Ones and Zeros Early Computers John Napier (1550

Early Computers

The definition of a Ones and Zeros computer is “a device that computes, …” The abacus could be considered the first Chapter 12 computer. Its effectiveness has withstood the test of time and it is still used in some parts of the world.

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John Napier (1550 - 1617) Napier’s Bones

In 1617, Scottish Napier's Bones were mathematician John multiplication tables Napier invented a inscribed on strips of technique of using wood or bone. numbering rods for Napier’s idea led to the multiplication. invention of the slide rule The rods became known in the mid 1600s by as "Napier's Bones." William Oughtred.

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The Pascaline The Pascaline

In 1642, Blaise Pascal As they agonized over the columns of figures, Pascal invented his "Pascaline" decided to build a machine that would do the job much faster and more accurately. calculator. His machine could add and subtract. Pascal, a mathematician The machine used a series of toothed wheels, which and philosopher, and his were turned by hand and which could handle numbers father, a tax official, were up to 999,999.999. compiling tax reports for Pascal’s device was also called the “numerical the French government in calculator” and was one of the world’s first mechanical adding machines. Paris.

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1 Gottfried Wilhelm von Leibniz The Step Reckoner

In 1671, Gottfried von This machine could add, subtract, multiply, divide and figure Leibniz invented the square roots through a series of stepped additions. "Stepped Reckoner" Although the machine did not become widely used, almost every which used a special type mechanical calculator build during the next 150 years was based of gear named the on it. stepped drum, also known as, the Leibniz Wheel. His device was not built until about 1694.

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Jacquard’s Loom Jacquard’s Loom

In 1805, Joseph-Marie Jacquard built a loom controlled Jacquard’s loom emphasized three concepts by punched cards. important in computer theory. Heavy paper cards linked in a series passed over a set of rods on a loom. First, the information could be coded on punched The pattern of holes in the cards determined which rods cards. were engaged, thereby adjusting the color and pattern of the product. Second, the cards could be linked in a series of instructions – essentially a program – allowing a Prior to Jacquard’s invention, a loom operator adjusted the loom settings by hand before each glide of the machine to do work without human intervention. shuttle, a tedious and time-consuming job. Third, the programs could automate jobs.

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The Stored Program Charles Babbage (1791 - 1871)

A program is a sequence of instructions for the Charles Babbage was computer to follow born in London, England. Also called “software” He entered Trinity College, Cambridge in Hardware: the chips, wires, switches, etc. on which the October 1810, and was software instructions are executed. recognized for his Primitive example: the Jacquard Loom exceptional mathematical The loom was the hardware abilities. The weaving pattern cards was the software Babbage is considered to The “program” was “stored” on punched cards be the “Father of Computing”

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2 Charles Babbage (1791 - 1871) Charles Babbage (1791 - 1871)

In 1827 Babbage became Lucasian Professor of On 13 July 1823 Babbage received a gold medal from Mathematics at Cambridge, a position he held for 12 the Astronomical Society for his development of the years, although he never taught. difference engine. He had become engrossed in the main passion of his life – the development of a mechanical computer. By 1834 Babbage had completed the first drawings of Babbage began to construct a small difference engine in the analytical engine, the forerunner of the modern 1819 and had completed it by 1822. electronic computer. Babbage illustrated what his small engine was capable Babbage discussed the concept of using punched cards of doing by calculating successive terms of the sequence n2 + n + 41. to control the operation of his calculating machine, but he was never able to build a full working model.

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Charles Babbage (1791 - 1871) Charles Babbage (1791 - 1871)

Babbage’s ideas embodied five key concepts of Although Babbage died before he could construct modern computers: his machine, such a computer has been built 1. Input device following Babbage's own design criteria. 2. Processor or number calculator In 1834 Babbage published his most influential 3. Control unit to direct the task to be performed and the sequence of calculations. work On the Economy of Machinery and 4. Storage place to hold number waiting to be Manufactures, in which he proposed an early processed form of what today we call operations research. 5. Output device

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Lady Augusta Ada (1815 - 1852) Lady Augusta Ada (1815 - 1852)

Augusta Ada was the Ada Byron was fascinated by a demonstration of daughter of the poet, Lord Babbage's Difference Engine. Byron. Babbage found that Ada understood and could explain Her mother encouraged the workings of his machine better than anyone else. her to study mathematics Babbage later toured Europe giving lectures on his more and music. advanced concept, the Analytical Engine, a machine She was a bright which he never fully constructed. mathematician who was Extensive notes were taken of some of these lectures in introduced to Charles French. Babbage in 1833.

