Final Report-1-(1)

A Project Report
On
Secure Scheme for Secret Sharing using Graph Theory
Submitted by
Shivangi Goel
13MSM006
Department of Applied Sciences
ITM University, Gurgaon
Haryana, India

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A Project Report
On
Secure Scheme for Secret Sharing using Graph Theory
submitted in partial fulfilment of the requirement for the award of the
degree in
MASTER OF SCIENCE (MATHEMATICS)
By
Shivangi Goel
13MSM006
Under the Supervision of
Prof. N.Chandramowliswaran
and
Dr. Gaurav Gupta
(Assistance Professor)
Department Of Applied Sciences
ITM University, Gurgaon
Haryana, India
MAY-2015

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DECLARATION
I certify that the work contained in this dissertation is original and has been
done by me in the Department of Applied Sciences, ITM University Gurgaon
under the guidance of my supervisor Dr. Gaurav Gupta and Prof.
N.Chandramowliswaran. The matter presented in this dissertation has not been
submitted by me for the award of any other degree of this or any other Institute.
I have conformed to the norms and guidelines given in the Ethical Code of
Conduct of the Institute. Whenever I have used information (theoretical
analysis and text) from other sources, I have given due credit to them by citing
them in the report.
Signature of the Student

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CERTIFICATE
This is to certify that the Dissertation report entitled “Secure Scheme
for Secret Sharing using Graph Theory” is bonafide work carried out by Ms.
Shivangi Goel, reg. No. 13MSM006 to ITM University Gurgaon, India under
my supervision and guidance and is worthy of consideration for the award of
degree in Master of Science (Mathematics) of the Institute.
Prof. N.CHANDRAMOWLISWARAN DR. GAURAV GUPTA
(Assistance Professor)
Date:

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To My Parents,
Mrs. Asha Goel & Mr. Shiv Kumar Goel
And
All My Teachers

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ACKNOWLEDGEMENTS
I would like to articulate my deep gratitude to my advisor and supervisor DR.
GAURAV GUPTA and DR. N.CHANDRAMOWLISWARAN for continuous support,
patience, moral support, motivation, enthusiasm and immense knowledge during my M.Sc.
research project. Furthermore, I would like to thank them for introducing me to the topic as
well as for the support on the way for the completion of this project report under the guidance
of professionals like them. I could not have imagined having a better guide and advisor for
my M.Sc. study. Besides my guide, I gratefully acknowledge all the faculty members of
Applied Science Department for many good suggestions during the progress of the project
work.
A project of this nature could never have been attempted without reference to and
inspiration from the works of others whose details are mentioned in the reference section. I
acknowledge my indebtedness to all of them. Last but not the least, my whole-heartedly
thanks to all of my friends who have patiently extended all sorts of help for accomplishing
this undertaking
SHIVANGI GOEL

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ABSTRACT
In recent years, Chinese remainder theorem (CRT)-based function sharing schemes are being
proposed in the literature for the security of operations taking place over a computer network.
It is important to ensure safety of such actions against terrible clients who may try to misuse
the frameworks. This thesis deals with two fundamental cryptographic tools that are helpful
in such contexts: Secret Sharing Scheme and key distribution schemes with the help of
system of two or more linear congruences. Both secret sharing schemes and key distribution
schemes are used in multiparty frameworks. Threshold schemes authorize a class of users to
share a secret by providing each user with an allotment of share. For the secret to be
recovered of any subset with cardinality t+1, a scheme with threshold t+1 is needed. In this
thesis, we constructed t-share holders by considering 2t prime numbers. Using the t-share
holders, we divide the secret S into t parts and all the t-shares are required to recover the
secret using CRT.

