Laplace and Inverse Laplace Transform Techniques in Cryptography Using the Series t sinh?rt

Abstract

Cryptography plays a vital role in securing information in modern digital communication systems. With the rapid expansion of electronic transactions and online services, ensuring secure data transmission has become increasingly essential. This paper proposes a novel mathematical approach to cryptography based on the Laplace Transform. In the proposed method, plaintext is converted into a numerical or functional representation and encrypted using the Laplace Transform, generating a transformed ciphertext. The original message is retrieved by applying the corresponding Inverse Laplace Transform. The use of mathematical transformation introduces additional computational complexity, thereby enhancing resistance against unauthorized access and attacks. The results indicate that the proposed technique improves data security while maintaining computational efficiency and flexibility. This approach demonstrates the potential of mathematical transforms in designing secure cryptographic systems. Potential applications include banking security, password protection, secure data transmission, and one-time password (OTP) systems.

Keywords

Cryptography, Laplace Transform, Hyperbolic Functions, Modular Arithmetic, Secure Communication.

Introduction

Information security is a fundamental requirement in modern digital communication systems, driven by the rapid growth of online transactions, cloud computing, and data exchange technologies. Ensuring the confidentiality, integrity, and authenticity of information has become increasingly important. Cryptography addresses these challenges by transforming plaintext into ciphertext using mathematically reliable techniques¹.

Traditional cryptographic methods are primarily based on modular arithmetic, substitution, and permutation operations. In recent years, there has been growing interest in exploring advanced mathematical tools to develop more secure and efficient encryption schemes. In this context, mathematical transformations offer a structured and systematic framework for data encoding and decoding². This creates a demand for innovative approaches to strengthen data security in modern communication systems.

In this paper, we present a cryptographic technique based on the Laplace Transform and its inverse. The proposed method represents plaintext as coefficients of a function involving the series t sinh rt,

 and applies the transform to generate encrypted data. The original message is accurately recovered using the corresponding Inverse Laplace Transform along with key values.

 

The proposed approach combines mathematical transformation and modular arithmetic to provide a systematic and mathematically rigorous framework for secure communication. It offers enhanced security through increased computational complexity while maintaining efficiency in implementation. Potential applications include digital communication networks, password protection, secure data transmission, and banking systems.

This approach provides a novel mathematical framework for encryption using Laplace Transform techniques³. It introduces an additional layer of complexity, making unauthorized decoding more difficult, and is flexible enough to be adapted to different types of data representations. Accurate recovery of the original message is ensured through the corresponding Inverse Laplace Transform. Overall, the method demonstrates effectiveness and reliability for secure communication systems.

2. PRELIMINARIES

Definition 2.1

Plaintext refers to the original readable message that can be understood without any transformation.

Definition 2.2

When a plaintext message is encoded using an appropriate encryption scheme, the resulting message is called ciphertext.

Definition 2.3

Encryption is the process of converting a readable message (plaintext) into an encoded form (ciphertext), whereas decryption is the reverse process that restores the ciphertext back to its original plaintext form. Both encryption and decryption rely on two fundamental elements: a well-defined algorithm and a corresponding key. The key plays a crucial role in controlling the transformation process, thereby ensuring the security of the communication.

Definition 2.4

In cryptography, modular arithmetic is used to convert numerical values into letters.

For an integer a,

a≡r  mod 26

where r

 is the remainder when a

 is divided by 26.

2.1. The Laplace Transform:

If f(t)

 is a function defined for all positive values of  t

, then the Laplace transform of ft

 is defined   as

Lft=Fs=0∞e-st ft dt

provided that the integral exists. Here the parameter s

 is a real or complex number.

The corresponding Inverse Laplace transform is

L-1Fs=f(t)

2.2. Linear Property

Laplace transform is a linear transform.

That is, if

Lf1t=F1s , Lf2t=F2s,……., Lfnt =Fns

Then

                 Lc1f1t+c2f2t+…+cnfnt=c1F1s+c2F2s+…+cnFns

where c1,c2,…cn

are constants.

Laplace Transform has diverse applications in fields such as mechanics, electrical circuits, beam analysis, heat conduction, wave propagation, transmission lines, signals and systems, control engineering, communication systems, hydrodynamics, and solar studies.

It is widely employed for solving differential equations, analysing systems, and developing mathematical models. Moreover, it plays a role in cryptography by supporting encryption, decryption, and secure data transformation. Hence, it stands as a fundamental tool in both engineering applications and information security.

Here we consider the following standard results of Laplace transform

1.  L{sinhkt}= ks2-k2 ,    s≥|k|
      L-1ks2-k2=sinhkt

2.  Ltn=n!sn+1 ,  n ? N      

     L-1n!sn+1=tn,        n ? N
3.  Ltnft=-ddsnF(s)
      L-1-ddsn F(s)=tnft

4.  Ltnekt=n!s-kn+1

       L-1n!s-kn+1=tnekt

where n=0,1,2,……

 are positive integers.