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3 Lady Augusta Ada (1815 - 1852) Lady Augusta Ada (1815 - 1852)

Ada translated the French notes into English and When Augusta was 19, she married the wealthy Lord added a lengthy addendum. King, Baron of Lovelace. At Babbage's request, she published her notes. Her husband had some knowledge of mathematics, but it She had a unique grasp of the concepts of was Ada who urged him to provide funding to Babbage programming subroutines, loops and jumps. when his government funding ceased. Ada also had an interest in gambling, and attempted to She is often referred to as the first programmer. apply some of Babbage's technology to that end, but She was instrumental in clarifying and preserving without great success. information on Charles Babbage and his work.

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Lady Augusta Ada (1815 - 1852) Mechanical Computers

In recognition of her contributions to the field of Mechanical calculators built from cogs and gears programming theory, the U.S. Department of Defense named their programming language for reducing Difference Engine software development and maintenance costs, known Analytical Engine (General Purpose) only as DoD-1 up to that point, "Ada" in her honor in May of 1979. The "Difference Engine" was massive steam powered Ada is one of the main characters in the “alternate mechanical calculator designed to print astronomical history” novel The Difference Engine by Bruce Sterling tables. and William Gibson, which imagines a world in which Babbage's machines were mass produced and the The Analytical Engine was a mechanical computer computer age started a century early. designed to run using punch cards.

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The Difference Engine The Analytical Engine

Designed to solve accuracy problem First true ancestor of the modern computer would have computed tables and output results directly onto printing plates key idea: automatically feed results back into the difference engine store numbers and intermediate results for any kind of function at all 1822: partial working model Gear-based 2 orders of difference 6-figure numbers Basic components store (memory) only computing machine Babbage ever completed mill (arithmetic unit) Government research grant (1823) control barrel (primitive program control unit) abandoned support in 1842 cycle time: around 3 seconds per operation The Scheutz difference engine (Sweden, 1854) program stored on punched cards idea came from Jacquard loom irony: British government bought one planned: steam power

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4 Electromechanical Computers Electromechanical Computers

Herman Hollerith Alan Turing Developed a punched card tabulating machine Computer theorist Used for the 1890 census Worked on the Colossus, used to decrypt German military messages, WW2 His company was one of several which began IBM Grace Hopper – worked as a coder on the Konrad Zuse Harvard Mark I Proposed use of vacuum tubes for switching of binary It used electrical relays circuits Sponsored by US Navy to compute navigational Hitler refused to fund his design tables

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Early Electronic Computers Early Electronic Computers

The ABC computer The ENIAC The Atanasoff-Berry Computer was the world's first electronic In 1946, John Mauchly and J Presper Eckert developed the digital computer. ENIAC I (Electrical Numerical Integrator And Calculator). It was built by John Vincent Atanasoff and Clifford Berry at The ENIAC computer was intended to be a general purpose Iowa State University to do math & physics calculations. one, but it was also designed for a very specific task, namely It incorporated several major innovations in computing compiling tables for the trajectories of bombs and shells. including the use of binary arithmetic, regenerative memory, Used 18,000 vacuum tubes, caused lights to dim in parallel processing, and separation of memory and computing Philadelphia neighborhoods when turned on. functions. Programmed by rewiring panels.

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Early Electronic Computers The von Neumann Architecture

The UNIVAC (UNIversal Automatic Computer) John von Neumann Built by John Mauchly and J Presper Eckert Inventor the concept of a stored-program computer (a computer whose program was stored in computer memory). Used by the Census Bureau in 1950s The von Neumann architecture is the basis of the digital This was the first computer to be produced commercially in computer as we know it today. the United States with 46 UNIVACs being built. A von Neumann Architecture computer has five parts: The UNIVAC computer was used to predict the results of the an arithmetic-logic unit (ALU) Eisenhower-Stevenson presidential race. The computer had correctly a control unit (CU) predicted that Eisenhower would win, and it was considered amazing memory that a computer could do what political forecasters could not. The UNIVAC quickly became a household name. some form of input/output (I/O) and a bus that provides a data path between these parts.

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5 The von Neumann Architecture Computers

A von Neumann Architecture computer performs the following Computers are built on two key principles steps: 1. Fetch the next instruction from memory at the address in the program Both instructions and data are represented by counter. numbers 2. Add the length of the instruction to the program counter. Instructions and data are stored in memory and are 3. Decode the instruction using the control unit. The control unit commands the rest of the computer to perform some operation. The instruction may read and written as numbers change the address in the program counter, permitting repetitive operations. The instruction may also change the program counter only if All computers use the binary number system some arithmetic condition is true, giving the effect of a decision, which (base-2) can be calculated to any degree of complexity by the preceding arithmetic and logic. basic nature of electronic circuits ON/OFF, current 4. Go back to step 1. flow/does not flow, 1 is +5V and 0 is 0V.