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CONTENTS
Introduction………………………………………………………………………………….(8)
Preliminary:-
(i) Linear Congruences……………………………...………………………………(9)
(ii) RSA-Algorithm………………………………………………………….…(9)-(10)
(iii) Public-Key Cryptography………………………………..………………..(11)-(12)
Literature………………………………………………………………………………(13)-(15)
Proposed Scheme:-
(i) Peterson Graph…….....................................................................................(16)-(19)
(ii) Heawood Graph………………………………………………..………….(20)-(23)
(iii) Asymmetric Graph……………………………………………………...…(24)-(29)
Future Work………………………………………………………………….……………..(30)
Conclusion………………………………………………………………………………….(31)
References…………………………………………………………………………….(32)-(34)

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Introduction
Graph theory is the study of graphs, which are mathematical structures used to model pair-
wise relations between objects. A "graph" in this context is made up of "vertices" or "nodes"
and lines called edges that connect them. A graph may be undirected, meaning that there is
no distinction between the two vertices associated with each edge, or its edges may
be directed from one vertex to another. They are used for defining the connectivity of
networks and for diagramming and understanding relationships of many kind. Graphs appear
in many domains, whenever it is useful to represent how things are either physically or
logically linked to one another in a network structure. The 13-node Arpanet shown below is
an example of a communication network, in which nodes are computers or other devices that
can relay messages, and the edges represent direct links along which messages can be
transmitted.

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PRELIMINARY
Linear Congruences
(i) Let (a, m) = 1. Then the linear congruence 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑚) has exactly one
solution.
(ii) Let (a,m) = d. then the linear congruence 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑚) has solutions iff 𝑑 𝑏⁄ .
(iii) Fermat’s Little Theorem (FLT)
If p is a prime, and p does not divide a, then 𝑎 𝑝−1
≡ 1(𝑚𝑜𝑑 𝑝).
Proof:- If p is a prime, the binomial coefficient
(
𝑛
𝑘
) =
𝑝(𝑝 − 1) … . (𝑝 − 𝑘 + 1)
𝑘!
≡ 0
becomes
(𝑥 + 𝑦)𝑝 ≡ 𝑥𝑝 + 𝑦𝑝 (𝑚𝑜𝑑 𝑝).
In particular, for y = 1,
(𝑥 + 1)𝑝 ≡ 𝑥𝑝 + 1 (𝑚𝑜𝑑 𝑝).
Claim→ For every non-negative integer x,
𝑥𝑝 ≡ 𝑥 (𝑚𝑜𝑑 𝑝).
By induction on x.
x = 0 clear.
x = 1 clear.
Assume the claim is true for x > 0. Then
(𝑥 + 1)𝑝 ≡ 𝑥𝑝 + 1 ≡ 𝑥 + 1(𝑚𝑜𝑑 𝑝).
Hence the claim is true for the next value x + 1. If (x, p) = 1, then
𝑥𝑝 ≡ 𝑥(𝑚𝑜𝑑 𝑝) ==> 𝑥. 𝑥𝑝 − 1 ≡ 𝑥. 1(𝑚𝑜𝑑 𝑝) ==> 𝑥𝑝 − 1 ≡ 1(𝑚𝑜𝑑 𝑝)
RSA-Algorithm
(Rivest , Shamir and Adleman)
RSA is one of the first practicable public-key cryptosystems that is based on the difficulty
of factoring the product of two large prime numbers, widely used for secure data
transmission. Here the encryption key is public and differs from the decryption key which is
kept secret. The RSA algorithm involves 3 steps:

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1) Key Generation
2) Encryption
3) Decryption
Key Generation:
RSA involves a public key and a private key.
 Choose two distinct prime numbers
p and q.
 Compute n = pq.
 Compute φ(n) = φ(p)φ(q) =(p − 1)(q − 1)
 Choose an integer e such that
1< e <φ(n) and (e, φ(n)) = 1
Determine d as d ≡ e−1
(mod φ(n)).
Encryption:
 Bob transmits his public key (n, e) to Alice and keeps the private key d secret. Alice
then wishes to send message ”M” to Bob.
 She first turns ”M” into an integer m, such that 0 ≤ m < n and then computes the
cipher text corresponding to
𝑐 ≡ 𝑚 𝑒
(𝑚𝑜𝑑 𝑛)
Decryption:
 Bob can recover m from c by using his private key exponent d via computing
𝑚 ≡ 𝑐 𝑑
(𝑚𝑜𝑑 𝑛)
 Given m, he can recover the original message ”M” by reversing the padding scheme.

