A Project Report

On

New Encryption and Decryption Techniques on ZN(3) and ZN

(4)

Submitted by

Aina Gupta

13MSM001

Department of Applied Sciences

ITM University, Gurgaon

Haryana, India

A Project Report

On

New Encryption and Decryption Techniques on ZN(3) and ZN

(4)

submitted in partial fulfillment of the requirement for the award of the degree

In

Master of Science (Mathematics)

by

Aina Gupta

13MSM001

Under the supervision of

Prof. N. Chandramowliswaran

and

Dr. Gaurav Gupta

(Assistant Professor)

Department of Applied Sciences

ITM University, Gurgaon,

Haryana, India

May 2015

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, text) from other sources, I

have given due credit to them by citing them in the report.

Signature of the student

CERTIFICATE

This is to certify that the Dissertation Report entitled “New Encryption and

Decryption Techniques on ZN(3)

and ZN(4)

” submitted by “Ms. AINA

GUPTA” to ITM University, Gurgaon, India is a record of bona fide project work

carried out by her under my supervision and guidance and is worthy of

consideration for the award of the degree in MASTER OF SCIENCE

(MATHEMATICS) of the Institute.

Prof. N. Chandramowliswaran Dr. Gaurav Gupta

(Assistant Professor)

Date:

To

Mummy - Papa

&

My brother

Pranav

ACKNOWLEDGEMENTS

I am thankful and grateful to my guide Dr. Gaurav Gupta and Prof. N.

Chandramowliswaran for giving me such a project which allows me to show my

dexterity and my adroitness. This topic may be different but there are many keys to

open this lock and I have used it. There are many books by which I am helped a lot

and some help provided by internet too.

Besides my guide, I would like to thank all the faculty members of Applied

Sciences, friends and family for their continuous support.

ABSTRACT

As the field of cryptography grows to incorporate new ideas and strategies

for encryption, decryption, key exchange problems and zero knowledge transfer

protocols lot of algebraic and Number theory methods have been heavily used. In

this dissertation we came up with new encryption, decryption bijective functions.

We generated a family of new encryption functions on Z(3)N and Z

(4)N .

Contents

1 Introduction 1

1.1 Why Cryptography? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Encryption and Decryption: . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Cryptography is partitioned into two classifications: . . . . . 5

2 Premilinaries 10

2.1 Bijective Functions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The gcd has the following properties: . . . . . . . . . . . . . . 11

2.3 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . 12

2.5 Definition and basic properties of congruences . . . . . . . . . . . . . 12

2.5.1 Properties of Congruence: . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Lemma (N. Chandramowliswaran et al.): . . . . . . . . . . . . . 14

2.6 Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Eulers phi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7.1 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Euler- Fermat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Fermats Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . 19

2.11 Möbius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.11.1 Properties: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Encryption and Decryption on Z(3)N and Z

(4)N 21

3.1 A: Find a new bijective map f : Z(4)N −→ Z

(4)N . . . . . . . . . . . . . . 21

3.1.1 Composition: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 B: Find a new bijective map f : Z(4)N −→ Z

(4)N . . . . . . . . . . . . . . . 27

2

3.2.1 f is bijective? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Composition: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 C: Find a new bijective map f : Z(3)N −→ Z

(3)N . . . . . . . . . . . . . . 32

3.3.1 Total Number of Bijective Functions . . . . . . . . . . . . . . . 37

3.4 D: Find a new bijective map f : Z(4)N −→ Z

(4)N . . . . . . . . . . . . . . 38

4 Conclusion and Future Work 44

4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 45

Chapter 1

Introduction

Cryptography is a logical blend of numerical hypothesis and computational ap-

plication which permits the classified exchange of data. It empowers you to store

delicate data or transmit it crosswise over unstable systems (like the Internet) so

that it can’t be read by anybody with the exception of the proposed beneficiary.

Cryptography is the accumulation of systems and methodologies for cover-

ing data in interchanges from the entrance by uninvited or unapproved gather-

ings. A consistent craftsmanship for managing this issue is known from right

on time vestige and it grew along the hundreds of years, generally in the cas-

ing in which two gatherings, say aristocrat and general imparted in composed

by sending one another messages which could just be comprehended when know-

ing some extra information, mystery keys and the points of interest for the tech-

nique of encoding and decoding the messages calculation. Calculations were fre-

quently collected from a gathering of helpful fundamental thoughts, known by

tradition.Today, cryptography is a foundation of the present day security advance-

ments used to secure data and assets on both open and shut systems.

Advanced cryptography is intensely taking into account numerical hypothe-

sis and software engineering practice; cryptographic calculations are composed

around computational hardness suspicions, making such calculations difficult to

break by and by any enemy. It is hypothetically conceivable to break such a frame-

work, yet it is infeasible to do as such by any known useful means. These plans

are subsequently termed computationally secure; hypothetical advances, e.g., en-

hancements in number factorization calculations, and speedier figuring innovation

1

require these answers for be consistently adjusted. There exist data hypothetically

secure plans that provably can’t be equaled the initial investment with boundless

figuring power−a case is the one−time pad yet these plans are more hard to exe-

cute than the best hypothetically flimsy however computationally secure compo-

nents.

An one-time Pad (OTP) is an encryption system that can’t be split if utilized

effectively. In this strategy, a plaintext is combined with an irregular mystery key

. At that point, every bit or character of the plaintext is encoded by consolidating

it with the relating bit or character from the pad utilizing measured expansion.

In the event that the key is really irregular, is at any rate the length of the plain-

text, is never reused in entire or to some extent, and is kept totally mystery, then

the subsequent ciphertext will be difficult to unscramble. It has additionally been

demonstrated that any figure with the ideal mystery property must utilize keys

with adequately the same necessities as OTP keys. Notwithstanding, functional

issues have kept one-time pad from being generally utilized.

1.1 Why Cryptography?

Cryptography is vital in ordinary life. When you purchase from a site, for exam-

ple, flipkart you are depending upon public key cryptography to keep your credit

card points of interest safe. At the point when your working programming con-

sequently redesigns over the web it too will utilize an public key calculation to

watch that the redesign it is going to introduce was truly distributed by the right

individuals, and not by somebody attempting to get into your PC.