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Number Systems Number Systems

Let’s review our decimal system (or base 10). Numbers can be represented in any base B (not We use 10 symbols (the digits), 0, 1, 2, 3, 4, 5, 6, just base 10). 7, 8, and 9, to represent all numbers. Using B symbols, 0, 1, 2, 3, 4, …, B – 1, to Each digit in a number has a precise place value represent all numbers. which is a power of 10. Each digit in a number has a precise place value Place values are powers of 10 starting on the which is a power of B. right with 100 and increasing toward the left, 101, Place values are powers of B starting on the right 102, 103, etc. with B0 and increasing toward the left, B1, B2, etc.

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Number Systems The Decimal Number System

The most common number systems associated Base 10 with computers are: Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 747510 3 2 1 0 Decimal Base = 10 10 10 10 10 Binary Base = 2 7 4 7 5 Octal Base = 8 The place value of each digit is determine by its position within the number. Hexadecimal (Hex) Base = 16 The digit 7 in the left-most position of 7475 counts for Let’s look at each one in more detail. 7000 and the digit 7 in the second position from the right counts for 70.

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6 The Binary Number System The Octal Number System

Base 2 Base 8 Digits: 0, 1 Place Values Digits: 0, 1, 2, 3, 4, 5, 6, 7 Place Values Example: 1574 Example: 10112 8 4 2 1 8 512 64 8 1 23 22 21 20 83 82 81 80 1 0 1 1 1 5 7 4

The digit 1 in the second position from the right The digit 7 in the second position from the right represents the value 2 and the digit 1 in the fourth represents the value 7×8 = 56 and the digit 1 in the position from the right represents the value 8. fourth position from the right represents the value 512.

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The Hexadecimal Number System The Binary Number System

Place Values Base 16 Computers use the binary number system or Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F base-2 to represent everything (numbers, letters, and instructions). Example: 1BAD16 4096 256 16 1 163 162 161 160 Although, Atanasoff was the first to build a 1 B A D modern computer that used this idea, he was not the first to think of it. The digit B in the third position from the right represents the value 11×256 = 2816 and the digit D in the first In fact the Egyptians used the powers of 2 in their position from the right represents the value 13. multiplication and division.

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Leibniz and Leibniz and The Binary Number System The Binary Number System Gottfried Wilhelm von Leibniz (co-father of Calculus) first Binary numbers had been known in India and in formally described the binary number system and its China 1500 years earlier. operations in writing in 1679. In India they were used in music to classify meter. Leibniz also philosophized about a computer based on a Leibniz believed that binary numbers represented binary numerical system. Creation In 1679 he wrote, "Despite its length, the binary system, The number 1 portraying God and 0 depicting in other words counting with 0 and 1, is scientifically the voidness. most fundamental system, and leads to new discoveries. He considered the binary 111 as an symbol of When numbers are reduced to 0 and 1, a beautiful order prevails everywhere." trinity.

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7 Bits and Bytes Bits and Bytes

What is a bit? What is a byte? A bit is a number (a place) in a set of binary numbers. A byte is 8 bits or 2 nibbles (4 bits). The bigger the number, the more bits there will be A byte is the amount of storage required to store a single A bit or binary digit is the smallest element a computer character, such as Q. can deal with and is either 1 or 0. It is the smallest data item in the microprocessor. A binary number consists of several bits. Written using variables, we have The right-most bit is called the LSB (least significant bit). b7 b6 b5 b4 b3 b2 b1 b0 The left-most bit is called the MSB (most significant bit) b7 is the MSB and also called the high order bit. or the HOB (high order bit). b0 is the LSB and also called the low order bit.

