Chapter 17: Information Science
Lesson Plan
 Binary Codes
 Encoding with Parity-Check
Sums
 Cryptography
 Web Searches and
Mathematical Logic
© 2013 W. H. Freeman and Company
For All Practical
Purposes
Mathematical Literacy in
Today’s World, 9th ed.
Chapter 17: Information Science
Binary Codes
 Mathematical Challenges in the Digital Revolution
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How to correct errors in data transmission
How to electronically send and store information economically
How to ensure security of transmitted data
How to improve Web search efficiency
Binary Code – A
 Binary Codes
system for coding
 Binary codes are the hidden language of
data made up of
computers, made up of two states, 0 and 1.
two states (or
 Examples of binary codes: Postnet code,
symbols).
UPC code, Morse code, Braille, etc.
 Other recent advancements, such as DVDs,
Blu Ray, fax machines, high definition TVs, cell
phones, use binary coding with data
represented as strings of 0’s and 1’s rather
than usual digits 0 through 9 and letters A
through Z.
Chapter 17: Information Science
Encoding with Parity-Check Sums
 Binary Coding
 Strings of 0’s and 1’s with extra digits for error correction can
be used to send full-text messages.
 Example: Assign the letter a the string 00001, b the string
00010, c the string 00100, and so on, until all letters and
characters are assigned a binary string of length 5.
For this example we can have:
25 = 2 × 2 × 2 × 2 × 2 = 32 possible binary strings.
 Error Detection and Correction via Binary Coding
 By translating words into binary code, error detection can be
devised so that errors in the transmission of the code can be
corrected.
 The messages are encoded by appending extra digits,
determined by the parity (even or odd sums) of certain portions
of the messages.
Chapter 17: Information Science
Encoding with Parity-Check Sums
 Parity-Check Sums
 Sums of digits whose parities determine the check digits.
 Even Parity – Even integers are said to have even parity.
 Odd Parity – Odd integers are said to have odd parity.
 Decoding
 The process of translating received data into code words.
 Example: Say the parity-check sums detect an error.
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The encoded message is compared to each of the possible correct
messages. This process of decoding works by comparing the
distance between two strings of equal length and determining the
number of positions in which the strings differ.
The one that differs in the fewest positions is chosen to replace the
message in error.
In other words, the computer is programmed to automatically correct
the error or choose the “closest” permissible answer.
Chapter 17: Information Science
Encoding with Parity-Check Sums
 Nearest-Neighbor Decoding Method
 A method that decodes a received message as the code word
that agrees with the message in the most positions provided that
there is only one such message.
 Assuming that errors occur independently, the nearest-neighbor
method decodes each received message as the one it most likely
represents.
 This method was first proposed in 1948 by Richard Hamming, a
mathematician at Bell Laboratories—one of the Hamming codes.
 Binary Linear Code
 A binary linear code consists of words composed of 0’s and 1’s
obtained from all possible messages of a given length by using
parity-check sums to append check digits to the messages. The
resulting strings are call code words.
 Think of the binary linear code as a set of n-digit strings in
which each string is composed of two parts—the message
part and the remaining check-digit part.
Chapter 17: Information Science
Encoding with Parity-Check Sums
 Weight of a Binary Code
 The minimum number of 1’s that occur among all
nonzero code words of that code.
 Variable-Length Code
 A code in which the number of symbols for each
code word may vary. Like in Morse code, the
letters that occur most frequently have the
shortest coding—similar to data compression.

Data Compression
 Encoding data process where the fewest symbols
represent the most frequently occurring data.
 Delta Encoding – A simple method of compression for sets
of numbers that fluctuate little from one number to the next.
 Huffman Coding – A widely used scheme for data compression created in
1951 by a graduate student, David Huffman. The code is made by using a
so-called code tree by arranging the characters from top to bottom according
to increasing probabilities.
Chapter 17: Information Science
Cryptography
 Crytptology
 In many situations, there is a desire for
security against unauthorized interpretation
of coded data (desire for secrecy).