Without cryptography money machines would not be conceivable, as the ma-

chines would not have the capacity to dependably speak with the bank PCs. With-

out cryptography, even the thought of electronic voting would not be conceivable.

Cryptographic thoughts can likewise be utilized as a part of making message

2

simpler to decipher without these thoughts we would not have mistake checking

capacities on correspondences lines, and the web would run significantly more

gradually. We would not have the capacity to decrease excess in a message and

thus pack it, accordingly invalidating prevalent archival projects, for example,

’zip’.

These thoughts, making messages minimized, mistake safe, secure (or the ma-

jority of the above) are identified with cryptography.

We need cryptography due to many reasons

Suppose Alice wants to send a message to Bob then things to be taken care of

are [12]

1. Authentication: Affirmation that the imparting substance is the one asserted

as it were Bob realizes that Alice could have sent the message he has recently got-

ten.

2. Access control: When verification is provided,access control is the adminis-

tration of controlling/restricting access to host frameworks and applications in a

system.

3. Data Confidentiality: Insurance of information from unapproved divulgence.

As such a message sent from Alice to Bob can’t be read by any other person.

4. Data Integrity: Confirmation that information got is as sent by an approved

element. As such Bob realizes that the message from Alice has not been altered

inside of travel.

5. Non-Repudiation: Security against disavowal by one of the gatherings in a

correspondence. As such it is outlandish for Alice to pivot later and say she didn’t

send the message.

3

Notwithstanding these,availability is likewise vital and hard to give. Assets

ought to dependably be accessible to those with fitting access benefits. An aggres-

sor ought not have the capacity to interfere, or pointlessly ease off, access to the

asset.

1.2 Encryption and Decryption:

In today’s advanced world, encryption is developing as a disintegrable piece of

all correspondence systems and data handling frameworks, for ensuring both put

away also, in travel information world .Security has turn into an progressively

critical component with the development of complex electronic correspondence.

Encryption is the change of plain information (known as plaintext) into indis-

cernible information (known as cipher content) through a calculation alluded to as

figure. Encryption is the change of information into a frame that is as near to incon-

ceivable as could reasonably be expected to peruse without the fitting information

(a key). Its objective is to guarantee security by keeping data avoided anybody

for whom it is most certainly not planned, even the individuals who have entry

to the encoded information.Decryption is the procedure of changing over cipher

text back to plain text.A cipher is a couple of calculations that make the encryption

and the decoding. The itemized operation of a cipher is controlled both by the cal-

culation and in every occurrence by a "key". This is a mystery (in a perfect world

known just to the communicants), ordinarily a short series of characters, which is

expected to decode the ciphertext.

The AES otherwise called the Rijndael calculation was chosen as a Standard

by National Foundation of Standards and Technology (NIST).Advanced Encryp-

tion Standard (Rijndael Block Cipher) turned into the new US Federal Information

Processing Standard on November 26, 2001[17, 5] to supplant the Data Encryption

Standard (DES) which was utilized for more than 20 a long time as a typical key

4

piece figure for FIPS.

Cryptography has a fundamental part in inserted frameworks outline. To-

day innovation going on profound submicron innovation as the quantity of gad-

gets and applications which send and get information are expanding quickly the

information exchange rates are getting to be higher. In numerous applications,

this information obliges a secured association which is generally accomplished by

cryptography.[14]

1.2.1 Cryptography is partitioned into two classifications:

1. Conventional Cryptography: In conventional cryptography, likewise called

secret key or symmetric-key encryption, one key is utilized both for encryption

and unscrambling. An amazingly straightforward illustration of traditional cryp-

tography is a substitution figure. A substitution figure substitutes one bit of data

for another. This is most as often as possible done by balancing letters of the alpha-

bet,for case: Julius Caesar’s cipher.The calculation is to balance the letter set and

the key is the quantity of characters to counterbalance it.

Case in point, on the off chance that we encode "SECRET" utilizing Caesar’s key

estimation of 3, we balance the letters in order so that the 3rd letter down (D) starts

the letter set.

So beginning with

ABCDEFGHIJKLMNOPQRSTUVWXYZ

also, sliding everything up by 3, you get

DEFGHIJKLMNOPQRSTUVWXYZABC where D=A, E=B, F=C, etc

Utilizing this plan, the plain text, "SECRET" scrambles as "VHFUHW." To per-

mit another person to peruse the cipher text, you let them know that the key is 3.

Clearly, this is exceedingly feeble cryptography by all accounts.

5

2. Public Key Cryptography: The issues of key circulation are comprehended

by Public key cryptography.Public key cryptography is a hilter kilter plan that uses

a couple of keys for encryption: an open key, which scrambles information, and a

relating private, or mystery key for decoding. You distribute your open key to

the world while keeping your private key secret.It is computationally infeasible to

reason the private key from general society key. Any individual who has an open

key can scramble data however can’t unscramble it. Just the individual who has

the relating private key can decode the data.

Public key cryptography relies on the presence of one-way capacities, or nu-

merical capacities that are anything but difficult to figure while their reverse ca-

pacity is generally hard to register.

Here are two basic cases:

1. Multiplication versus factorization: Suppose we have two prime numbers,

5 and 11, and that we need to ascertain the product; it ought to take no time to

figure that esteem, which is 55. Presently assume, rather, we have a number, 55,

and we need which combine of prime numbers are duplicated together to acquire

that number. We will in the long run concoct the arrangement yet while figuring

the product took seconds, factorization will take longer. The issue turns out to be

much harder on the off chance that we begin with primes that have 500 digits or

somewhere in the vicinity, in light of the fact that the product will have 1000 digits.

2. Exponentiation versus logarithms: Suppose we need to take the number 4 to

the 5th power; once more, it is moderately simple to figure 45 = 1024. Anyhow, we

have the number 1024 and needs to know the two whole numbers that are utilized,

x and y so that logx1024 = y, it will take more time to locate the two numbers.

While the samples above are unimportant, they do speak to two of the use-

ful sets that are utilized with open key cryptography; in particular, the simplicity

6

of duplication and exponentiation versus the relative trouble of factorization and

figuring logarithms, separately.

A few samples of open key cryptosystems are Elgamal (named for its creator,

Taher Elgamal), RSA (named for its creators, Ron Rivest, Adi Shamir, and Leonard

Adleman).

