CS 241 Data Organization Ciphers March 22, 2018 Cipher In - - PowerPoint PPT Presentation

CS 241 Data Organization Ciphers

March 22, 2018

Cipher

• In cryptography, a cipher (or cypher) is an

algorithm for performing encryption or decryption.

• When using a cipher, the original information is

known as plaintext, and the encrypted form as ciphertext.

• The encrypting procedure of the cipher usually

depends on a piece of auxiliary information, called a key.

• A key must be selected before using a cipher to

encrypt a message.

• Without knowledge of the key, it should be

difficult, if not nearly impossible, to decrypt the resulting ciphertext into readable plaintext.

Substitution Cipher

• In cryptography, a substitution cipher is a

method of encryption by which units of plaintext are replaced with ciphertext according to a regular system.

• Example: case insensitive substitution cipher

using a shifted alphabet with keyword ”zebras”:

• Plaintext alphabet:

ABCDEFGHIJKLMNOPQRSTUVWXYZ

• Ciphertext alphabet:

ZEBRASCDFGHIJKLMNOPQTUVWXY

flee at once. we are discovered! Enciphers to SIAA ZQ LKBA. VA ZOA RFPBLUAOAR!

Other substitution ciphers

Caesar cipher Shift alphabet by fixed amount. (Caesar apparently used 3.) ROT13 Replace letters with those 13 away. Used to hide spoilers on newsgroups. pigpen cipher Replace letters with symbols.

Substitution Cipher: Encipher Example

OENp(ENTE#X@EN#zNp(ENCL]pEnN7p-pE;8N]LN} dnEdNp#Nz#duN-Nu#dENXEdzE9pNCL]#L8NE;p-b @];(N0G;p]9E8N]L;GdENn#uE;p]9Nld-L/G]@]p _8NXd#|]nENz#dNp(EN9#uu#LNnEzEL;E8NXd#u# pENp(ENQELEd-@NOE@z-dE8N-LnN;E9GdENp(EN^ @E;;]LQ;N#zN<]bEdp_Np#N#Gd;E@|E;N-LnN#Gd NT#;pEd]p_8Nn#N#dn-]LN-LnNE;p-b@];(Np(]; N5#L;p]pGp]#LNz#dNp(ENCL]pEnN7p-pE;N#zN) uEd]9-D

Breaking a Substitution Cipher

In English,

• The most common character is the space: “ ”.
• Letters in order of frequency are:

ETAONRISHDLFCMUGYPWBVKXJQZ

• Letter pairs in order of frequency are: TH HE

AN RE ER IN ON AT ND ST ES EN OF TE ED OR TI HI AS TO

• Doubled letters in order of frequency are: LL

EE SS OO TT FF RR NN PP CC MM

Substitution Cipher: plaintext & Map

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defense, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America. =51 e=39 t=28

• =24

[ N] [!%] ["Z] [#1] [$f] [%=] [&r] [’I] [( ] [)U] [*,] [+a] [,8] [-m] [.D] [/y] [0P] [1’] [2\] [33] [4h] [5?] [6t] [7K] [8"] [9W] [:.] [;c] [<:] [=o] [>F] [?{] [@R] [A)] [B^] [C5] [Dj] [EA] [Fv] [GM] [H$] [IY] [J0] [Ke] [L<] [Mq] [NH] [O}] [PT] [Q+] [R‘] [S7] [Tl] [UC] [Vx] [WO] [X&] [Y[] [Z2] [[g] [\>] []s] [^J] [_!] [‘V] [a-] [bb] [c9] [dn] [eE] [fz] [gQ] [h(] [i]] [j4] [ki] [l@] [mu] [nL] [o#] [pX] [q/] [rd] [s;] [tp] [uG] [v|] [wS] [x*] [y_] [z6] [{k] [|B] [}w]

Substitution Map Generator: Globals

#include <stdio.h> #define ASCII_START 32 #define ASCII_END 126 #define MAP_SIZE 94 char map[MAP_SIZE ];

• Program that creates a substitution map based
• n the first letter of the plaintext message.
• The program creates a one-to-one map of all

printable ASCII characters.

Substitution Map Generator: buildMap

void buildMap(char seed) { int i; int m=94; int a=189; int c=53; int n=seed; for (i=0; i<MAP_SIZE; i++) { n = (a*n + c) % m; map[i] = n+ASCII_START; } }

• seed: first number in the

pseudo-random sequence.

• m, a, c: constants chosen to

give pseudo-random behaviour.