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Public Key Cryptography (Dr. N.Chandramowliswaran)
"Public Key Cryptography" using Finite key of Primes with Combinatorial conditions
for secret sharing.
Example 1: Let "G" be a given simple connected graph
Let "T" be a Secret Spanning tree of G.
Assume that there exists a finite set of primes (distinct) { p1, p2, p3, p4, p5, p6, } with
the following conditions:
[The basic problem is, here, how to determine such a finite set of very large ODD
primes with the following condition]
𝑓: 𝑉(𝐺)
1−1,𝑂𝑁𝑇𝑂
→ {𝑝1, 𝑝2, 𝑝3, 𝑝4, 𝑝5, 𝑝6}𝑤𝑖𝑡ℎ 𝑓(𝑣𝑖) = 𝑝𝑖[1 ≤ 𝑗 ≤ 6]
(∏ 𝑝𝑖
6
𝑗=1
, ∑ 𝑝𝑖 𝑝𝑗)
{𝑣 𝑖,𝑣 𝑗}∈𝐸(𝑇)
= 1 − − − −−→ (1)
In this example, we have the below graph T as a spanning tree of G.

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(∏ 𝑝𝑖
6
𝑗=1
, 𝑝1 𝑝2 + 𝑝1 𝑝4 + 𝑝1 𝑝5 + 𝑝1 𝑝6 + 𝑝3 𝑝4) = 1 − − − −−→ (2)
(2) Is equivalent to
(p1, p3p4) = 1 -----→ (I)
(p2, p1p4 + p1p5 + p1p6 + p3p4) = 1 -----→ (ii)
(p3, p1p2 + p1p4 + p1p5 + p1p6) = 1 -----→ (iii)
(p4, p1p2 + p1p5 + p1p6) = 1 -----→ (IV)
(p5, p1p2 + p1p4 + p1p6+ p3p4) = 1 -----→ (v)
(p6, p1p2 + p1p4 + p1p5 + p3p4) = 1 -----→ (VI)
Now, Applying “Cryptography” on this
Suppose “S” is the given “Secret” with the following conditions:
(p3p4)S ≡ degT(v1) (mod p1)
(p2, p1p4 + p1p5 + p1p6 + p3p4)S ≡ degT(v2)(mod p2)
(p1p2 + p1p4 + p1p5 + p1p6)S ≡ degT(v3)(mod p3)
(p1p2 + p1p5 + p1p6)S ≡ degT(v4)(mod p4)
(p1p2 + p1p4 + p1p6+ p3p4)S ≡ degT(v5)(mod p5)
(p1p2 + p1p4 + p1p5 + p3p4)S ≡ degT(v6)(mod p6)
The above system has only one solution i.e. mod (∏ 6𝑗=1 pi)

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Literature
Key distribution is the main problem in cryptographic systems, and important component of
the security subsystem of distributed systems, communication systems, and data networks. If
users of a group wish to communicate using symmetric encryption, they must share a
common key. Secret sharing/secret splitting is the procedure of assigning a secret amongst a
class, each of whom is given a share of the secret. When different types of shares are merged
together then only the secret can be retrieved; individual shares are of no use on their own.
Secret sharing was invented independently by Adi Shamir [1] and George Blakley [6] in
1979. Secret sharing schemes are ideal for storing information that is highly sensitive and
highly significant. Examples: encryption/decryption keys, bank accounts password etc. Each
and every section of information must be kept safe and secure. The revelation of these
information is very dangerous as someone can misuse it. However, it is also paramount that
they are not to be lost. Historical methods for encryption are inappropriate for attaining
remarkable confidentiality and authenticity. This is because when storing the encryption key,
one must decide on between keeping a single copy of the key in one location for maximum
secrecy or keeping multiple copies of the key in different locations for greater reliability [18].
Raising genuineness or authenticity of the key by storing numerous replicas at various
positions lowers secrecy by providing more chances for a copy to fall into the wrong hands.
Secret sharing strategy deals with this issue, and allows effective secrecy and authenticity to
be attained. A secure secret sharing strategy assigns shares so that any person with lower than
t-shares has no additional knowledge about the secret than someone with 0-shares. For
example the secret sharing strategy in which the secret expression “security” is split into the
segments as
se−−−−−−, −−cu−−−−, −−−−ri−− and −−−−−− ty.
Anyone with 0-shares understands only that the password is made up of eight characters. He
would have to guess the password from 268
= 208 billion possible combinations. A person
with one share, however, would have to estimate only the six characters of the password,
from 266
= 308 million combinations, and so on as more persons collaborate. Therefore this
network is not a secure secret sharing scheme, as a person with less than t-secret shares is
able to lower the problem of attaining the inner secret without first needing to recover all of
the significant portions. More generally, (n; k) secret sharing is the problem of distributing a
secret number “S” among n people so that no k − 1 of them have any information about “S”