Cryptanalysis : [15] The objective of cryptanalysis is to discover some shortcom-

ing or instability in a cryptographic plan, consequently allowing its subversion

or avoidance. It is a typical confusion that each encryption strategy can be bro-

ken. Regarding his WWII work at Bell Labs, Claude Shannon demonstrated that

the one-time pad figure is unbreakable, if the key material is genuinely irregular,

never reused, kept mystery from all conceivable aggressors, and of equivalent or

more noteworthy length than the message. Most ciphers, aside from the one-time

pad, can be broken with enough computational exertion by savage power assault,

yet the measure of exertion required may be exponentially reliant on the key size,

when contrasted with the exertion required to make utilization of the cipher .

Now, I would like to come to my pivotal topic, that is, Algebra. As the field

of cryptography grows to incorporate new ideas and methods, the cryptographic

applications of number theory have likewise widened. Notwithstanding logical

number theory, increasing use has been made of algebraic number hypothesis and

algebraic geometry. Key dissemination is a chief issue in cryptographic structures

and is a genuine piece of the security subsystem of correspondence structures and

data frameworks. If customers of a social event wish to impart using symmetric

encryption,they must share an ordinary key. Secret sharing insinuates the frame-

work for scattering a riddle amongst a group, each of whom is appointed with

shares of secret. The puzzle can be recovered exactly when sufficient number of

shares are combined together. Secret sharing was created autonomously by Adi

7

Shamir[1] and George Blakley[2]. Secret sharing arrangements is ideal for secur-

ing information that is exceptionally sensitive. If customers of a social occasion

wish to pass on using symmetric encryption, they have to share a commonplace

key. A protected secret sharing arrangement disperses confers so that anyone less

than p shares has no extra information about the secret than some individual with

0 shares.

A percentage of the late applications of number theory and algebraic methods

to cryptography - the number field seive strategy for factoring large numbers, uti-

lizing bijective capacities with the end goal of encryption and decoding. We get

from it what we desire for and it is very useful in all the manner.

Why do we need bijective functions for Encryption? An encryption calculation

is not so much a well defined function, in the numerical sense. Regardless, it is

fundamental that under no situation the calculation delineate diverse inputs to

the same yield else unscrambling is inconceivable. In this way, if the encryption

calculation is deterministic, it is no less than an injective function. Also, if the de-

coding calculation can acknowledge any string as a ciphertext, then the encryption

calculation should likewise be surjective, and in this case it is bijective.

case in point

In the event that Alice needs to transmit a vital private message to Bob, it would

be shrewd of her to scramble it with an encryption capacity,

e :< messages >→< encodedmessage > (1.1)

Obviously, this capacity must be invertible for deciphering to be conceivable and

is subsequently a bijection. Its converse is the decoding capacity d(.).

8

Trapdoor functions: An open key cryptosystem has the property that somebody

who knows just how to encipher can’t utilize the enciphering key to locate the

decoding key without a restrictively extensive calculation. At the end of the day

the enciphering capacity g : X −→ Y is easy to figure once the enciphering key

E is known, yet it is hard practically speaking to register the converse capacity

g−1 : Y −→ X. That is, from the outlook of practical processability, g is not in-

vertible (without some extra data - the translating key D). Such a capacity g is

known as a trapdoor capacity. That is, a trapdoor capacity g is a capacity which is

anything but difficult to process however whose opposite g−1 is difficult to figure

without having some extra assistant data past what is important to register g. The

backwards g−1 is anything but difficult to figure, nonetheless, for somebody who

has this data D.

N. Chandramowliswaran et.al proposed a new key generation algorithms us-

ing theory of numbers in 2012[11], using Pells equation in 2013 [7] , using given

set of primes 2014 [6] , using non-homogenous equations [9]. A large portion of

the specialized terms can be found there or in the course readings, for example,

Apostol (1976)[13], Koblitz (1994)[10], Stallings[16], An Introduction to the Theory

of Numbers by Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery [3].

Many basic concepts can be found in the above mentioned books or in the talk

given by N. Chandramowliswaran [8] or Number Theory as given by Naoki Sato

[4].

9

Chapter 2

Premilinaries

2.1 Bijective Functions:

A bijection is a capacity between the components of two sets, where each com-

ponent of one set is matched with precisely one component of the other set, and

each component of the other set is combined with precisely one component of the

first set. There are no unpaired components.Mathematically, a bijective function

f : A −→ B is a one-to-one (injective) and onto (surjective) mapping of a set A to a

set B.

A bijection from the set A to the set B has a converse capacity from B to A. On

the off chance that A and B are limited sets, then the presence of a bijection implies

they have the same number of components.

For a mapping between 2 sets A and B to be a bijection, four properties must

hold:

• every component of A absolute necessity be combined with no less than one

component of B

• no component of A may be combined with more than one component of B

• every component of B must be combined with no less than one component

of A

• no component of B may be combined with more than one component of A

10

Example: The function f(x) = x from the set of integers to integers is injective

and surjective. Thus it is also bijective.

2.2 GCD

Theorem: Given any two integers a and b, there is one and only one number d

with the following properties:

• d ≥ 0

• d|a and d|b

• e|a and e|b =⇒ e|d

Note. d = 0 if, and only if a = b = 0 . Otherwise d ≥ 1

• The number d of the above theorem is called the greatest common divisor

(GCD) of a and b.

• It is denoted by (a, b)

• If (a, b) = 1 then a and b are said to be relatively prime.

2.2.1 The gcd has the following properties:

1. (a, b) = (b, a) (commutative)

2. (a, (b, c)) = ((a, b), c) (associative)

3. (ac, bc) = |c|(a, b) (distributive)

4. (a, 1) = (1, a) = 1 and (a, 0) = (0, a) = 0

5. If a|bc and if (a, b) = 1 , then a|c

11

6. Let (a, b) = 1 and d be any divisor of a + b , that is, d|a + b. then (a, d) =

(b, d) = 1

7. If (a, b) = 1 =⇒ (am, bn) = 1 for m ≥ 1 and n ≥ 1

8. (a, b) = 1,(a, c) = 1 and (b, c) = 1 if, and only if (abc, ab+ ac+ bc) = 1

2.3 Prime Numbers

Definition : An integer p is called prime if p > 1 and if the only positive divisors

of p are 1 and p. When an integer p is not prime, we say that p is composite.