• for loop body: generates a

pseudo-random number between 0 and m, maps character i+ASCII_START to

n+ASCII_START

Substitution Map Generator: printMap

void printMap(void) { int i; for (i=0; i<MAP_SIZE; i++) { printf("[%c%c] ", i+ASCII_START , map[i]); if ((i % 16) == 15) printf("\n"); } printf("\n"); }

Substitution Map Generator: main

void main(void) { char c=getchar (); buildMap(c); printMap (); int i = 0; while (c!= EOF) { if (c < ASCII_START) break; if (c > ASCII_END) break; if (i % 40 == 0) printf("\n"); printf("%c", map[c-ASCII_START ]); c=getchar (); i++; } printf("\n"); }

Book Cipher

A book cipher is a cipher in which the key is some aspect of a book or other piece of text. A message is typically encoded by three numbers for each letter:

• Page
• Line
• Word

Where the encoded letter is the first letter of the specified word. It is typically essential that both correspondents not

• nly have the same book, but the same edition.

One-time pad

• Random key (or “pad”) is at least as long as

the text.

• Add plaintext to key (modulo 26) to encrypt.
• Subtract key from ciphertext to decrypt.
• Destroy the key after use.
• Problems:
• Truly random key is hard to produce
• How to exchange the key securely?
• Never ever use the key again.
• Modern stream ciphers mimic the one-time

pad.

Modern Cipher Categories

Stream Ciphers Applied to a continuous stream of

• symbols. Algorithms applied to small

blocks (1 to 16 bytes) are often still called stream ciphers. Block Ciphers Applied to blocks of symbols. Symmetric Key Algorithms The same key is used for both encryption and decryption. For example: RC4 (used in Secure Sockets Layer (SSL) ). Asymmetric Key Algorithms A different key is used for both encryption and decryption. For example: RSA which uses public / private key pairs.

RSA: Key Generation

• 1. Choose two distinct prime numbers p and q.

The primes should be chosen uniformly at random and should be of similar bit-length.

• 2. Compute the divisor, n = pq, to be used in the

modulus operation.

• 3. Compute φ(pq) = (p − 1)(q − 1). The totient,

φ, of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n.

• 4. Choose an integer e such that 1 < e < φ(pq),

and e and φ(pq) are coprime.

• 5. Find d: (ed − 1) can be evenly divided by

φ(p − 1)(q − 1).

• 6. The public key consists of the modulus n and e.
• 7. The private key consists of the modulus n and

d.

Linear Congruential Generator (LCG)

A Linear Congruential Generator is one of the oldest and best known pseudorandom number generator algorithms: Xn+1 = (aXn + c) mod m where Xn is the sequence of pseudorandom values, and

• Modulus: m, 0 < m
• Multiplier: a, 0 < a < m
• Increment: c, 0 ≤ c < m
• Seed: X0, 0 ≤ X0 < m

LCG example

Example: Xn+1 = (7Xn + 11) mod 18, X0 = 0. X1 = (7(0) + 11) mod 18 = 11 X2 = (7(11) + 11) mod 18 = 16 X3 = (7(16) + 11) mod 18 = 15 X4 = (7(15) + 11) mod 18 = 8

LCG with Pseudorandom Behavior

Xn+1 = (aXn + c) mod m

• All of the values, a, c, m, and Xn used in an

LCG are integers.

• The sequence of integers in an LCG can never

have a period greater than m. Why?

• Some values of the modulus, multiplier and

increment yield a sequence with maximum period and with good pseudorandom behavior.

LCG with Pseudorandom Behavior

An LCG will have a full period if and only if:

• c and m are coprime (have no common factor

> 1)

• a − 1 is divisible by all prime factors of m,
• a − 1 is a multiple of 4 if m is a multiple of 4.

LCGs in Common Use

Source m a c Numerical Recipes 232 1664525 1013904223 Borland C/C++ 232 22695477 1 glibc (used by GCC) 231 1103515245 12345 ANSI C: Watcom, Digital Mars, CodeWarrior, IBM Visu- alAge C/C++ 231 1103515245 12345 Borland Delphi, Virtual Pascal 232 134775813 1 Microsoft Visual/Quick C/C++ 232 214013 (343FD16) 2531011 (269EC316) Microsoft Visual Basic (6 and earlier) 224 1140671485 (43FD43FD16) 12820163 (C39EC316) RtlUniform from Native API 231 − 1 2147483629 (7FFFFFED16) 2147483587 (7FFFFFC316) Apple CarbonLib 231 − 1 16807 MMIX by Donald Knuth 264 6364136223846793005 1442695040888963407 Newlib 264 6364136223846793005 1 VAX’s MTH$RANDOM, old versions of glibc 232 69069 1 Java’s java.util.Random 248 25214903917 11 LC53 in Forth 232 − 5 232 − 333333333 Note: LCG’s do not always return all of the bits in the values they produce. Example: Java produces 48 bits, but only returns the 32 most significant. table from http://en.wikipedia.org/wiki/Linear_congruential_generator