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but k of them can determine “S”. Shamir’s secret sharing does this by giving the nth
party
f(n), where f is an appropriately chosen polynomial. The dealer Bob chooses random field
elements a1, a2, . . . , ak−1, and uses the polynomial f(t) = s + a1t + a2t2
+ ⋅ ⋅ ⋅ + ak−1tk−1
. He
gives the nth
person f(n). For k people to reconstruct the secret, they just combine their shares
and use Lagrange interpolation to find the unique degree k − 1 polynomial passing through
the k points. (Lagrange interpolation works over any field. In practice, a large finite field
would probably be used.) The secret is just the constant term. Any k – 1 people have no
knowledge about S. They have k − 1 points on the polynomial, but there is a degree k − 1
polynomial going through their k − 1 points and (0, t) for any integer t, so their combined
knowledge reveals nothing about S. The current trend towards cloud computing means that
more and more data is being processed and stored by online resources beyond physical and
logical control of the owner. It is a simple task for an antagonist to obstruct, replicate and
cache any data sent across a public network and from this point on, confidentiality is
determined solely by the original encryption scheme. Hence re-encryption is worthless, and
even deletion of data cannot be guaranteed. Current cryptographic schemes in wide
deployment today, such as RSA, Dife Hellman, and AES, do not guarantees over long term
confidentiality.
One of the standard topics in a first course in number theory is the Euler ɸ function, with ɸ(n)
defined as the number of positive integers less than n and relatively prime to n. A famous
theorem involving ɸ is that suppose a and b are any two positive integers with (a, b) = 1 then
aɸ(b)
+ bɸ(a)
≡ 1(mod ab). The Chinese Remainder Theorem (CRT) can also be used in secret
sharing [18], there are two secret sharing schemes that make use of the Chinese Remainder
Theorem, Mignotte’s and Asmuth-Bloom’s Schemes. They are threshold secret sharing
schemes, in which the shares are generated by reduction modulo the integers mi, and the
secret is recovered by essentially solving the system of congruences using the Chinese
Remainder Theorem.

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Theorem 1 (Chinese Remainder Theorem)
Suppose that m1, m2, . . . , mr are pair-wise relatively prime positive integers, and let a1, a2, . .
. , ar be integers. Then the system of congruences, x ≡ ai (mod mi) for 1 ≤ i ≤ r, has a unique
solution modulo M = m1 ×m2× . . . ×mr which is
given by:
𝑥 ≡ 𝑎1 𝑚1 𝑦1 + 𝑎2 𝑚2 𝑦2 + … … … + 𝑎 𝑟 𝑚 𝑟 𝑦𝑟 (𝑚𝑜𝑑 𝑀), 𝑤ℎ𝑒𝑟𝑒 𝑀𝑖 = 𝑀/𝑚𝑖
And 𝑦𝑖 ≡ (𝑀𝑖)−1(𝑚𝑜𝑑 𝑚𝑖)𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑟
Here, a key is broadcasted/shared among the various different share holders in the system that
are under the process of encryption and decryption. The aim is to preserve the authenticity of
the vertices i.e. nodes that exist in the system. Here, the shares are properly assigned by
selecting 2t prime numbers and then it is divided among the corresponding nodes for which it
is created.

