104/21/23

MATH 224 – Discrete MathematicsIntegers and Computers

Integers and number theory are important building blocks for computer science. (Number theory deals with the properties of integers.)

Why are integers so important in computer science?

Consider the memory of a computer. What does it store? Just sequences of zeros and ones. When put all together the total memory may be interpreted as one large binary integer.

How are integers most commonly represented in memory? What is the 32 bit representation of 500? What is the 32 bit representation of -500?

204/21/23

MATH 224 – Discrete MathematicsIntegers and Computers

How are letters (characters) stored inside a computer’s memory?

How are floating point numbers (i.e. numbers with fractional parts) stored?

How are text files; Word documents; C++ programs; and machine instructions stored in memory?

At the lowest level, digital computers can only work with numbers.

How is audio stored with integers?

How are photographs stored with integers? Can color be represented by an integer?

How are videos stored as integers?

304/21/23

MATH 224 – Discrete MathematicsInteger Terminology

Integers may be classified as either prime numbers or composite numbers. What is a prime number, and what is a composite number?

The following is often referred to as the Fundamental Theorem of Arithmetic.

Every positive integer larger than 1 may be expressed as a single prime number or a unique product of primes. This product is called the prime factorization of an integer.

Those numbers that are not primes, having two or more primes in their prime factorization, are called composite numbers.

What about negative integers less than 1. Do they have a unique prime factorization? Is there a way to make the factorization unique?

404/21/23

MATH 224 – Discrete MathematicsDivision Algorithm (uniqueness of modulus)

The so called division algorithm is not really an algorithm, but instead states a property of integers. It states that when an integer a is divided by a positive integer d, the non-negative remainder r is unique. Note that d may not be equal to zero and r is restricted to a value 0 ≤ r < d. What if d is negative?

Also note that r = a modulus d or in C++, r = = a % d.

If d is a divisor of a what is the value of a % d?

504/21/23

MATH 224 – Discrete MathematicsOther Important Terminology

Greatest common divisor: Given two integers, the largest number that divides both evenly is called the greatest common divisor. Give an example of the use of the greatest common divisor.

Relatively prime: Two numbers that do not have any common divisor other than 1 are said to be relatively prime.

Least common multiple: Given two integers, the smallest number that is divisible by both of the integers is called the least common multiple. Give an example of the use of the least common multiple.

604/21/23

MATH 224 – Discrete MathematicsModular Arithmetic (the % operator in C++)

Modular Arithmetic deals with the remainder when one integer is divided by another. So for example 12 ≡ 5 mod 7. Or in C++ (12 % 7 = = 5) is a true proposition. There is an entire theory built on the modulus operator. In fact a set of values modulus a prime number is sometimes called a field under the operations of addition and multiplication.

For example 5*3 ≡ 1 mod 7, which implies that 3 is equivalent to 1/5 in arithmetic modulo 7, and is called the multiplicative inverse. What is the multiplicative inverse of 4 in mod 7 arithmetic?

And 5 + 2 ≡ 0 mod 7, which implies that 2 is equivalent to –5 in arithmetic modulo 7. Thus 2 is the additive inverse of 5 in mod 7 arithmetic.

704/21/23

MATH 224 – Discrete MathematicsModular Arithmetic Applications in CS

Some of these concepts related to modular arithmetic are used in CS.

•Modular arithmetic is used to specify negative integers in a way that makes computation easier for computers,

•In the area of security modular arithmetic is used for encryption and authentication. What is encryption? Give an example of where it is used.

•Simple modular arithmetic is used to create hash tables for storing data such as the symbol table for a program in C++.

•Modular arithmetic is used to create circular arrays for implement queues. What is a circular array and what is a queue in computer science?

•Modular arithmetic is used to generate random numbers by most random number generators. These are called pseudo-random since they are not truly random.

804/21/23

MATH 224 – Discrete MathematicsInteger Representation Using Base Two

Why is base 2 used by most computers?

Negative integers are represented using the twos complement of the positive number. For example, with an eight bit integer negative 32 corresponds to 224. Eight bit numbers are viewed as numbers mod 256 since the integers from 0 to 255 may be presented. Note that 32 + 224 % 256 = = 0. As binary numbers

32 is 0010 0000 and 224 is 1110 0000. Now if we add these together we have

0010 0000 +1110 0000 1 0000 0000With eight bit numbers the one on the right is just dropped. To make this work all integers with a leading one are interpreted as negative numbers.

Why does mod 256 not work for the multiplicative inverse?

904/21/23

MATH 224 – Discrete MathematicsInteger Representation Using Octal and Hexadecimal

For humans it is often easier to work with either octal, base 8, or hexadecimal, base 16 numbers.

Every digit of an octal number represents three bits in a binary representation and every bit of a hexadecimal representation represents four bits in a binary representation. Can you explain why these statements are true?

For example the binary number 0101 0001 has the hexadecimal representation of 5116,

and 1111 1011 has hexadecimal representation FB16 . What are these numbers in

decimal?

Finally, how are floating point numbers represented? Are binary numbers used to represent floating point? In what way does it make sense to say that floating point numbers look like integers in memory?

1004/21/23

MATH 224 – Discrete MathematicsThe Euclidean Algorithm for the GCD

The Euclidean Algorithm is named after the Greek mathematician Euclid who is most well know for his axioms of plane geometry. Euclid’s algorithm finds, the greatest common divisor for two positive integers. Below is a version of this algorithm in C++

int function gcd (int a, int b) {int x = a, y = b, r;while (!(y = = 0)) {

r = x % y;x = y;y = r

}return x;

} // end gcd

Lets apply this function to 60 and 210. What happens to the values of x and y after the first pass through the while loop?

1104/21/23

MATH 224 – Discrete MathematicsPublic Key Encryption Using RSA

One of the most significant uses of modular arithmetic is encrypting messages so that only the person having a special key is able to determine the original message. Below are the basic steps of the RSA algorithm.

First everyone has the following information.

• An integer n = pq where p and q are very large prime numbers, but only the value of n is know by everyone.

• An integer e that is relatively prime to (p−1)(q−1)

Second only the person receiving the message has the values of p and q. If p and q are very large, even with a very fast computer it will take an extremely long time to compute them from the information provided.

1204/21/23

MATH 224 – Discrete MathematicsPublic Key Encryption Using RSA

Now with the values of n and e, a message may be encrypted by converting the message to integers M, perhaps using ASCII. The integers are modified by means of the following formula.

C = Me mod n

Note that there are algorithms for doing modular exponentiation quickly. After the conversion the integers corresponding to C are sent.

The receiver decrypts the message using the formula

M = Cd mod n

The value of d is the multiplicative inverse of e mod (p−1)(q−1). The value of d is not difficult to calculate if p and q are know, but the values of p−1 and q−1 are needed to find d.

See the text for a proof that this actually works and an example of how these formulas are used.