Bipartite Graph, Cryptography, Encoding & Decoding

1. Application
of
Mathematics
S.SWATHI SUNDARI, M.Sc.,M.Phil.,
Assistant Professor of Mathematics
V.V.Vanniaperumal College for Women
Virudhunagar
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2. DATA TRANSFER USING BIPARTITE GRAPH
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● ● ●
● ● ●

3. DATA TRANSFER IS…
Transfer of data in a secret way of using
some techniques and it’s a safety issue in
the current world.
This can be done in Cryptography.
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4. CRYPTOGRAPHY IS…
The science or study of
techniques of secret writing and
message hiding.
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5. Encryption Decryption
The process of converting
Plaintext into Ciphertext.
PLAINTEXT
CIPHERTEXT
The process of converting
Ciphertext into Plaintext.
CIPHERTEXT
PLAINTEXT
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Terminology

6. Terminology
KEY
The secret, like a password, which is used
to encrypt and decrypt information.
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7. For example
Plaintext letters : A B C ,..., Z
Ciphertext letters : C D E ,..., B
ENCRYPTION DECRYPTION
HELLO (Plaintext) JGNNQ (Ciphertext)
JGNNQ (Ciphertext) HELLO (Plaintext)
It is an example of “Secret Key Cryptography”.
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8. Types of Keys
Based upon the keys, encryption and decryption
can be done by using two types of keys:
Secret key Cryptography
Public key Cryptography
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9. Secret Key Cryptography:
This type of cryptography technique uses
just a single key.
Public Key Cryptography:
In this type of cryptography, one key is used to
encrypt, and a matching key is used to decrypt.
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10. Mathematic Application in
Data Transfer
Graph Theory is growing as a promising field
for Data Transfer.
We propose a method of message
encryption as a graph.
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11. What is…
Graph:
It consists of a pair G =(V,E), where
●V= V(G)= set of vertices.
●E= E(G)= set of edges.
Example:
Here, V={ 1,2,3} 1 ●
E= { }
G = (V,E) is a (3,2) graph. 2● ●3
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12. Weighted Graph:
A number (weight) is assigned to each edge on a
graph, ( weights might represent costs, lengths or
capacities, etc., depend upon a problem)
Multigraph:
Lines joining the same points are called
“Multigraph”.
1●
2● ●3
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Example:

13. Independent Sets:
An independent set is a set of vertices in a graph,
no two of which are adjacent.
Bipartite Graph:
A graph G is called a “Bigraph” or “Bipartite
Graph” if V can be patitioned into two disjoint
subsets and such that every line of G joins a
point of to a point of .
( , ) is called a “Bipartition” of G.
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1 ● 2 ● 3 ● = {1,2,3}
={4,5,6}
Example:
4 ● 5 ● 6 ●

14. Construction of Encryption Table:
First we decide the number of characters (S) required for
the message encryption.
We can randomly fix the number of rows & columns of
the table, Note: The number of cells available in the table is
atleast of length of S.
Assign numbers 1,2,3…k, to the columns and numbers
k+1,k+2,…m, to the rows, where
k = number of columns k ≤ 9,
m = number of rows.
Distribute the characters in S randomly in the table.
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15. Encryption Table:
For normal message, we use the 26 alphabets and blank space.
Each character in the cell receives a number value.
For Example:
A receives value 14.
U receives value 310.
Note:
The first character represents
a the column number, remaining
the row number.
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1 2 3
4 A B C
5 D E F
6 G H I
7 J K L
8 M N O
9 P Q R
10 S T U
11 V W X
12 Y Z Space

16. Graph Construction from Number Sequence
Let M be the message to be encrypted of length k.
Convert each character in M into it’s corresponding
number values using Table.
Let the resulting number sequence be M1.
Let M1 be represented as .
Where,
are numbers.
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,

17. We construct a graph G as follows
Vertices set of G:
Number of vertices in G = Number of distinct row
numbers + Column numbers used to generate M1.
Each vertex receive it’s corresponding row and
column value as it’s label.
Edge set of G:
Draw edges between the vertex pairs
.
Let us label these edges as
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18. Number of edges in G = length of M.
and are always
independent sets. So, the graph G is always a Bipartite Graph.
Edge Weights:
Assign random numbers as
the edge weights to the edges
so that
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19. Encryption Algorithm
Let M: I CAN be the message to be encrypted.
Step-1: Convert each character in M into it’s corresponding
number values using Table to generate M1.
For the message M, M1: 36 312 34 14 28
Step-2 : Construct the graph corresponding to the
sequence M1. For M1,
Vertex Set = {1, 2, 3, 4, 6, 8, 12}
Edge Set = { (3 6), (3 12), (3 4), (1 4), (2 8) }
= { }
Edge Weights = { 10, 15, 24, 56, 78 }
assigned to the edges
respectively.
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1 2 3
4 A B C
5 D E F
6 G H I
7 J K L
8 M N O
9 P Q R
10 S T U
11 V W X
12 Y Z Space

20. Step - 3: Send G to the receiver.
For decrypting the message we reverse the procedure.
The resulting graph G is as follows:
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1● 2● 3●
4● 6 ● 8● 12●
1524
10
56
78

21. Decryption Algorithm
Suppose the received graph is as follows:
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2● 3●
5● 4● 7● 8● 12●
16
4420
78
65
35
92

22. Step -1: Arranging the edge weights in increasing order,
we generate sequence as follows:
{ 16, 20, 35, 44, 65, 78, 92}
Step -2: Picking the corresponding vertex labels from the
graph, we generate the sequence as follows:
{ 24, 25, 3 12, 34, 38, 38, 37 }
Step –3: From the table, the message is decrypted as
follows,
BE COOL
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1 2 3
4 A B C
5 D E F
6 G H I
7 J K L
8 M N O
9 P Q R
10 S T U
11 V W X
12 Y Z Space

23. Applications of Data Transfer Using
Graphs
Electronic Money :
Encryption is used in electronic money
schemes to protect conventional transaction data like
account numbers, transaction amounts and digital
signatures.
Braille System :
In this system, encryption & decryption of any
message using Graph Theory and Braille numbers.
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