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PROPOSED SCHEME
1) Peterson Graph:
Petersen graph is an undirected graph with 10 vertices and 15 edges which is most commonly
drawn as a pentagon with a pentagram inside, with five spokes. It has even number of
vertices so, perfect matching is possible. The Petersen graph has a Hamiltonian path but
no Hamiltonian cycle. It has chromatic number 3, meaning that its vertices can be colored
with three-colors but not with two, such that no edge connects vertices of the same colors.
Now, I am using Peterson graph as a network where the nodes are the login ID and the edges
are the users.
Organising the issue of less no. of Login ID in Petersen Graph as a network.
1) Given system has 10 Login ID and 15 users.
2) Every user can use at most two Login IDs.

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3) Each Login ID is utilized by exactly three users.
4) Nodes (vertices) of the graph G are represented as Login IDs and edges as users.
5) If a user − k using Login IDs Log IDp and Log IDq, then join them by an edge.
6) Conflict users are those having/sharing common Log ID and those who do not share
any common Log ID are the non-conflict users Eg;- Conflict users: user−3, user−4
and user−9, they share common Log ID4. Non-Conflict users: user−4, user−6 and
user–14, they do not have any common Log ID.
7) Defining V (G) = {vi = Log IDi ∣ 1 ≤ i ≤ 10}
8) Defining E(G) = {k = user k ∣ 1 ≤ k ≤ 15}
9) Defining f(vi) = f(Log IDi) = σ(i), where σ is a tranformation on the set of numbers
{1, 2, . . . , 10}. For each Log IDi we have a different σ(i). [18]
10) Now we shall define graceful labelling ‘g’ on the set {σ(1), σ(2), . . . , σ(10)} g : {σ(i)
: 1 ≤ i ≤ 10} → {0, 1, 2, . . . , q − 1, q} Let
User-k
Log IDr●――――――――●logIDs
g[user k] =∣ g(σ(r)) − g(σ(s)) ∣∈ {1, 2, . . . , q} where 1 ≤ r, s ≤ 10, r≠ s
11) g : E(G)→ {1, 2, . . . , q} and g is kept confidential, and g[user k] is defined
for each user k.
12) g[user k] is the user–ID
User-k
13) User k has two Login IDs i.e. (σ(r), σ(s)) and the whole system is kept confidential.
14) P : V (G) → {p1, p2, . . . , p10} where pi, 1 ≤ i ≤ 10 are distinct odd primes with q <
min{pi}, 1 ≤ i ≤ 10, q < pk∀k ( P is kept hidden) g[user k] is known 1 ≤ k ≤ 15.
15) Defining ek : (ek , (pr − 1)(ps − 1)) = 1 ( ek kept hidden) and mk ≡ (g[user k])ek (mod
prps)P[Log IDr] = pr,P[Log IDs] = ps, 1 ≤ r, s ≤ 10, r ≠s
16) Now the user (edges) set is break apart into subset of Non-Conflict users i.e. the set
of Independent Edges.
17) M1 = {user − 6, user − 7, user − 8, user − 9, user − 10} :
User − 6 ←→ {Log ID2, Log ID10}

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User − 10←→ {Log ID3, Log ID9}
M2 = {user − 1, user − 3, user − 11, user − 13}:
User – 13 ←→ {Log ID7, Log ID10}
M3 = {user − 2, user − 4, user − 12, user − 14}:
M4 = {user − 5, user − 15}:
Now, Defining congruence equations for the set M1, M2, M3 and M4 as follows:
A ≡ m6 (mod p2p10)
A ≡ m7 (mod p1p6)
A ≡ m8 (mod p5p7)
A ≡ m9 (mod p4p8)
A ≡ m10 (mod p3p9)
A has a distinct solution mod( p1p2 . . . . . . p10)
Therefore, M1 Non-Conflict users share the common secret A.
B ≡ m1 (mod p1p2)
B ≡ m3 (mod p4p5)
B ≡ m11 (mod p6p9)
B ≡ m13 (mod p7p10)
B has a distinct solution mod( p1p2 p4 p5p6 p7 p9 p10)
Therefore, M2 Non-Conflict users share the common secret B.