• If a prime p does not divide a, then (p, a) = 1

• If a prime p divides ab, then p|a or p|b

• Any two consecutive integers are relatively prime.

2.4 Fundamental theorem of arithmetic

Every integer n > 1 can be represented as a product of prime factors in only one

way, apart from the order of the factors.

2.5 Definition and basic properties of congruences

Given integers a,b,m with m > 0. We say that a is congruent to b modulo m, and

we write

a ≡ b(Modm) (2.1)

if m divides the difference a− b or m|(a− b).

The number m is called the modulus of the congruence.

12

Example: 20 ≡ 3(Mod17)

2.5.1 Properties of Congruence:

1. Congruence is an equivalence relation,i.e.,

• a ≡ a(Modm) (reflexivity)

• a ≡ b(Modm) =⇒ b ≡ a(Modm) (symmetry)

• a ≡ b(Modm) and b ≡ c(Modm) =⇒ a ≡ c(Modm) (transitivity)

2. If a ≡ b(Modm) and c ≡ d(Modm). then

3. ax+ cy ≡ bx+ dy(Modm) for all integers x and y.

4. ac ≡ bd(Modm)

5. an ≡ bn(Modm) for every positive integer n.

6. f(a) ≡ f(b)(Modm) for every polynomial f with integer coefficients.

7. if c > 0 then a ≡ b(Modm) if and only if ac ≡ bc(Modmc).

8. Assume a ≡ b(Modm). If d|m and d|a then d|b.

9. If a ≡ b(Modm) then (a,m) = (b,m)

10. If a ≡ b(Modm) and if 0 ≤ |b− a| ≤ m , then a = b.

11. If a ≡ b(Modm) and a ≡ b(Modn) where (m,n) = 1,then a ≡ b(Modmn).

12. Let a,b,c be positive integers such that

(a, b) = 1, (a, c) = 1, (b, c) = 1 (2.2)

13

13. Let x,y be any given integers.

x ≡ y(Moda), x ≡ y(Modb), x ≡ y(Modc) (2.3)

then

x ≡ y(Modabc) (2.4)

14. Let p,q be two distinct primes. Then

pq−1 + qp−1 ≡ 1(Modpq) (2.5)

We can generalize the above result for any three primes p,q,r with

qr−1 + rq−1 ̸≡ 0(Modp) (2.6)

rp−1 + pr−1 ̸≡ 0(Modq) (2.7)

pq−1 + qp−1 ̸≡ 0(Modr) (2.8)

2.5.2 Lemma (N. Chandramowliswaran et al.):

[6]

Let p,q and r be three given distinct odd primes. Then there exist integers k1, k2

and k3 such that

k1p(qr−1 + rq−1) + k2q(p

r−1 + rp−1) + k3r(pq−1 + qp−1) + 2 ≡ 0(Modpqr) (2.9)

14

Proof: Define X = pq−1 + qp−1 + pr−1 + rp−1 + qr−1 + rq−1 − 2.

Then,

X ≡ (qr−1 + rq−1)(Modp) (2.10)

X ≡ (pr−1 + rp−1)(Modq) (2.11)

and,

X ≡ (pq−1 + qp−1)(Modr) (2.12)

By CRT,the above system of congruences has exactly one solution modulo the

product pqr.

Define M = pqr then Mp = M/p = qr,Mq = M/q = pr and Mr = M/r = pq.

Since (Mp, p) = 1, then there is a unique Mp1 such that MpMp1 = 1(Modp).

Similarly there are unique Mq1 and Mr1 such that MqMq1 = 1(Modq) and MrMr1 =

1(Modr).

Consider

X ≡ ((pq−1 + qp−1)MrMr1 + (pr−1 + rp−1)MqMq1 + (qr−1 + rq−1)MpMp1)(Modpqr)

(2.13)

that is,

pq−1+qp−1+pr−1+rp−1+qr−1+rq−1−2 ≡ ((pq−1+qp−1)MrMr1+(pr−1+rp−1)MqMq1

(2.14)

+(qr−1 + rq−1)MpMp1)(Modpqr)

15

=⇒ −2 ≡ ((pq−1 + qp−1)(MrMr1 − 1) + (pr−1 + rp−1)(MqMq1 − 1)+ (2.15)

(qr−1 + rq−1)(MpMp1 − 1))(Modpqr)

Thus,

k1p(qr−1 + rq−1) + k2q(p

r−1 + rp−1) + k3r(pq−1 + qp−1) + 2 ≡ 0(Modpqr) (2.16)

2.6 Linear Congruences

A equation of the form

ax ≡ b(Modm) (2.17)

where a,b,m are positive integers and x is a variable is called a Linear Congruence.

Example 2x ≡ 3(Mod4) has no solutions, since 2x − 3 is odd for every x and

therefore cannot be divisible by 4.

Theorem 1: Assume (a,m) = 1. Then the linear congruence

ax ≡ b(Modm) (2.18)

has exactly one solution.

Theorem 2: Assume (a,m) = d. Then the linear congruence

ax ≡ b(Modm) (2.19)

has solutions, if and only if, d|b.

16

Theorem 3: If (a, b) = d there exist integers x and y such that

ax+ by = d (2.20)

2.7 Eulers phi function

Φ(n) is an arithmetic function that counts the number of positive integers less than

or equal to n that are relatively prime to n.

• Φ(n) = |{k : 1 ≤ k ≤ n, gcd(n, k) = 1}|

• Φ(n) = nΠp|n(1− 1/p) where the product is over the distinct prime numbers

dividing n.

• Φ(pk) = pk − pk−1 = pk − 1(p− 1) = pk(1− 1/p)

Examples:

• let n = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9.

The other six numbers in the range 1 ≤ k ≤ 9 , that is, {1, 2, 4, 5, 7, 8} are

relatively prime to 9.

Therefore, Φ(9) = 6.

As another example, gcd(1) = 1 since gcd(1, 1) = 1.

• Φ(36) = Φ(2232) = 36(1− 1/2)(1− 1/3) = 36.1/2.2/3 = 12

2.7.1 Properties:

1. The phi function is a multiplicative function, meaning that if two numbers m

and n are relatively prime (gcd(m,n) = 1), then

Φ(mn) = Φ(m)Φ(n) (2.21)

17

2. The phi function is important mainly because it gives the order of the multi-

plicative group of integers modulo n (the group of units of the ring Z/nZ).

