Modulo 2 Arithmetic

Introduction

These notes describe how to go about modulo 2 addition, subtraction and division.

Modulo 2 Arithmetic

Modulo 2 arithmetic is performed digit by digit on binary numbers. Each digit is considered independently from its neighbours. Numbers are not carried or borrowed.

Addition

Modulo 2 addition is performed using an exclusive OR (xor) operation on the corresponding binary digits of each operand. The following table describes the xor operation:

A B A xor B

0 0 0

0 1 1

1 0 1

1 1 0

A	B	A xor B
0	0	0
0	1	1
1	0	1
1	1	0

We can add two binary numbers, X and Y as follows:


    (X) 10110100
    (Y) 00101010 +
    (Z) 10011110

Subtraction

Modulo 2 subtraction provides the same results as addition. This can be illustrated by adding the numbers X and Z from the addition example.


    (X) 10110100
    (Z) 10011110 +
    (Y) 00101010

The addition example shows us that X + Y = Z so Y = Z - X.

However, the subtraction example shows us that Y = Z + X.

As neither Z nor X is zero, the addition and subtraction operators must behave in the same way.

Division

Modulo 2 division can be performed in a manner similar to arithmetic long division. Subtract the denominator (the bottom number) from the leading parts of the enumerator (the top number). Proceed along the enumerator until its end is reached. Remember that we are using modulo 2 subtraction. For example, we can divide 100100111 by 10011 as follows:


                 10001 remainder 100
       10011|100100111
             10011 
                 10111
                 10011 
                   100

This has the effect that X/Y = Y/X. For example:


                 1 remainder 1011
       11001|10010
             11001 
              1011


                 1 remainder 1011
       10010|11001
             10010 
              1011

References:

A.S. Tanenbaum (1996). Computer Networks. Prentice Hall ISBN 0-13-394248-1. See section 3.2.2.

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