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C ≡ m2 (mod p1p5)
C ≡ m4 (mod p3p4)
C ≡ m12 (mod p6p8)
C ≡ m14 (mod p7p9)
C has a distinct solution mod( p1p3 p4 p5p6 p7 p8 p9)
Therefore, M3 Non-Conflict users share the common secret C.
D ≡ m5 (mod p2p3)
D≡ m15 (mod p8p10)
D has a distinct solution mod( p2 p3 p8 p10)
Therefore, M4 Non-Conflict users share the common secret D.

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2) Heawood Graph:
Heawood graph is an undirected graph which is composed of 14 vertices and 21 edges.
The graph is cubic, bipartite and also all cycles in the graph have at least six edges.
Heawood graph has 24 perfect matchings. The figure below shows that the vertices of the
graph placed on a cycle, with the internal diagonals of the cycle forming a matching.
Heawood graph can be partitioned into three perfect matchings (i.e. 3-color its edges) in
eight different ways by sub-dividing the cycle edges into two matchings. Through
symmetry of the graph, we can transform every two perfect matching into every two
Hamiltonian cycle and vice-versa. Now, we use Heawood graph as a network of login ID
and users. For each vertex, we are assigning a very large prime number i.e. p1, p2, . . . , p14
and for edges we are using numbers i.e. 1,2,3, . . . ,21.
2) Every user can use at most two Login ID.
3) Each Login ID is utilized by exactly three users.
4) Nodes (vertices) of the graph G are represented as Login IDs and edges as users
5) If a user−k using Login IDs Log IDp and Log IDq, then join them by an edge.

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6) Conflict users are those having/sharing common Log ID and those who do not
share any common Log ID are the non-conflict users Eg:- Conflict users: user−1,
user−2 and user−15, they share common Login ID Log ID1 Non-Conflict users:
user−4, user−7 and user–11, they do not have any common Log ID.
9) Defining f(vi) = f(Log IDi) = σ(i), where σ is a transformation on the set of
numbers {1, 2, . . . , 14}. For each Log IDi we have a different σ(i)
10) Now we shall define graceful labelling ‘g’ on the set {σ(1), σ(2), . . . , σ(14)} g :
{σ(i) : 1 ≤ i ≤ 14} → {0, 1, 2, . . . , q − 1, q} Let
User-k
11) g : E(G)→ {1, 2, . . . , q} and g is kept confidential and g[user k] is defined for
each user k.
12) g[user k] is user–ID
User-k
User k has two Login IDs i.e.(σ(r), σ(s)) and the whole system is kept
confidential.
13) P : V (G) → {p1, p2, . . . , p14} where pi, 1 ≤ i ≤ 14 are distinct odd primes with q
< min{pi}, 1 ≤ i ≤ 14, q < pk∀k ( P is kept hidden) g[user k] is known 1 ≤ k ≤ 21.
14) Defining kj : (ek , (pr − 1)(ps − 1)) = 1 ( ek kept secret) and mk ≡ (g[user k])ek
(mod prps)P[Log IDr] = pr,P[Log IDs] = ps, 1 ≤ r, s ≤ 14, r ≠s
15) Now the user (edges) set is break apart into subset of Non-Conflict users i.e. the
set of Independent Edges.
M1 = {user − 1, user −3, user −5, user −7, user – 9, user − 11, user − 13}:

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M2 = {user − 2, user −4, user −6, user −8, user –10, user − 12, user − 14}:
M3 = {user − 15, user −16, user −17, user −18, user – 19, user − 20, user − 21}:
Now, defining congruence equations for the set M1, M2 and M3 as follows:
A ≡ m1 (mod p1p14)
A ≡ m3 (modp2p3)
A ≡ m5 (mod p4p5)
A ≡ m7 (mod p6p7)
A ≡ m9 (mod p8p9)
A ≡ m11 (mod p10p11)
A ≡ m13 (mod p12p13)
A has a distinct solution mod( p1p2 . . . . . . p14)
B ≡ m2 (mod p1p2)
B ≡ m4 (mod3p4)
B ≡ m6 (mod p5p6)