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The Binary Number System Converting Binary to Decimal

Simplest number system is base 2, or binary Example: Convert 1011.1012 to decimal Uses the 2 digits (“bits”) 0 and 1 Solution: 1011.1012 Used exclusively in computers (ON/OFF switches, = (1×23) + (0×22) + (1×21) + (1×20) magnetized/demagnetized memory elements) + (1×2–1) + (0×2–2) + (1×2–3) A typical binary number is 1011.101 2 = 8 + 2 + 1 + 0.5 + 0.125 The subscript 2 denotes the base – the base should be included if it is not 10 = 11.625 Exercise: Convert 110001.0112 to decimal

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Converting Decimal to Binary Decimal to Binary - A Better Way

We’ll begin by converting integers Previous method is awkward for large numbers. Example: Convert 183 to binary A better method is to repeatedly divide by 2, writing down the quotient and remainder at each step, until the Solution: Note that the powers of 2 are 1, 2, 4, 8, 16, 32, quotient is zero. 64, 128, 256, 512, …. Now write down the remainders in reverse order – this is Now write 183 using just these powers. the binary form of the integer Thus 183 = 128 + 55 Example: Convert 91 to binary = 128 + 32 + 23 = 128 + 32 + 16 + 7 Exercise: Convert 212 to binary

= 128 + 32 + 16 + 4 + 2 + 1 = 101101112 Answer: 212 = 110101002

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8 Decimal to Binary - A Better Way Addition in Base 2

Operation Quotient Remainder The addition table 91 ÷ 2 = 45 1 appears at the right. + 0 1 45 ÷ 2 = 22 1 Basically, it is pretty 22 ÷ 2 = 11 0 simple: 11 ÷ 2 = 5 1 0 + 0 = 0 and 0 0 1 5 ÷ 2 = 2 1 1 + 0 = 0 + 1 = 1 2 ÷ 2 = 1 0 It is important to remember that 1 + 1 = 0 1 ÷ 2 = 0 1 1 1 10 with a 1 carry. 91 = 10110112

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Binary Addition Addition of Binary Numbers

(a)0 (b) 0 Start at the least significant digit and add digits + 0 + 1 (bits) in the same column and carry anything over 0 1 1 (the largest binary digit) to be added to the column of the next most significant bits. (c)1 (d) 1 For example: 111 + 0 + 1 1011 1 1 0 + 1001 Carry Bit 10100

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Binary Addition Examples Binary Complements (a)1011 (b)1010 (c) 1011 There are two kinds of complements one’s + 1100 + 100 + 101 complement and two’s complement. 10111 1110 10000 One’s complement is simply done by changing the 1’s to 0’s and the 0’s to 1’s. (d)101 (e) 10011001 Examples: + 1001 + 101100 0 1 0 1 1 0 0 1 becomes 1 0 1 0 0 1 1 0 1110 11000101 1 1 1 0 0 1 0 1 becomes 0 0 0 1 1 0 1 0

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9 Two’s Complement Binary Subtraction

The two’s complement of a binary number is Binary subtraction is implemented by adding the two’s obtained by first complementing the number and complement of the number to be subtracted. then adding 1 to the result. Example: 1001110 1101 two’s 1101 – 1001 complement + 0111 0110001 One’s Complement of 1001 10100 + 1 If there is a carry in the HOB then it is ignored. Thus, the 0110010 Two’s Complement answer is 0100.

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Multiplication in Base 2 Binary Multiplication Examples (a) (c) The multiplication table 1011 1001 appears at the right. × 0 1 × 11 × 101 (b) 110 Basically, it is pretty 1011 1001 simple, 0 times anything × 101 10110 0 is 0 and 1 times anything 110 is the anything. 0 0 0 100001 100100 0 That is, 101101 1 × 1 = 1 and 11000 1 0 1 0 × 0 = 0 × 1 = 1 × 0 = 0 11110

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Cryptology Cryptography Terminology

Cryptography Encryption: convert plaintext message into a ciphertext that looks like gibberish The art of making codes. MEET ME AT THE LIBRARY becomes Using encryption to conceal text. iQA/AwUBO8RZysYL3oijlaCiEQI3OwCgm7Uzwx Cryptanalysis UW26KR/emgIBs+FavKAdgAoN4F The art of breaking codes. Decryption: convert ciphertext back into the original The breaking of secret writing. plaintext iQA/AwUBO8RZysYL3oijlaCiEQI3OwCgm7Uzwx Cryptology UW26KR/emgIBs+FavKAdgAoN4F becomes The study of encryption and decryption. MEET ME AT THE LIBRARY

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10 Cryptography Terminology Example

Encryption and decryption are based on an algorithm What’s the message? and a key I have been asked to speak to you today about sometimes two keys, one for encryption and one for cryptology. I will be brief because we do not have a decryption tremendous amount of time. You will be able to follow The algorithm is assumed to be public knowledge, but this lecture, I assure you. At the very least, you will the key is secret learn something about cryptology. From Caesar’s cipher to Lewis Carrol’s Vigenère cipher, we will look Cryptanalysis is deciphering a text without knowing the at them all. key Specialty of GCHQ, NSA, etc