3. The phi function also plays a key role in the definition of the RSA encryption

system.

2.8 Euler- Fermat Theorem

Assume (a,m)=1. Then we have

aΦ(m) ≡ 1(Modm) (2.22)

It also follows that

aΦ(m)+1 ≡ a(Modm) (2.23)

Example: If a = 5 ,m = 6 and (5, 6) = 1 Then Φ(6) = (2 − 1) ∗ (3 − 1) = 2 So,

5Φ(6) = 52 = 25 and 25 ≡ 1(Mod6)

2.9 Fermats Little Theorem

If p is a prime number, then for any integer a, the number ap − a is an integer

multiple of p.

ap ≡ a(Modp) (2.24)

If a is not divisible by p, Fermat’s little theorem is equivalent to the statement

that ap−1 − 1 is an integer multiple of p, or in symbols

ap−1 ≡ 1(Modp) (2.25)

18

Example: take p = 7, a = 2

27 = 128 ≡ 2(Mod7)

and since 2 does not divide 7, therefore,

27−1 = 26 = 64 ≡ 1(Mod7)

2.10 Chinese Remainder Theorem

If m1,m2, .......,mk are pair wise relatively prime positive integers, and if a1, a2, ....., ak

are any integers, then the simultaneous congruences

x ≡ a1(Modm1), x ≡ a2(Modm2), ..., x ≡ ak(Modmk) (2.26)

have a solution, and the solution is unique modulo m,

where m = m1m2.......mk

Example:

m1 ≡ 2(Mod3),m2 ≡ 1(Mod5) (2.27)

then x(unique) ∈ {1, 2, 3, ..., 14}

here x= 11

2.11 Möbius function

The Möbius function µ is defined as follows:

• µ(1) = 1

• µ(p1.p2.p3.......pk) = (−1)k where p1.p2.p3.......pk are all distinct primes.

• µ(n) = 0, otherwise

therefore, µ(n) = 0 if,and only if n has a square factor > 1.

19

2.11.1 Properties:

1. For any positive integer N > 1,

Σd,d|Nµ(d) = 0

2. µ(N) = ΣNk=1,(k,N)=1e

2Πik/N

20

Chapter 3

Encryption and Decryption on Z(3)N and Z

(4)N

3.1 A: Find a new bijective map f : Z(4)N −→ Z

(4)N

STEP 1: Let N be a given positive integer.

Define the matrix A an B as

A =

a b 0 0

0 0 a b

c d 0 0

0 0 c d

here (a,N) = 1,(b,N) = 1,(c,N) = 1,(d,N) = 1

B =

m1 b12 b13 b14

b21 b22 m3 b24

b31 b32 b33 m4

b41 m2 b43 b44

where bi,j ∈ {0, 1, 2, ....., N − 1}

and message m as

m =

m1

m2

m3

m4

where mi,j ∈ {0, 1, 2, ....., N − 1}

21

Define C = AB

Cij =

a b 0 0

0 0 a b

c d 0 0

0 0 c d

m1 b12 b13 b14

b21 b22 m3 b24

b31 b32 b33 m4

b41 m2 b43 b44

STEP 2:

f

m1

m2

m3

m4

=

am1 + bb21

ab32 + bm2

cb13 + dm3

cm4 + db44

f is bijective?

f is one-one?

f

m1

m2

m3

m4

= f

n1

n2

n3

n4

am1 + bb21

ab32 + bm2

cb13 + dm3

cm4 + db44

=

an1 + bb21

ab32 + bn2

cb13 + dn3

cn4 + db44

On comparing

am1 + bb21 = an1 + bb21 =⇒ am1 = an1 =⇒ m1 = n1 (3.1)

22

ab32 + bm2 = ab32 + bn2 =⇒ bm2 = bn2 =⇒ m2 = n2 (3.2)

cb13 + dm3 = cb13 + dn3 =⇒ dm3 = dn3 =⇒ m3 = n3 (3.3)

cm4 + db44 = cn4 + db44 =⇒ cm4 = cn4 =⇒ m4 = n4 (3.4)

m1

m2

m3

m4

=

n1

n2

n3

n4

Therefore, f is one-one

f is onto because Z(4)N is finite and f is one-one

STEP 3: Calculating f−1

f

m1

m2

m3

m4

=

am1 + bb21

ab32 + bm2

cb13 + dm3

cm4 + db44

f−1

m1

m2

m3

m4

=

n1

n2

n3

n4

now

23

an1 + bb21 = m1 =⇒ an1 = m1 − bb21 =⇒ n1 = a1m1 − a1bb21 (3.5)

where a1a ≡ 1(ModN)

ab32 + bn2 = m2 =⇒ bn2 = m2 − ab32 =⇒ n2 = b1m2 − b1ab32 (3.6)

where b1b ≡ 1(ModN)

cb13 + dn3 = m3 =⇒ dn3 = m3 − cb13 =⇒ n3 = d1m3 − d1cb13 (3.7)

where d1d ≡ 1(ModN)

cn4 + db44 = m4 =⇒ cn4 = m4 − db44 =⇒ n4 = c1m4 − c1db44 (3.8)

where c1c ≡ 1(ModN)

f−1

m1

m2

m3

m4

=

n1

n2

n3

n4

=

a1m1 − a1bb21

b1m2 − b1ab32

d1m3 − d1cb13

c1m4 − c1db44

m1

m2

m3

m4

= f

n1

n2

n3

n4

=

an1 + bb21

ab32 + bn2

cb13 + dn3

cn4 + db44

Example: Take N = 4. Define A and B as

24

A =

1 2 0 0

0 0 1 2

3 1 0 0

0 0 3 1

B =

x 2 1 2

2 0 z 2

3 1 1 w

3 y 3 1

then,

f(x, y, z, w) = (x+ 4, 1 + 2y, 3 + z, 3w + 1) (3.9)

It can be easily proved that f is bijective and inverse can be easily computed.