Linear Congruential Generator in C

#include <stdio.h> void main (void) { unsigned long m = 1 << 31; unsigned long a = 1103515245; unsigned long c = 12345; unsigned long x = 123456; int i; for (i=1; i <=100; i++) { x = (a*x + c) % m; printf("%20lu\n", x); } }

136235578099065 15519887047211639486 13838650204930187039 3346717808905085548 3049859088482533429 7398829542172257482 15952872801263783483 9897407542008592728 12436645181455133361 8967167257519066006 15551416184959193879 14392011937321128708 7012146815830884589 837494824685357858 9089108537697995187 4486637365379619696 10706175307392302825 3333036225721595758 15798734497693109775 1184521088365346460 762097951296315557

Toy Linear Congruential Generator

#include <stdio.h> void main (void) { int m = 18; int a = 7; int c = 1; int x = 0; int i; for (i=1; i <=28; i++) { printf("%4d\n", x); x = (a*x + c) % m; } } 1 8 3 4 11 6 7 14 9 10 17 12 13 2 15 16 5

Poor Linear Congruential Generator

void main(void) { int m = 18; int a = 6; int c = 1; int x = 0; int i; for (i=1; i <=14; i++) { printf("%4d\n", x); x = (a*x + c) % m; } }

This LCG does NOT have a full period. How can this be known before computing it? (6 × 7 + 1) mod 18 (42 + 1) mod 18 43 mod 18 7 1 7 7 7 7 7 7 7 7 7 7

Question: Linear Congruential Generator

#include <stdio.h> void main(void) { int m = 10; int a = 2; int c = 1; int x = 7; int i; for (i=1; i <=4; i++) { printf("%d ", x); x = (a*x + c) % m; } printf("\n"); }

The output is: A: 7 5 1 3 B: 7 5 7 4 C: 7 5 6 7 D: 7 5 2 6 E: 7 5 3 7

XOR Cipher Lab

The cipher used in this project:

• Encrypts plain text messages consisting solely
• f the printable ASCII characters.
• Decrypts ciphertext consisting solely of the

printable ASCII characters.

• The printable ASCII characters are 8-bit codes

in the range of values from 32 to 126 (00100000 through 01111110). See http://en.wikipedia.org/wiki/ASCII

• Any 8-bit sequences outside of this range

constitutes invalid data.

Cipher Record Format

Each cipher record must be of the form: Action lcg m , lcg c , Data \n 1 char 1-20 char 1 char 1-20 char 1 char any # of char 1 char

Action: Must be either ‘e’ or ‘d’ specifying encryption or decryption respectively. lcg m: Specifies a 64-bit positive integer used for m of a Linear Congruential Generator. Must be decimal digits. lcg c: Specifies a 64-bit positive integer used for c of a Linear Congruential Generator. Must be decimal digits. Data: Printable ASCII character data to be encrypted or decrypted. Can be empty or arbitrarily long. Note: with the given algorithm there will be no need to keep the full line of data in memory.

Cipher Algorithm: Summary

• 1. Determine the Linear Congruential Generator

specified by the given key. This is done only

• nce per line of input.
• 2. Read 1 byte of data b
• 3. Generate random value x
• 4. Compute encrypted byte as b XOR (x mod

128)

• 5. Deal with any non-printable ASCII characters.
• 6. Print the resulting character(s) to the standard
• utput stream.
• 7. Return to step 2 and continue until the end of

the line

Cipher Algorithm: Initialization

Given a 128-bit symmetric key consisting of:

• m : a 64-bit positive integer used as an LCG

modulus,

• c : a 64-bit positive integer used as an LCG

increment. Define a Linear Congruential Generator: Xn+1 = (aXn + c) mod m Where

• X0 = c
• a = 1 + 2p, if 4 is a factor of m, otherwise,

a = 1 + p.

• p = product of m’s unique prime factors

Reading Data Bytes

• Within each record, the data portion is the set
• f characters between the second comma and

’\n’.

• When encrypting: Use getchar() to read 1

byte of data to encrypt.

• When decrypting: Reading in an encrypted

byte may require reading 2 bytes from standard input since some character codes have 2-byte ciphertext representations.

• Any character in the data segment not in the

range of printable ASCII characters (32 to 126) is an error.

Non-printable ASCII characters

This cipher algorithm can generate target bytes that are outside the range of printable ASCII.

• When encrypting, if a generated byte e is:

< 32 Replace with two bytes: ’*’ and ’?’+e. = 127 Replace with two bytes: ’*’ and ’!’. =’*’ Replace with two bytes: ’*’ and ’*’.