 Hence came cryptology, which is the study
of how to make and break secret codes.
Cryptology –
The study of how
to make and
break secret
codes.
 Encryption
 The process of encoding data (or simply disguising the data) to
protect against unauthorized interpretation.
 In the past, encryption was primarily used for military and
diplomatic transmission.
 Today, encryption is essential for securing electronic transactions
of all kinds. Here are some examples:
 Web sites allowed to receive/transfer encrypted credit-card numbers
 Schemes to prevent hackers from charging calls to your cell phone
 Various schemes used to authenticate electronic transactions
Chapter 17: Information Science
Cryptography
Types of Cryptosystems:
 Caesar Cipher
 A cryptosystem used by Julius Caesar whereby each letter is
shifted the same amount. Not much effort to “crack” this code!
 Modular Arithmetic
 A more sophisticated scheme for transmitting messages secretly.
 This method of encrypting data is based on addition and
multiplication involving modulo, n.
 For any positive integer a and n, we define a mod n (“a modulo
n” or “a mod n”) to be the remainder when a is divided by n.

Decimation Cipher
 Multiply each position by a fixed number and use modular
arithmetic rather than adding a fixed number to the position as
in the Caesar cipher.
Chapter 17: Information Science
Cryptography
 Vigenére Cipher
 A cryptosystem that uses a key word to determine how much
each letter is shifted.
 Key word – A word used to determine the amount of shifting for each
letter while encoding a message.
 RSA Public Key Encryption Scheme – A method of
encoding that permits each person to announce publicly the means
by which secret messages are to be sent to him or her.
 In honor of Rivest, Shair, and Adleman, who discovered it. The
method is practical and secure because no one knows an
efficient algorithm for factoring large integers (about 200 digits
long).
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Chapter 17: Information Science
Cryptography
 RSA Public Key Encryption Scheme – A method of
encoding that permits each person to announce publicly the means
by which secret messages are to be sent to him or her.
 In honor of Rivest, Shair, and Adleman, who discovered it. The
method is practical and secure because no one knows an
efficient algorithm for factoring large integers (about 200 digits
long).
 Cryptogram – A cryptogram is a sentence (or message) that has
been encrypted.
 Cryptography is the basis for popular word puzzles, called
cryptograms, found in newspapers, puzzle books, and Web sites.
Chapter 17: Information Science
Web Searches and Mathematical Logic
 Web Searches
 The number of Web pages indexed by large Internet search
engines, such as Google, are in the billions.
 The algorithm used by the Google search engine, for instance,
ranks all pages on the Web using interrelations to determine their
relevance to the user’s search.
 Factors such as frequency, location near the top of the page of key
words, font size, and number of links are taken into account.
 Boolean Logic
 A branch of mathematics that uses operations to connect
statements, such as the connectives: AND, OR, NOT.
 Boolean logic was named after George Boole (1815–1864), a
nineteenth-century mathematician.
 Boolean logic is used to make search engine queries more
efficient.
Chapter 17: Information Science
Web Searches and Mathematical Logic
 Expression
 In Boolean logic, an expression is simply a statement that is either
true or false.
 Complex expressions can be constructed by connecting individual
expression with connectives: AND, OR, and NOT.
 Connectives, their math notations, and meanings:
AND conjunction ^, means to find the results with all of the words.
OR disjunction v, means to find the results of at least one of the words.
NOT
¬, means to find the results without the words.
 Two expressions are said to be logically equivalent if they have the
same value, true or false.
 Truth Tables – Tabular representations of an expression in which
the variables and the intermediate expressions appear in columns,
and the last column contains the expression being evaluated.
Chapter 17: Information Science
Web Searches and Mathematical Logic
 Internet Protocol (IP) Address
 An address of four numbers between 0 and 255 separated by
dots assigned to routers, computers, printers and fax machines
that allows those linked electronically to uniquely identify and
communicate with each other.
 From the IP address, we can determine the network address
of the computer, which specifies the network or subnet that
the computer is part of.
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Network address is determined from a companion number to the
IP address called the subnet mask.
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