Hence, the message (0, 2, 1, 3) can be encrypted as (4, 5, 4, 10)

3.1.1 Composition:

(fog)

m1

m2

m3

m4

= f

a2m1 + b2b21

a2b32 + b2m2

c2b13 + d2m3

c2m4 + d2b44

=

a1a2m1 + a1b2b21 + b1b21

a1b32 + b1a2b32 + b1b2m2

c1b13 + d1c2b13 + d1d2m3

c1c2m4 + c1d2b44 + d1b44

Inverse of Composition:

(fog)−1

m1

m2

m3

m4

=

n1

n2

n3

n4

a1a2n1 + a1b2b21 + b1b21 = m1 =⇒ a1a2n1 = m1 − a1b2b21 − b1b21 (3.10)

=⇒ n1 = A2A1m1 − A2A1a1b2b21 − A2A1b1b21 (3.11)

25

where A1a1 ≡ 1(ModN) and A2a2 ≡ 1(ModN)

a1b32 + b1a2b32 + b1b2n2 = m2 =⇒ b1b2n2 = m2 − a1b32 − b1a2b32 (3.12)

=⇒ n2 = B2B1m2 −B2B1a1b32 −B2B1b1a2b32 (3.13)

where B1b1 ≡ 1(ModN) and B2b2 ≡ 1(ModN)

c1b13 + d1c2b13 + d1d2n3 = m3 =⇒ d1d2n3 = m3 − c1b13 − d1c2b13 (3.14)

=⇒ n3 = D2D1m3 −D2D1c1b13 −D2D1d1c2b13 (3.15)

where D1d1 ≡ 1(ModN) and D2d2 ≡ 1(ModN)

c1c2n4 + c1d2b44 + d1b44 = m4 =⇒ c1c2n4 = m4 − c1d2b44 − d1b44 (3.16)

=⇒ n4 = C2C1m4 − C2C1c1d2b44 − C2C1d1b44 (3.17)

where C1c1 ≡ 1(ModN) and C2c2 ≡ 1(ModN)

(fog)−1

m1

m2

m3

m4

=

n1

n2

n3

n4

=

A2A1m1 − A2A1a1b2b21 − A2A1b1b21

B2B1m2 −B2B1a1b32 −B2B1b1a2b32

D2D1m3 −D2D1c1b13 −D2D1d1c2b13

C2C1m4 − C2C1c1d2b44 − C2C1d1b44

26

m1

m2

m3

m4

= (fog)

n1

n2

n3

n4

3.2 B: Find a new bijective map f : Z(4)N −→ Z

(4)N

STEP 1: Let N be a given positive integer.

Define the matrix A an B as

A =

a 0 0 b

0 a b 0

c 0 0 d

0 c d 0

here (a,N) = 1,(b,N) = 1,(c,N) = 1,(d,N) = 1

B =

b11 b12 m3 b14

b21 m2 b23 b24

b31 b32 b33 m4

m1 b42 b43 b44

where bi,j ∈ {0, 1, 2, ....., N − 1}

and message m as

m =

m1

m2

m3

m4

where mi,j ∈ {0, 1, 2, ....., N − 1}

27

Define C = AB

Cij =

a 0 0 b

0 a b b

c 0 0 d

0 c d 0

b11 b12 m3 b14

b21 m2 b23 b24

b31 b32 b33 m4

m1 b42 b43 b44

STEP 2:

f

m1

m2

m3

m4

=

ab11 + bm1

am2 + bb32

cm3 + db43

cb24 + dm4

3.2.1 f is bijective?

f is one-one?

f

m1

m2

m3

m4

= f

n1

n2

n3

n4

ab11 + bm1

am2 + bb32

cm3 + db43

cb24 + dm4

=

ab11 + bn1

an2 + bb32

cn3 + db43

cb24 + dn4

On comparing

ab11 + bm1 = ab11 + bn1 =⇒ bm1 = bn1 =⇒ m1 = n1 (3.18)

am2 + bb32 = an2 + bb32 =⇒ am2 = an2 =⇒ m2 = n2 (3.19)

28

cm3 + db43 = cn3 + db43 =⇒ cm3 = cn3 =⇒ m3 = n3 (3.20)

cb24 + dm4 = cb24 + dn4 =⇒ dm4 = dn4 =⇒ m4 = n4 (3.21)

m1

m2

m3

m4

=

n1

n2

n3

n4

Therefore, f is one-one

f is onto because Z(4)N is finite and f is one-one

therefore, f is bijective

STEP 3: Calculating f−1

f

m1

m2

m3

m4

=

ab11 + bm1

am2 + bb32

cm3 + db43

cb24 + dm4

f−1

m1

m2

m3

m4

=

n1

n2

n3

n4

now

ab11 + bn1 = m1 =⇒ bn1 = m1 − ab11 =⇒ n1 = b1m1 − b1ab11 (3.22)

29

where b1b ≡ 1(ModN)

an2 + bb32 = m2 =⇒ an2 = m2 − bb32 =⇒ n2 = a1m2 − a1bb32 (3.23)

where a1a ≡ 1(ModN)

cn3 + db43 = m3 =⇒ cn3 = m3 − db43 =⇒ n3 = c1m3 − c1db43 (3.24)

where c1c ≡ 1(ModN)

cb24 + dn4 = m4 =⇒ dn4 = m4 − cb24 =⇒ n4 = d1m4 − d1cb24 (3.25)

where d1d ≡ 1(ModN)

f−1

m1

m2

m3

m4

=

n1

n2

n3

n4

=

b1m1 − b1ab11

a1m2 − a1bb32

c1m3 − c1db43

d1m4 − d1cb24

m1

m2

m3

m4

= f

n1

n2

n3

n4

=

ab11 + bn1

an2 + bb32

cn3 + db43

cb24 + dn4

3.2.2 Composition:

(fog)

m1

m2

m3

m4

= f

a2b11 + b2m1

a2m2 + b2b32

c2m3 + d2b43

c2b24 + d2m4

=

a1b11 + b1a2b11 + b1b2m1

a1a2m2 + a1b2b32 + b1b32

c1c2m3 + c1d2b43 + d1b43

c1b24 + d1c2b24 + d1d2m4

30

Inverse of Composition:

(fog)−1

m1

m2

m3

m4

=

n1

n2

n3

n4

a1b11 + b1a2b11 + b1b2n1 = m1 =⇒ b1b2n1 = m1 − a1b11 − b1a2b11 (3.26)