• When decrypting, if a generated byte p is a

non-printing ASCII character, then print the specified error message and read to the end of line.

Output Format

• For every line of input, one line of output must

be sent to the standard output stream.

• If the input line is invalid, then the output has

the form:

("%5d) %s Error\n", inputLineNumber, trash) where

trash is any string no longer than twice the length any data part of the line.

• If the input line is valid, then the output has

the form:

("%5d) %s\n", inputLineNumber, outStr)

Where outStr is the encrypted or decrypted data.

Example: Read Line

Input: e126,25,Byte\n

• Given Key: m = 126 and c = 25
• Calculate: a = 43

(2 × 3 × 7+1 since m = 126 = 2 × 3 × 3 × 7)

• LCG: Xn+1 = (43(Xn) + 25) mod 126 with

X0 = 25

• Data: Byte

B 1 1 y 1 1 1 1 1 t 1 1 1 1 e 1 1 1 1

LCG Example: Generating random values

You’ll be generating

• ne value at a time in

your program, I’m just demonstrating the sequence here. In this example, we have Xi mod 128 = Xi since m = 126 < 128 This will not be the case with larger LCG values. i Xi Xi mod 128 25 25 1 92 92 2 75 75 3 100 100 4 41 41 5 24 24 6 49 49 7 116 116 8 99 99 . . . . . . . . .

Example: Encrypting

Data: B 66 1 1 Xi mod 128 25 1 1 1 encrypted [ 91 1 1 1 1

Example: Encrypting

Data: B 66 1 1 Xi mod 128 25 1 1 1 encrypted [ 91 1 1 1 1 Data: y 121 1 1 1 1 1 Xi mod 128 92 1 1 1 1 encrypted % 37 1 1 1

Example: Encrypting

Data: B 66 1 1 Xi mod 128 25 1 1 1 encrypted [ 91 1 1 1 1 Data: y 121 1 1 1 1 1 Xi mod 128 92 1 1 1 1 encrypted % 37 1 1 1 Data: t 116 1 1 1 1 Xi mod 128 75 1 1 1 1 encrypted ? 63 1 1 1 1 1 1

Example: Encrypting

Data: B 66 1 1 Xi mod 128 25 1 1 1 encrypted [ 91 1 1 1 1 Data: y 121 1 1 1 1 1 Xi mod 128 92 1 1 1 1 encrypted % 37 1 1 1 Data: t 116 1 1 1 1 Xi mod 128 75 1 1 1 1 encrypted ? 63 1 1 1 1 1 1 Data: e 101 1 1 1 1 Xi mod 128 100 1 1 1 encrypted *@ 1 1

Example: Encrypting

Data: B 66 1 1 Xi mod 128 25 1 1 1 encrypted [ 91 1 1 1 1 Data: y 121 1 1 1 1 1 Xi mod 128 92 1 1 1 1 encrypted % 37 1 1 1 Data: t 116 1 1 1 1 Xi mod 128 75 1 1 1 1 encrypted ? 63 1 1 1 1 1 1 Data: e 101 1 1 1 1 Xi mod 128 100 1 1 1 encrypted *@ 1 1 Byte encrypts to [%?*@

Finding Unique Prime Factors - Algorithm

• 1. Let n be the number of which to find the prime

factors.

• 2. Start with 2 as a test divisor.
• 3. If the test divisor squared is greater than the

current n, then the current n is either 1 or

• prime. Save it if prime and return.
• 4. If the remainder of n divided by the test divisor

is zero, then:

• a. The test divisor is a prime factor of n. Save it.
• b. Replace n with n divided by the test divisor.
• c. Repeat b until test divisor is no longer a factor of n.
• 5. If, in step 4, the remainder of n divided by the

test divisor was not zero, then increment the test divisor.

• 6. Loop to step 3.

How Many Prime Factors Can n have?

Write a function that finds the unique prime factors

• f a number, n, and stores each factor in an array.

How big does the array need to be?

• If n is a 64-bit integer then the largest number

it can hold is 264 − 1.

• Since 2 is the smallest prime, the largest 64-bit

number must have less than 64 factors.

How Many Prime Factors Can n have?

Write a function that finds the unique prime factors

• f a number, n, and stores each factor in an array.

How big does the array need to be?

• If n is a 64-bit integer then the largest number

it can hold is 264 − 1.

• Since 2 is the smallest prime, the largest 64-bit

number must have less than 64 factors.

• But only Unique Prime Factors are needed.
• The product of the first 15 primes is:

2×3×5×7×11×13×17×19×23×29×31×37× 41 × 43 × 47 = 11, 682, 905, 869, 181, 336, 790. Times 53 > 264 − 1.