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Null Cipher Types of Ciphers

A null-cipher is a type of hidden message where the real Transposition Ciphers message is camouflaged in an innocent sounding message. The Rail Fence, The Twisted Path, and Scrambling with a Key Word. A famous example of a null cipher sent by a German spy in WWII: Monoalphabetic Substitution Ciphers Apparently neutral’s protest is Caesar, Key Word, The Pigpen Cipher, Random thoroughly discounted and ignored. Isman hard hit. Blockade issue affects Polyalphabetic Substitution Ciphers pretext for embargo on by products, ejecting suets and vegetable oils. Date Shift, Porta’s Digraphic, Playfair, and Vigenère

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Simple Code Machines Transposition Ciphers

Typewriter Codes A transposition cipher is one that does not A Telephone Dial Code change any letters of the original message (the “plaintext”). The Scytale It rearranges the letters according to a secret The Alberti Disk system. Thomas Jefferson’s Wheel Cipher The simplest transposition cipher is made by just Grilles writing the message backward. The Triangle Code

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11 Transposition Ciphers The Rail Fence Cipher

Example: Count the number of letters. THE EAGLE HAS LANDED If the number is not a multiple of 4, add enough Encoded: dummy letters (nulls) at the end to make it a DEDNAL SAH ELGAE EHT multiple of 4. Easy to decode, harder if the original word order Example: is kept, but reverse the letters of each word. MEET ME AFTER THE TOGA PARTY EHT ELGAE SAH DEDNAL Add a Z to make it 24 letters long.

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The Rail Fence Cipher The Rail Fence Cipher

Write the message by printing every other letter a Encoding and decoding is simpler and more trifle lower on the page. accurate if you divide the cipher text into groups MEET ME AFTER THE TOGA PARTY Z of 4. That is why we added the nulls Z at the end.

M E M A T R H T G P R Y Variations include copying the two rows in reverse order or copying one row forward and the E T E F E T E O A A T Z other backward. Other varieties involve using more than two rows. MEMA TRHT GPRY ETEF ETEO AATZ

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3-Line Rail Cipher The Twisted Path Cipher

Variation by writing the letters in a zigzag. Uses a rectangular grid or matrix which is a MEET ME AFTER THE TOGA PARTY Z checkerboard of empty squares. The message is written in the cells from left to M M T H G R right, taking the rows from top to bottom E T E F E T E O A A T Z M E E T M E E A R T P Y A F T E R T H E T O G A MMTH GRET EFET EOAA TZEA RTPY P A R T Y Z

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12 The Twisted Path Cipher The Twisted Path Cipher

Then trace on the matrix a particular path agreed Copy the letters along the path starting at the upon in advance by everyone who will be using beginning and going forward or at the end and the code. working backward. M E E T M E M E E T M E A F T E R T A F T E R T H E T O G A H E T O G A P A R T Y Z P A R T Y Z ETAZ YGRM TEOT RTTE EFEA PHAM ETAZ YGRM TEOT RTTE EFEA PHAM

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The Twisted Path Cipher Scrambling with a Key Word

You can use a spiral starting at any corner cell This is a variation on the twisted path cipher. and whirl inward, clockwise or counterclockwise, Instead of using regular paths, broken or or you can begin at one of the central cells and continuous, a key word is used for mixing up the spiral outward. columns of the matrix in a completely haphazard M E E T M E way. A F T E R T First the message is written in the cells according H E T O G A to an agreed upon plan. P A R T Y Z

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Scrambling with a Key Word Scrambling with a Key Word

To scramble the columns, number them 1 to 6, The number corresponds to its position in the but mix up the digits. alphabet. Our key word would be: 546132 Write the six digits above the columns. Since numbers are not easy to remember, we 5 4 6 1 3 2 use a key word. M E E T M E Any word, with no two letters alike, can be the O G A P A A key. T Z Y T R F We will use the word: MIRAGE E H T R E T

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13 Monoalphabetic Scrambling with a Key Word Substitution Ciphers The digits tell us the order to follow in copying the The order of the letters stays the same but for columns from top down. each letter a different letter, or some kind of symbol, is used. Copy the first column headed 1, then the column headed 2, and so on to column 6. Something is substituted for every letter of the message. The ciphered text will be: Monoalphabetic (or single alphabet) means that TPTR EAFT MARE EGZH MOTE EAYT for every letter, one and only one letter or symbol is substituted.