3. MAIN RESULTS

3.1 Method of Encryption

We consider standard expansion

   sinh rt=rt+r3t33!+r5t55!+?+r2i+1t2i+12i+1!+?+

                   =i=0∞rt2i+1(2i+1)!
tsinh rt=rt2+r3t43!+r5t65!+r7t87!+…+…
                       =i=0∞r2i+1t2i+2(2i+1)!

where r∈N

 is a constant with N

 is the set of natural numbers.

Step 1

Select the message to be sent that is plaintext.

Step 2

The plaintext can be organized as a finite sequence of numbers as allocate 0 to A and 1 to B and so on Z will be 25.

Step 3

Our plaintext is CIRCLE

By above step our plaintext is equivalent to 2, 8, 17, 2, 11, 4.

Step 4

Write the numbers as coefficients of Gi

, Therefore our plaintext is

        G0=2,  G1=8, G2=17,  G3=2, G4=11, G5=4,

        Gn=0 for n≥6

Step 5

Write these numbers as the coefficient in   tsinhrt 

where r is a constant.

Let us consider

              ft=Gtsinh2t

              ft = tG0.2t+G123t33!+G225t55!+G327t77!+G429t99!+G5211t1111!

                         =G02t21!+G123t43!+G225t65!+G327t87!+G429t109!+G5211t1211!

             ft  = 22t21!+823t43!+1725t65!+227t87!+1129t109!+4211t1211!

Step 6

Taking Laplace transform on both sides

             Lf(t)}=L{Gtsinh2t}

=L22t21!+823t43!+1725t65!+227t87!+1129t109!+4211t1211!

                       ?Ltn=n!sn+1

Applying Laplace Transform each term separately

   L2.2t21!=2×21!×2s3

                      L2.2t21!=8s3

milarly,

                       L8.23t43!=256s5

                       L17.25t65!=3264s7

                       L2.27t87!=2048s9

                      L11.29t109!=56320s11

                      L4.211t1211!=98304s13

∴We get,

  Lft=8s3+256s5+3264s7+2048s9+56320s11+98304s13

Step 7

Therefore resultant values are      8    256     3264    2048     56320       98304

The resultant values are reduced using modulo 26

That is,
8= 8 mod 26            = 8

                                256= 256 mod 26       =22

2048=2048 mod 26     =20

                             3264=3264 mod 26      =14

56320=56320 mod 26  =20

                            98304= 98304 mod 26 =24

nce we get our ciphertext as I W O U E Y which is equivalent to 8  22   14   20   4    24 

with key ki for i=0,1,2,……

Step 8

To find the key values,

Let Original value be Ki

 and Remainder be ri

Then,
                      Key=Ki-ri26

K_i=8,       ?    r?_i=8       ⇒  (8-8)/26           =0
              K_i=256,       r_i=22    ⇒  (256-22)/26    =9
              K_i=3264,    r_i=14     ⇒  (3264-14)/26=125

Similarly, we can find key values for remaining.

Therefore, the key values are  0, 9, 125, 78, 2166, 3780

Hence the message C I R C L E is encrypted as I W O U E Y with key is 0, 9, 125, 78, 2166, 3780

Theorem 3.1

The given plaintext in terms of Gi, i=1,2,3…,

 under Laplace Transform of Gtsinhrt

 (that  is by writing them as a coefficient of tsinhrt,

 and then taking Laplace Transform)

can be converted to ciphertext Gi',

where                                                                                                                                     

                  Gi'=qi-26ki  ,   for i=0,1,2,3,……,

and

                 qi=r2i+12i+2Gi    for i=0,1,2,….

                                                r=1,2,3,….

with key,
                         Ki-ri26

3.2 Method of Decryption

Received ciphertext is I W O U E Y with keys 0   9    125    78    2166    3780

 Step 1

The ciphertext can be organized as a finite sequence of numbers as allocate 0 to A and 1 to B and so on Z will be 25.

Step 2

The ciphertext IWOUEY is equivalent to 8    22   14   20   4    24

Step 3

Write the numbers as coefficients of Gi',

 G0'=8, G1'=22, G2'=14, G3'=20, G4'=4, G5'=24, Gn'=0 for n≥6

Step 4

dd the corresponding key values to obtain qi

From theorem,

      qi=Gi'+26ki,   ki for i=0,1,2… as 0   9   125   78   2166   3780   

      q0=G0'+26k0=  8+260          =8

      q1=G1'+26k1=22+269         =256

 q2=G2'+26k2=14+26125    =3264

       q3=G3'+26k3 =20+2678     =2048

       q4=G4'+26k4 = 4+262166  =56320

       q5=G5'+26k5=24+263780 =98304

ep 5

Form the transformed expression using qi

,

G-dds2s2-22 =i=0∞qis2i+3

                                      =8s3+256s5+3264s7+2048s9+56320s11+98304s13

Step 6

Apply Inverse Laplace Transform to the expression.

=L-18s3+256s5+3264s7+2048s9+56320s11+98304s13

Step 7

Apply Inverse Laplace Transform

                                 L-11sn+1=tnn!

We know that,

f(t)=G02t21!+G123t43!+G225t65!+G327t88!+G429t1010!+G5211t1211!

Taking Inverse Laplace Transform on each term separately

                       L-18s3=8×t22!