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B ≡ m8 (mod p7p8)
B ≡ m10 (mod p9p10)
B ≡ m12 (mod p11p12)
B ≡ m14 (mod p13p14)
B has a distinct solution mod( p1p2 . . . . . . p14)
C ≡ m15 (mod p1p6)
C ≡ m16 (mod2p11)
C ≡ m17 (mod p3p8)
C ≡ m18 (mod p4p13)
C ≡ m19 (mod p5p10)
C ≡ m20 (mod p7p12)
C ≡ m21 (mod p9p14)
C has a distinct solution mod(p1p2 . . . . . . p14)

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3) Aysmmetric Graph:
It is an undirected graph with 25 vertices and 42 edges. It has odd number of vertices,
therefore we cannot obtain a perfect matching or 1-factor for this graph because perfect
matching is a graph containing 𝑛 2⁄ edges where n is the no. of vertices. Now, we are
using this graph as a network where the nodes are the login ID and the edges are the
users. For each vertex, we are assigning a very large prime number i.e. p1, p2, . . . , p25 and
for edges we are using numbers i.e. 1,2,3, . . . ,42.
2) Every user can use at most two Login ID.
3) Here the distribution is not symmetric i.e. some login ID are used by 5 users, some are
used by 4 users and some are used by 3 users.
4) Representing the Login IDs by the nodes (vertices) of the graph G and users by the
edges.
5) If there is a user−k using Login ID’s Log IDp and Log IDq, then join them by an edge.

25
6) Conflict users are those having/sharing common Log ID and those who do not share
any common Log ID are the non-conflict users Eg:- Conflict users: user−5, user−6
and user−20, they share common Login ID Log ID6 Non-Conflict users: user−3,
user−7 and user–11, they do not have any common Log ID.
9) Defining f(vi) = f(Log IDi) = σ(i), where σ is a transformation on the set of numbers
{1, 2, . . . , 25}. For each Log IDi we have a different σ(i) [18].
10) Now we shall define graceful labelling ‘g’ on the set {σ(1), σ(2), . . . , σ(25)} g : {σ(i)
: 1 ≤ i ≤ 25} → {0, 1, 2, . . . , q − 1, q} Let
User-k
11) g : E(G)→ {1, 2, . . . , q} and g is kept confidential and g[user k] is defined for each
user k.
12) g[user k] is user–ID
User-k
user k has two Login IDs i.e. (σ(r), σ(s)) and the Entire Network is kept secret.
13) P : V (G) → {p1, p2, . . . , p10} where pi, 1 ≤ i ≤ 25 are distinct odd primes with q <
min{pi}, 1 ≤ i ≤ 10, q < pk∀k ( P is kept hidden) g[user k] is known 1 ≤ k ≤ 42.
14) Define ek : (ek , (pr − 1)(ps − 1)) = 1 ( ek is kept hidden) and
mk ≡ (g[user k])ek (mod prps)P[Log IDr] = pr,P[Log IDs] = ps, 1 ≤ r, s ≤ 25, r ≠s
15) Now the user (edges) set is break apart into subset of Non-Conflict users i.e. the set of
Independent Edges.
16) M1 = {user − 1, user − 3, user − 5, user − 7, user – 9, user − 11,
user − 28, user − 31, user − 34} :
user − 1 ←→ {Log ID1,Log ID2}
user – 9 ←→ {Log ID9,Log ID10}

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M2 = {user − 2, user − 4, user − 6, user − 8, user – 10, user − 12,
user − 35, user − 29, user − 32} :
M3 = {user − 13, user − 15, user − 18, user − 20, user – 23,
user − 25,user − 27, user − 17, user − 22} :
M4 = {user − 36, user − 30, user − 33, user − 16, user – 21} :
M5 = {user − 14, user − 24, user − 19, user − 37} :