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Monoalphabetic Caesar Cipher Substitution Ciphers One of the simplest and oldest substitution Also known as shift ciphers. ciphers is created by writing the alphabet forward, A key number tells you how far to shift a second the underneath, the alphabet backward: alphabet when it is written underneath the first ABCDEFGHIJKLMNOPQRSTUVWXYZ one. ZYXWVUTSRQPONMLKJIHGFEDCBA Suppose the key number is 3. Each letter stands for the letter directly below(or Write the alphabet in a row, put your pencil on A above) it. and count 3 letters to the right, starting on B and Example: GSRH RH ZM VCZNKOV ending on D.

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Caesar Cipher Caesar Cipher

Put D below A and continue to the right with E, F, G, .. To encode a message, find the letter in the top Until you reach Z, then go back to A and finish the row and substitute for it the letter immediately alphabet. below. A=0, B=1, C=2, …, Y=24, Z=25. To decode, find the letter in the bottom row and Encrypt: C = (P + 3) mod 26 write the letter above it. Decrypt: P = (C – 3) mod 26 Note: Sometimes a word becomes another word COLD in a 3-shift Plaintext Letter A B C D E F G H … PECAN in a 4-shift Ciphered text Letter D E F G H I J K … SLEEP in a 9-shift

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14 Caesar Cipher Example General Caesar Cipher

ATTACK AT DAWN DWWDFN DW GDZQ Mathematically, the encryption and decryption functions can be described as follows: Breaking the Caesar Cipher can simply be done by The sender encodes each plain letter P using the testing all possible shifts. key b as follows: Since we used an alphabet of length 26 we have to test 26 shifts. C = (P + b) mod 26. Encryptions and decryptions should be performed by a The recipient decodes each cipher letter C using computer. again the key b as follows: For that purpose, we number each letter a=0, b=1, c=2, ..., y=24, z=25. P = (C – b) mod 26.

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Key Word Ciphers Multiplication Cipher

A simple substitution cipher using the keyword Similar to a shift cipher but instead of adding the JUPITER key we multiply by it. Write the alphabet in a row, then underneath, The number we multiply by is our key a. write JUPITER, followed by all the other letters in alphabetical order. Thus, our encryption function may be written as ABCDEFGHIJKLMNOPQRSTUVWXYZ C = a * P mod 26 JUPITERABCDFGHKLMNOQSVWXYZ where P denotes the value of the plaintext letter Note: Key words can not have duplicate letters. and C denotes the value of the ciphertext letter.

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Multiplication Cipher Multiplication Cipher Example

You can't just multiply by any key a; multiplying Let’s use a = 3, so our formula is by some numbers won't result in a usable cipher. C = (3 * P) mod 26 The key a must have an inverse modulo 26. A=0, B=1, C=2, D=3, E=4,…, Y=24, Z=25. The only numbers that work are those that are Multiply by 3 and modulo 26 yields “relatively prime” to 26. A=0, B=3, C=6, D=9, E=12,…, Y=20, Z=23. Which means they don't have any prime factors in 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 common with 26. 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 Thus, you can use any odd number except 13. A D G J M P S V Y B E H K N Q T W Z C F I L O R U X

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15 Affine (Linear) Cipher Linear Cipher Example

Affine means linear, so this cipher takes on the Let’s use a = 5 and b = 13, so our formula is same form as a line: C = (5 * P + 13) mod 26 C = (a * P + b) mod 26 A=0, B=1, C=2, D=3, E=4,…, Y=24, Z8. Note that when a = 1 you have a shift cipher, and Multiply by 5 and add 13 and modulo 26 yields when b = 0 we have a multiplication cipher. A=13, B=18, C=23, D=2, E=7,…, Y=3, Z=8.

Again here the greatest common factor for a and 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 26 has to be 1. 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 N S X C H M R W B G L Q V A F K P U Z E J O T Y D I

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The Pigpen Cipher The Pigpen Cipher

The Pigpen Cipher was used by Freemasons in The alphabet is written in the grids shown, and then the 18th Century to keep their records private. each letter is enciphered by replacing it with a symbol that corresponds to the portion of the pigpen grid that It was also reportedly used by the Confederate contains the letter. States in the Civil War. The cipher does not substitute one letter for another but rather it substitutes each letter for a symbol.