To obtain the original coefficients Gi

Let C be integer coefficient

                         C×21!=8×t22!

                                  C=8×t22×1×12

                                C=2t2

                      L-18s3=2t2

  Similarly, we obtain

                        L-1256s5  =8t4
                        L-13264s7=17t6

        L-12048s9 =2t8

                       L-156320s11=11t10

                       L-198304s13=4t12

 Step 8

stituting these coefficients in f(t)

 we get              
     ft=Gtsinh2t

     f(t)=22t21!+823t43!+1725t65!+227t88!+1129t1010!+4211t1211!

Step 9
Convert the coefficient back to letters
That is,  2    8    17    2     11      4  is equivalent to CIRCLE
Hence the ciphertext I W O U E Y is decrypted as the plaintext C I R C L E with key 0, 9,     
125, 78, 2166, 3780.
Theorem 3.2

The given ciphertext string in terms of Gi

, i=1,2,3….

 with the given key ki for

i= 0,1,2,…,

 under Inverse Laplace transform of

G-dds2s2-22=i=0∞qis2i+3

an be converted to plaintext Gi

, where

Gi = 26ki +Gi'r2i+12i+2  ,   i=0,1,2,….

and       qi=26ki +Gi' ,     for i=0,1,2,…..

 GENERALIZATION

We now extend the results obtained in section 3 for more generalized functions. Let N

 be the set of natural numbers. For encryption of the given message string in terms of  Gi

. We                   consider the function

                               ft= Gi tjsinhrt,       where  r,j ∈ N

Following the same procedure discussed in section 3, we take the Laplace transform of ft.

Then the given message string Gi

 can be converted into the ciphertext Gi' ,

where

Gi'=Gir2i+12i+22i+3…2i+j+1mod 26

Let

        qi= Gi r2i+12i+22i+3….2i+j+1,  i=0,1,2,3…

and  the key is defined by

                              ki=qi-Gi'26     for i=0,1,2….

Theorem 4.1

The given plaintext string in terms of  Gi

,  i=1,2,3….

 under the Laplace transform of           Gtjsinhrt 

(that is, by writing them as coefficients of tjsinhrt 

and then taking the       Laplace transform) can be converted into ciphertext Gi',

 where

Gi'=qi-26ki  ,   for i=0,1,2,3,……,

and

               qi= Gi r2i+12i+22i+3….2i+j+1

with key

ki=qi-Gi'26       for i=0,1,2,….

Theorem 4.2

The given ciphertext string in terms of Gi'

,  i=1,2,3…

 with key ki

, for i=0,1,2,...

              under the Inverse Laplace transform of
G-ddsjr(s2-r2)=i=0∞qis2i+j+2,   i=0,1,2…

can be converted back into the plaintext  Gi ,

 where                         
          Gi =26ki  +Gi'r2i+12i+22i+3….2i+j+1  ,  i=0,1,2…

and

                                qi=26ki +Gi' ,     for i=0,1,2,…..

 4.1 Encryption algorithm

 i. Treat every letter in the plaintext message as a number, so that using the mapping

A=0, B=1, C=2, …, Z=25.

 ii. The plaintext message  Gi

? is organized as a finite sequence of numbers based on the above conversion.  Only consider  Gi

 till the length of input string, ie., i=0 to n-1

iii. Consider a suitable function f(t)

. Take Laplace transform and obtain the required formula for encryption. Hence each character in the input string converts to a new position Gi'

 iv. Key value for each character can be obtained from the corresponding relation

 v. Send Gi'

and ki

? as pair to the receiver.

4.2 Decryption algorithm

i. The receiver obtains the encrypted pair Gi'

and ki

ii. Use the key values to recover the original numerical sequence corresponding to the plaintext message.

iii. Arrange the obtained numbers as a finite sequence.

iv. Convert each number of the sequence into the corresponding alphabet using the mapping A=0,B=1, C=2,…, Z=25

v. The resulting string obtained is the original plaintext message.
5. ILLUSTRATIVE EXAMPLES

Suppose the original message be string ‘CIRCLE’. Using the results of section 4, we can   convert it to

1. ‘MEMOOA’  for  r=1, j=2

2. ‘SKEGIA’      for r=2, j=3

3. ‘SMCSAA’    for r=3, j=4

4. ‘QUGMUM’  for r=4, j=1

5. ‘YOOEAA’    for r=5, j=5

6. CONCLUSION

This paper presents a novel cryptographic technique based on the Laplace Transform and its inverse for secure data communication. The method converts plaintext into an encrypted mathematical form and accurately reconstructs the original message through the inverse process. By combining mathematical transformations with modular arithmetic, the approach ensures a structured and reliable encryption framework. The use of series representation enhances the complexity and strength of the system. The proposed technique is suitable for applications such as secure messaging, banking security, and digital data protection. Furthermore, it offers a promising direction for developing advanced mathematical models in cryptography.

7. ACKNOWLEDGEMENTS

The authors express their sincere gratitude to the PG & Research Department of Mathematics,

A. V. C. College (Autonomous), for their support and encouragement in completing this work.

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