27
M6 = {user − 38, user − 40} :
M7 = {user − 39, user − 42} :
M8 = {user − 26, user − 41} :
Now, Defining congruences equations for the set M1 , M2 , . . . , M8 as follows:
A ≡ n1 (mod p1p2)
A ≡ n3 (mod p3p4)
A ≡ n5 (mod p5p6)
A ≡ n7 (mod p7p8)
A ≡ n9 (mod p9p10)
A ≡ n11 (mod p11p12)
A ≡ n28 (mod p13p14)
A ≡ n31 (mod p16p17)
A ≡ n34 (mod p19p20)
A has a distinct solution mod ( p1p2 . . . . . . p20)
B ≡ n2 (mod p2p3)
B ≡ n4 (mod p4p5)
B ≡ n6 (mod p6p7)
B ≡ n8 (mod p8p9)

28
B ≡ n10 (mod p10p11)
B ≡ n12 (mod p12p1)
B ≡ n35 (mod p20p21)
B ≡ n29 (mod p14p15)
B ≡ n32 (mod p17p18)
B has a distinct solution mod ( p1p2 . . . . . . p21)
C ≡ n13 (mod p12p13)
C ≡ n15 (mod p2p14)
C ≡ n18 (mod p4p14)
C ≡ n20 (mod p6p17)
C ≡ n23 (mod p8p19)
C ≡ n25 (mod p10p20)
C ≡ n27 (mod p11p22)
C ≡ n17 (mod p3p23)
C ≡ n22 (mod p7p24)
C has a distinct solution mod ( p2p3 p4 p6 p7 p8 p10 p11 p12p13 p14 p16 p17 p19 p20 p22 p23 p24).
D ≡ n36 (mod p21p13)
D ≡ n25 (mod p15p16)
D ≡ n27 (mod p18p19)
D ≡ n17 (mod p22p3)
D ≡ n22 (mod p7p23)
D has a distinct solution mod ( p3 p7 p13 p15 p16 p18 p19 p21 p22 p23)
Therefore, M4 Non-Conflict users share the common secret D.
E ≡ n14 (mod p1p22)
E ≡ n25 (mod p9p24)
E ≡ n27 (mod p5p23)
E ≡ n17 (mod p21p25)
E has a distinct solution mod ( p1 p5 p9 p21 p22 p23 p24 p25)
Therefore, M5 Non-Conflict users share the common secret E.

29
F ≡ n38 (mod p15p25)
F ≡ n40 (mod p22p24)
F has a distinct solution mod ( p15p22p24 p25)
Therefore, M6 Non-Conflict users share the common secret F.
G ≡ n39 (mod p18p25)
G ≡ n42 (mod p23p24)
G has a distinct solution mod ( p18p23p24 p25)
Therefore, M7 Non-Conflict users share the common secret G.
H ≡ n26 (mod p11p24)
H ≡ n41 (mod p22p23)
H has a distinct solution mod ( p11p22p23 p24)
Therefore, M8 Non-Conflict users share the common secret H.

30
FUTURE WORK
1) Testing of the proposed scheme.
2) Generating Algorithms using proposed edge decompositions.
3) Application of proposed algorithms.
4) Development of new algorithms based on the decomposition of vertices.

31
CONCLUSION
This thesis dealt with two fundamental cryptographic tools that are useful in such contexts:
Secret Sharing Scheme and Key Distribution Schemes. Both Secret Sharing Schemes and key
distribution schemes are used in multiparty frameworks. Secret Sharing Schemes enables
some predetermined sets of parties to recover a given secret. These schemes make it possible
to store secret information in a system, such that only good subsets can recover the
information or the data. Furthermore, by utilizing these schemes we can allow only better
subsets to perform action in a system. The approach used in this thesis for secret sharing is to
break the secret into various primes and assign it to the associated share holders in the
system. Also if any one of the share is missing then it is unable to decrypt the secret by any
attacker. Hence forth this is a new opening in the field of cryptosystem and can be use for
various network problem.

32
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35
PROFILE
Shivangi Goel
Qualification:- pursuing Master of Science (Mathematics)
B.Sc (H) Mathematics
Contact Address:- E-5/58 Sector-16, Rohini, Delhi-110089
Email:- sg051292@gmail.com
Phone No. :- 9999493925

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