1. Tutorial on Programming with Python: Galois Field
Kishoj Bajracharya
Asian Institute of Technology
June 20, 2011
Contents
1 Galois Field 2
1.1 To import class FField for Galois Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 To Create Galois Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Polynomial Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Result of example1.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Co-efficient of Polynomial Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Result of example2.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 Conversion from co-efficient of polynomial to element . . . . . . . . . . . . . . . . . . . . 4
1.8 Result of example3.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Operation Over Galois Field 5
2.1 Addition Operation over Galois Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Result of example4.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Multiplication Operation over Galois Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Result of example5.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Operations over Galois Field: Another Approach . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Result of example6.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1

2. 1 Galois Field
1.1 To import class FField for Galois Field
1. Browse the URI http://www.mit.edu/˜emin/source_code/py_ecc and see the latest
release of the file “py ecc-1-4.tar.gz” and download it.
2. Extract it and keep all the files in a separate folder named “Field”.
3. Use the import command to use the class FField from the file name “ffield.py”
import sys
sys.path.append(‘./Field’)
import ffield
1.2 To Create Galois Field
To create the Galois field F = GF (2n )
F = ffield.FField(n)
Create the field GF (23 )
F = ffield.FField(3)
1.3 Polynomial Representation
The polynomial representation of any number a can be obtained using following codes
F.ShowPolynomial(a)
Returns: string
Note: the value of a lies between 0 to 2n − 1
Following example named “example1.py” shows how do we use python programming language to get poly-
nomial from galois field.
1 # example1.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3)
8
9 # Show the polynomial representation from 0-2ˆ3-1 i.e. 0-7
10 print ’For 0, polynomial: ’ + F.ShowPolynomial(0)
11 print ’For 1, polynomial: ’ + F.ShowPolynomial(1)
12 print ’For 2, polynomial: ’ + F.ShowPolynomial(2)
13 print ’For 3, polynomial: ’ + F.ShowPolynomial(3)
14 print ’For 4, polynomial: ’ + F.ShowPolynomial(4)
15 print ’For 5, polynomial: ’ + F.ShowPolynomial(5)
16 print ’For 6, polynomial: ’ + F.ShowPolynomial(6)
17 print ’For 7, polynomial: ’ + F.ShowPolynomial(7)
2

3. 1.4 Result of example1.py
Fig1: Result of example1.py
1.5 Co-efficient of Polynomial Representation
The coefficient of the polynomial representation of any number a can be obtained using following codes
F.ShowCoefficients(a)
Returns: list
Note: the value of a lies between 0 to 2n − 1
Following example named “example2.py” shows how do we use python programming language to get the
co-efficient of a polynomial from galois field.
1 # example2.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3)
8
9 # Show the coefficient of the coefficient of polynomial representation of a,b,c,d,e,f,g,h
10 print ’For 0, coefficient of polynomial: ’ + str(F.ShowCoefficients(0))
11 print ’For 1, coefficient of polynomial: ’ + str(F.ShowCoefficients(1))
12 print ’For 2, coefficient of polynomial: ’ + str(F.ShowCoefficients(2))
13 print ’For 3, coefficient of polynomial: ’ + str(F.ShowCoefficients(3))
14 print ’For 4, coefficient of polynomial: ’ + str(F.ShowCoefficients(4))
15 print ’For 5, coefficient of polynomial: ’ + str(F.ShowCoefficients(5))
16 print ’For 6, coefficient of polynomial: ’ + str(F.ShowCoefficients(6))
17 print ’For 7, coefficient of polynomial: ’ + str(F.ShowCoefficients(7))
3