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Pigpen Cipher Example The Polybius Checkerboard

Decode the following message: Polybius was an ancient 1 2 3 4 5 Greek writer who devised a method of substituting 1 A B C D E 2-digit numbers for each letter. 2 F G H I J Using the matrix at the 3 K L M N O Answer: This cipher was once used by right, substitute for each freemasons. letter the numbers 4 P Q R S T marking the row and column. 5 U V W X Y/Z

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16 The Polybius Checkerboard Random Substitution Ciphers

Always put the row number first. A random substitution cipher is one that is For example, the number for K is 31. constructed without a plan. The word PUMPKIN would become Merely write the alphabet, and next to each letter 14 - 51 – 33 – 41 – 31 – 24 – 34. you put any letter, number or symbol. Note that both Y and Z are written in the last cell Popular Examples: to divide the letters evenly. Arthur Conan Doyle’s “The Adventure of the Dancing Men” The context of the message should make it clear which of the two letters is intended. Edgar Allen Poe’s “The Gold Bug”

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Breaking Substitution Ciphers Breaking Substitution Ciphers

Are monoalphabetic ciphers secure? Monoalphabetic ciphers can all be broken. At First Glance: The reason why such ciphers can be broken is 26! possible encryptions. that although letters are changed the underlying At one decipherment per microsecond it would letter frequencies are not! take 1000 years to test all 26! decipherments by If the plain letter “a” occurs 10 times, its cipher brute force. letter will do so 10 times. Answer – No! Therefore, any monoalphabetic cipher can be In a long message, letter frequencies betray the text. broken with the aid of letter frequency analysis.

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Breaking Substitution Ciphers Polyalphabetic Ciphers

Cryptanalyst's Tools: Monoalphabetic ciphers – one symbol for each Letter frequency data letter. Prefix/suffix lists Polyalphabetic ciphers are more difficult to break. ant-, inter-, post-, -ate, -ing Poly means many. Letter pair/triple lists That is, throughout the ciphertext different -re-, -th-, -en-, -de-, -ion-, -ive-, -ble- symbols can stand for the same letter and the Common pattern lists same symbol can stand for different letters. -eek-, -oot-, -our-

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17 Polyalphabetic Ciphers Simple Example

The Reason Create two ciphers using a multiplication and a To make substitution ciphers more secure. shift cipher. The reason behind using polyalphabetic ciphers is to Alternate ciphers while encrypting or decrypting flatten frequency distributions! the text. Advantages of polyalphabetic ciphers: For example: Flattens letter frequencies. Double letter pairs not so obvious. Odd positions use C = (3 * P) mod 26 Prefix tables become more complicated. Even positions use: C = (5 * P + 13) mod 26

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Porta’s Digraphic Cipher Porta’s Digraphic Cipher

Invented by Giovanni Battista Porta in 1563. The formula for each letter pair L1-L2 is A digraphic cipher is one in which pairs of letters, C = L1 * 26 + L2 instead of individual letters, provide the basis of Assuming A = 0, B = 1, …, Z = 15. We get the cipher text. In the Porta Cipher, a single symbol is substituted AA = 0, AB = 1, AC = 2, …, ZZ = 675 for every pair of letters in the message. Example: Help me! To use this cipher you need an enormous 26×26 becomes HE-LP-ME which is 186-301-316. matrix.

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The Playfair Cipher The Playfair Cipher

Invented by Charles Wheatstone, famous for The cipher replaces each pair of letters in the musical instruments and one of the pioneers of plaintext with another pair of letters, so it is a type the telegraph. of digraph cipher. Named for his friend Baron Lyon Playfair. The matrix is smaller than that used in the Porta Wheatstone created the cipher for sending secret cipher. messages by standard telegraph codes. To construct the matrix, pick a keyword and write It was used for many years by the British Army it into a 5×5 square, omitting repeated letters and and Australia used it in World War II. combining I and J in one cell.

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18 The Playfair Cipher The Playfair Cipher

In this example, we use B O X E R Follow the keyword with the rest of the alphabet’s the keyword BOXER and letters in alphabetical order. write it into the square by rows. A C D F G Encryption depends on the type of digraph (letter pair). Any other pattern will do H I/J K L M including writing it by The digraphs fall into one of three categories: columns or writing it in a both letters are in the same row spiral starting at one N P Q S T both letters are in the same column corner and ending in the center. U V W Y Z the letters share neither a row nor a column

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The Playfair Cipher The Playfair Cipher

If the two letters are in If both letters are in the same row B O X E R different rows and in they are replaced by the letters immediately to the different columns right of each one. If a letter is at the end of a row, it is A C D F G each letter is replaced by replaced by the letter at the beginning. the letter in the same row that is also in the column H I/J K L M If both letters are in the same column occupied by the other they are replaced by the letter immediately beneath letter. N P Q S T each one. If a is at the bottom of a column, it is For example, AT replaced by the letter at the top. becomes GN. U V W Y Z