=⇒ n1 = B2B1m1 −B2B1a1b11 −B2B1b1a2b11 (3.27)

where B1b1 ≡ 1(ModN) and B2b2 ≡ 1(ModN)

a1a2n2 + a1b2b32 + b1b32 = m2 =⇒ a1a2n2 = m2 − a1b2b32 − b1b32 (3.28)

=⇒ n2 = A2A1m2 − A2A1a1b2b32 − A2A1b1b32 (3.29)

where A1a1 ≡ 1(ModN) and A2a2 ≡ 1(ModN)

c1c2n3 + c1d2b43 + d1b43 = m3 =⇒ c1c2n3 = m3 − c1d2b43 − d1b43 (3.30)

=⇒ n3 = C2C1m3 − C2C1c1d2b43 − C2C1d1b43 (3.31)

where C1c1 ≡ 1(ModN) and C2c2 ≡ 1(ModN)

c1b24 + d1c2b24 + d1d2n4 = m4 =⇒ d1d2n4 = m4 − c1b24 − d1c2b24 (3.32)

31

=⇒ n4 = D2D1m4 −D2D1c1b24 −D2D1d1c2b24 (3.33)

where D1d1 ≡ 1(ModN) and D2d2 ≡ 1(ModN)

(fog)−1

m1

m2

m3

m4

=

n1

n2

n3

n4

=

B2B1m1 −B2B1a1b11 −B2B1b1a2b11

A2A1m2 − A2A1a1b2b32 − A2A1b1b32

C2C1m3 − C2C1c1d2b43 − C2C1d1b43

D2D1m4 −D2D1c1b24 −D2D1d1c2b24

m1

m2

m3

m4

= (fog)

n1

n2

n3

n4

3.3 C: Find a new bijective map f : Z(3)N −→ Z

(3)N

STEP 1: Let N be a given positive integer. Define the matrix A and B as follows.

A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

where ai,j ∈ {0, 1, 2, ....., N − 1}

B =

b11 m1 m3

m2 m2 b23

m1 b32 m3

where bi,j ∈ {0, 1, 2, ....., N − 1}

32

and message m as

m =

m1

m2

m3

where mi,j ∈ {0, 1, 2, ....., N − 1}

Define C = AB

Cij =

a11 a12 a13

a21 a22 a23

a31 a32 a33

b11 m1 m3

m2 m2 b23

m1 b32 m3

STEP 2

f

m1

m2

m3

=

a11b11 + a12m2 + a13m1

a21m1 + a22m2 + a23b32

a31m3 + a32b23 + a33m3

=

d1

d2

d3

a13m1 + a12m2 = d1 − a11b11 (3.34)

a21m1 + a22m2 = d2 − a23b32 (3.35)

(a31 + a33)m3 = d3 − a32b23 (3.36)

that is,

a13 a12

a21 a22

m1

m2

=

d1 − a11b11

d2 − a23b32

where

33

((a13a22 − a12a21), N) = 1 (3.37)

and

(a31 + a33, N) = 1 (3.38)

In General the function f can be written as:

f : Z(3)N −→ Z

(3)N (3.39)

f(x, y, z) = (α+ ax+ by, β + cx+ dy, γ + kz) (3.40)

f is bijective? f is one-one?

Suppose,

f(x, y, z) = f(x1, y1, z1)

(α + ax+ by, β + cx+ dy, γ + kz) = (α + ax1 + by1, β + cx1 + dy1, γ + kz1) (3.41)

=⇒ α + ax+ by = α+ ax1 + by1 =⇒ ax+ by = ax1 + by1 (3.42)

and

=⇒ β + cx+ dy = β + cx1 + dy1 =⇒ cx+ dy = cx1 + dy1 (3.43)

a(x− x1) + b(y − y1) = 0(ModN) (3.44)

and

c(x− x1) + d(y − y1) = 0(ModN) (3.45)

34

that is,

a b

c d

x− x1

y − y1

=

0

0

Since

((ad− bc),ModN,N) = 1

therefore,

a b

c d

is invertible

x− x1 = 0 =⇒ x = x1 (3.46)

and

y − y1 = 0 =⇒ y = y1 (3.47)

also,

γ + kz = γ + kz1 =⇒ kz = kz1 =⇒ z = z1 (3.48)

that is, (x, y, z) = (x1, y1, z1)

f is one-one

f is onto because Z(3)N is finite and f is one-one.

Therefore, f is bijective.

Calculating f−1:

f−1(x, y, z) = (x1, y1, z1) (3.49)

35

(x, y, z) = f(x1, y1, z1) (3.50)

=⇒ (x, y, z) = (α + ax1 + by1, β + cx1 + dy1, γ + kz1) (3.51)

that is,

ax1 + by1 = x− α (3.52)

cx1 + dy1 = y − β (3.53)

and

kz1 = z − γ (3.54)

a b

c d

x1

y1

=

x− α

y − β

from here x1, y1 can be easily calculated.

and,

z1 = k−1(z − γ) = k1(z − γ) (3.55)

where, kk1 ≡ 1(ModN)

Example: Take N=3. then,

f : Z(3)N −→ Z

(3)N (3.56)

take α = 1, β = 2, γ = 0

36

then,

f(x, y, z) = (1 + ax+ by, 2 + cx+ dy, kz) (3.57)

now,

(ad− bc(Mod3), 3) = 1 and (k, 3) = 1

therefore, choosing a = 1, b = 1, c = 1, d = 2, k = 2

then,

f(x, y, z) = (1 + x+ y, 2 + x+ 2y, 2z) (3.58)

It can be easily proved that f is bijective and inverse can be easily computed.

Hence, a message (1, 2, 1) is encrypted as (4, 7, 2)

3.3.1 Total Number of Bijective Functions

Let N be a given fixed positive integer such that N ≥ 3

f : Z(3)N −→ Z

(3)N (3.59)

f(x, y, z)α,β,γ;a,b,c,d;,k = (α + ax+ by, β + cx+ dy, γ + kz)

Here ((ad− bc)(ModN), N) = 1 and (k,N) = 1

α, β, γ, a, b, c, d, r, k ∈ ZN

|f[α,β,γ;a,b,c,d;r,k] : Z(3)N −→ Z

(3)N | = N3Φ(N).Φ2(N) = N8Πp|N(1− 1/p)2.(1− 1/p2)

(3.60)

where p is a prime such that p|N

Φ(N) = NΠp|N(1− 1/p)

and

Φ2(N) = N4Πp|N(1− 1/p)(1− 1/p2)

37

3.4 D: Find a new bijective map f : Z(4)N −→ Z

(4)N

STEP 1: Let N be a given positive integer. Define the matrix A and B as follows.