4. 1.6 Result of example2.py
Fig2: Result of example2.py
1.7 Conversion from co-efficient of polynomial to element
The coefficient(list) of the polynomial can be converted to the element(integer)
F.ConvertListToElement(list)
Returns: integer
Note: the value of a lies between 0 to 2n − 1
Following example named “example3.py” shows how do we use python programming language to convert
co-efficient of a polynomial to element over a galois field.
1 # example3.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3)
8
9 # Converting list into the element
10 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,0,0,0]))
11 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,0,0,1]))
12 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,0,1,0]))
13 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,0,1,1]))
14 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,1,0,0]))
15 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,1,0,1]))
16 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,1,1,0]))
17 print "List: [0,0,0,0], Element: " + str(F.ConvertListToElement([0,1,1,1]))
18
19 # Next method to convert
20 print F.ConvertListToElement(F.ShowCoefficients(0))
21 print F.ConvertListToElement(F.ShowCoefficients(7))
4

5. 1.8 Result of example3.py
Fig3: Result of example3.py
2 Operation Over Galois Field
2.1 Addition Operation over Galois Field
Following example named “example4.py” shows how do we use python programming language to add
polynomials over a galois field.
1 # example4.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3)
8
9 print ’For 2 + 5’
10 x = F.Add(2,5)
11 print ’Number: ’ + str(x)
12 print ’Polynomial: ’ + F.ShowPolynomial(x)
13 print ’List: ’ + str(F.ShowCoefficients(x))
14 print ’’
15
16 print ’For 3 + 5’
17 x = F.Add(3,5)
18 print ’Number: ’ + str(x)
19 print ’Polynomial: ’ + F.ShowPolynomial(x)
20 print ’List: ’ + str(F.ShowCoefficients(x))
21 print ’’
22
23 print ’For 6 + 5’
24 x = F.Add(6,5)
25 print ’Number: ’ + str(x)
26 print ’Polynomial: ’ + F.ShowPolynomial(x)
27 print ’List: ’ + str(F.ShowCoefficients(x))
28 print ’’
5

6. 2.2 Result of example4.py
Fig4: Result of example4.py
2.3 Multiplication Operation over Galois Field
Following example named “example5.py” shows how do we use python programming language to multiply
polynomials over a galois field.
1 # example5.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3)
8
9 print ’For 2 * 5’
10 x = F.Multiply(2,5)
11 print ’Number: ’ + str(x)
12 print ’Polynomial: ’ + F.ShowPolynomial(x)
13 print ’List: ’ + str(F.ShowCoefficients(x))
14 print ’’
15
16 print ’For 3 * 5’
17 x = F.Multiply(3,5)
18 print ’Number: ’ + str(x)
19 print ’Polynomial: ’ + F.ShowPolynomial(x)
20 print ’List: ’ + str(F.ShowCoefficients(x))
21 print ’’
22
23 print ’For 6 * 5’
24 x = F.Multiply(6,5)
25 print ’Number: ’ + str(x)
26 print ’Polynomial: ’ + F.ShowPolynomial(x)
27 print ’List: ’ + str(F.ShowCoefficients(x))
28 print ’’
6

7. 2.4 Result of example5.py
Fig5: Result of example5.py
2.5 Operations over Galois Field: Another Approach
Following example named “example6.py” shows how do we use python programming language to add and
multiply polynomials over a galois field by another approach.
1 # example6.py
2 import sys
3 sys.path.append(’./Field’)
4 import ffield
5
6 # Create the field GF(2ˆ3)
7 F = ffield.FField(3) #GF(2ˆ3)
8
9 a = ffield.FElement(F,0)
10 b = ffield.FElement(F,1)
11 c = ffield.FElement(F,2)
12 d = ffield.FElement(F,3)
13 e = ffield.FElement(F,4)
14 f = ffield.FElement(F,5)
15 g = ffield.FElement(F,6)
16 h = ffield.FElement(F,7)
17
18 # Operation in a Galois Field:
19 # Addition Operation
20 p = c + f
21 print p
22 p = d + f
23 print p
24 p = g + f
25 print p
26 print ’’
27
28 # Multiplication Operation
29 q = c * f
30 print q
31 q = d * f
7

8. 32 print q
33 q = g * f
34 print q
2.6 Result of example6.py
Fig6: Result of example6.py
8