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The Playfair Cipher Example The Playfair Cipher

Encrypt: B O X E R “MEET ME AT ONE AM.” ME ET ME AT ON EA MX A C D F G First, break it up into two-letter groups. ME ET ME AT ON EA MX H I/J K L M If both letters in a pair are the same, insert an X between them. LR RS LR GN BP BF KR N P Q S T If there is only one letter in the last group, add an X to it. Now we encrypt each two-letter group. U V W Y Z

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19 The Playfair Cipher The Vigenere Cipher

To decrypt the message, simply reverse the Named for Blaise de Vigenere a 16th century process: Frenchman. If the two letters are in different rows and columns, The Vigenere cipher is a polyalphabetic cipher take the letters in the opposite corners of their based on using successively shifted alphabets, a rectangle. different shifted alphabet for each of the 26 If they are in the same row, take the letters to the left. English letters. If they are in the same column, take the letters above The procedure is based on the table shown on each of them. the next slide and the use of a keyword.

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The Vigenere Cipher The Vigenere Cipher

Note that each row of the One Method of Using the Vigenere Table: matrix corresponds to a 1. Choose a key. Caesar Cipher. 2. Break text into groups of five characters. The first row is a shift of 0 3. Write the key in repeating fashion. the second is a shift of 1 and 4. Use letter of key to establish column. the last is a shift of 25. 5. Use plaintext to establish row. For a better matrix click 6. Encrypt by using intersection of row and column. here.

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The Vigenere Cipher Example The Vigenere Cipher

The keyword is CIPHER. A computer program would implement the The message is MEET ME AT ONE AM. following algorithm: C I P H E R C I P H E R C Code the letters as numbers (A=0, B=1, etc.) The key is keyword M E E T M E A T O N E A M To Encrypt: Add the keyword to the plaintext (letter by O M T A Q V C B D U I R O letter) Encrypted becomes OMTA QVCB DDUI RO. To Decrypt: Subtract keyword from the ciphertext

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20 The Date Shift Cipher Operations Research

This is a variation on the Vigenere cipher Operations Research (OR) is the use of the scientific sometimes called a Gronsfeld cipher. methodology in studying systems whose design or operation require human decision making. It uses the digits of a key number instead of a the OR provides the means for making the most effective letters of keyword. decisions - some of which are mainly concerned with The easiest key number to use is the date. design, while others are mainly operational in nature. Everything else is essentially the same as the OR is interdisciplinary, drawing on (and contributing to) Vigenere cipher. the techniques from many fields, including mathematics, engineering, economics and the physical sciences.

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Operations Research Operations Research Topics

OR practitioners have successfully solved a wide variety Linear Algebra Decision Making Under of real world problems. Linear Programming and Uncertainty Most importantly, new applications are continually Duality Decision Trees arising, most notably in computer and telecommunication Game Theory Multi-criteria Decision technology, in the financial and economic community, in Dynamic Programming Making medicine, in education, to name just a few. Critical Path Method Sensitivity Analysis Many of these new applications originate from relatively Graph Theory Inventory Theory and recent societal problems, such as food and energy Production production and distribution, health maintenance, Goal Programming environmental pollution control, and software production. Branch and Bound

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References References

Binary Numbers, John Rieman, September 28th, 2001, 11:12pm, D. Kahn, The Codebreakers, Macmillan Publishing http://www.cs.colorado.edu/~l3d/courses/CSCI1200- 96/binary.html Company, 1976. Binary Number System, Erik Ostergaard, September 28th, 2001, A. Menezes, P. van Oorschot, and S. Vanstone, 11:14pm, http://www.danbbs.dk/~erikoest/binary.htm Handbook of Applied Cryptography, CRC Press, 1997. “Bit (computer)”, Microsoft® Encarta® Online Encyclopedia 2001, ©1997-2000 Microsoft Corporation, September 28th, 2001, M. Gardner, Codes, Ciphers, and Secret Writing, Dover 11:17pm, Publications, Inc., 1972. http://encarta.msn.com/find/Concise.asp?z=1&pg=2&ti=76155185 L. Smith, Cryptography the Science of Secret Writing, 1 Dover Publications, Inc., 1943.

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21 Web Sites

Check out Computer History web site. The Black Chamber web site. The GCHQ web site. The Crypto Tutorial web site. A Cryptographic Compendium The SSH website Check out Operations Research Tutor web site.

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