first, then divide up ciphertext and revert to multiple simple ciphers Polygraphic substitution ciphers: Examples: Playfair, Hill Cipher Have larger alphabet: in analysis consider digraphs and trigraphs To increase security, need to apply multiple operations: Multiple iterations and more complex operations – takes longer Rotor machines: mechanical devices to implement this Example: Enigma Semester IFN642

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