A =

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

where ai,j ∈ {0, 1, 2, ....., N − 1}

B =

m1 b12 m1 m4

b21 m2 m2 m4

m2 m1 m3 b34

m3 m3 b43 m4

where bi,j ∈ {0, 1, 2, ....., N − 1}

and message m as

m =

m1

m2

m3

m4

where mi,j ∈ {0, 1, 2, ....., N − 1}

Define C = AB

Cij =

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

m1 b12 m1 m4

b21 m2 m2 m4

m2 m1 m3 b34

m3 m3 b43 m4

38

STEP 2

f

m1

m2

m3

m4

=

a11m1 + a21b21 + a31m2 + a41m3

a12b12 + a22m2 + a32m1 + a42m3

a13m1 + a23m2 + a33m3 + a43b43

a14m4 + a24m4 + a34b34 + a44m4

=

d1

d2

d3

d4

a11m1 + a31m2 + a41m3 = d1 − a21b21 (3.61)

a32m1 + a22m2 + a42m3 = d2 − a12b12 (3.62)

a13m1 + a23m2 + a33m3 = d3 − a43b43 (3.63)

(a14 + a24 + a44)m4 = d4 − a34b34 (3.64)

that is,

a11 a31 a41

a32 a22 a42

a13 a23 a33

m1

m2

m3

=

d1 − a21b21

d2 − a12b12

d3 − a43b43

where

(a11(a22a33 − a42a23)− a31(a32a33 − a42a13) + a41(a32a23 − a22a13), N) = 1 (3.65)

and

(a14 + a24 + a44, N) = 1 (3.66)

39

In General the function f can be written as:

f : Z(4)N −→ Z

(4)N (3.67)

f(x, y, z, w) = (α+ ax+ by + cz, β + dx+ ey + fz, γ + ix+ jy + kz, δ + pw) (3.68)

f is bijective? f is one-one?

Suppose,

f(x, y, z, w) = f(x1, y1, z1, w1)

(α + ax+ by + cz, β + dx+ ey + fz, γ + ix+ jy + kz, δ + pw) = (3.69)

(α + ax1 + by1 + cz1, β + dx1 + ey1 + fz1, γ + ix1 + jy1 + kz1, δ + pw1) (3.70)

that is,

α + ax+ by + cz = α+ ax1 + by1 + cz1 (3.71)

=⇒ ax+ by + cz = ax1 + by1 + cz1 (3.72)

β + dx+ ey + fz = β + dx1 + ey1 + fz1 (3.73)

=⇒ dx+ ey + fz = dx1 + ey1 + fz1 (3.74)

40

and

γ + ix+ jy + kz = γ + ix1 + jy1 + kz1 (3.75)

=⇒ ix+ jy + kz = ix1 + jy1 + kz1 (3.76)

that is,

a(x− x1) + b(y − y1) + c(z − z1) = 0(ModN) (3.77)

d(x− x1) + e(y − y1) + f(z − z1) = 0(ModN) (3.78)

and

i(x− x1) + j(y − y1) + k(z − z1) = 0(ModN) (3.79)

that is,

a b c

d e f

i j k

x− x1

y − y1

z − z1

=

0

0

0

Since

(a(ek − fj)− b(dk − fi) + c(dj − ei)), N) = 1 (3.80)

therefore,

41

a b c

d e f

i j k

is invertible

x− x1 = 0 =⇒ x = x1 (3.81)

y − y1 = 0 =⇒ y = y1 (3.82)

and

z − z1 = 0 =⇒ z = z1 (3.83)

also,

δ + pw = δ + pw1 =⇒ pw = pw1 =⇒ w = w1 (3.84)

that is, (x, y, z, w) = (x1, y1, z1, w1)

f is one-one

f is onto because Z(4)N is finite and f is one-one.

Therefore, f is bijective.

Calculating f−1:

f−1(x, y, z, w) = (x1, y1, z1, w1) (3.85)

(x, y, z, w) = f(x1, y1, z1, w1) (3.86)

42

=⇒ (x, y, z) = (α+ax1+by1+cz1, β+dx1+ey1+fz1, γ+ix1+jy1+kz1, δ+pw1) (3.87)

that is,

ax1 + by1 + cz1 = x− α (3.88)

dx1 + ey1 + fz1 = y − β (3.89)

ix1 + jy1 + kz1 = z − γ (3.90)

and

pw1 = w − δ (3.91)

a b c

d e f

i j k

x1

y1

z1

=

x− α

y − β

z − γ

from here x1, y1, z1 can be easily calculated.

and,

w1 = p−1(w − δ) = p1(w − δ) (3.92)

where, pp1 ≡ 1(ModN)

43

Chapter 4

Conclusion and Future Work

4.1 Conclusion

This thesis gives the fundamental methods to construct bijective functions on Z(3)N

and Z(4)N . Essential ideas regarding encryption and decryption have been pro-

vided. We have proposed novel encryption and decoding procedures utilizing

bijective functions, sparse matrices and simultaneous mathematical equations. Be-

sides, by utilizing these plans we can permit just better subsets to perform activity

in a framework. Every procedure is a novel in its own particular manner, which

may be suitable for diverse applications. The more challenging aspect is to find an

alternative public key cryptosystem.

4.2 Future Work

We will extend our result from Z(3)N , Z

(4)N to Z

(p)N . Likewise, we will figure out the

general formula for compositions, compute the general equation for the number

of bijective functions. f : Z(4)N −→ Z

(4)N and extend it to Z

(p)N .

We will attempt to discover new and proficient trap door functions.

44

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46

PROFILE

AINA GUPTA

Qualification: - Pursuing M.Sc. (Mathematics)

B.Sc. (Hons.) Mathematics

Contact Address: - H. No - 179, Sector-23, Gurgaon, Haryana

Email: - ainagupta01@gmail.com

Phone No: - 08447351589