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A Novel Method of Encryption using Modified RSA Algorithm and Chinese Remainder Theorem


Academic year: 2023

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This is to certify that the thesis entitled “A NEW METHOD OF ENCRYPTION USING MODIFIED RSA ALGORITHM AND CHINESE REMAINDER THEOREM” submitted by SANGEETA PATEL and PARTHA PRITTAM NAYAK partially meets the requirements for the award of a Bachelor of Technology Degree in Electronics &. In this world of cryptography, it is now common knowledge that the weakest link lies in the implementation of cryptographic algorithms. This project covers the implementation of RSA algorithms with and without Chinese Remainder Theorem and also uses the Variable Radix Number System.

One way to speed things up is to break things up, calculate modulo p and modulo q using the Chinese remainder theorem. This project aims to implement RSA algorithm using Chinese Remainder Theorem as well as devise a modification whereby it would be increasingly difficult to decrypt a given encrypted message by using a Variable Radix system to encrypt the given message in the first lap. After the credentials are accepted, the authentication process is started to ensure that the requesting party has the permissions to perform the required functions.

One of the keys, the private key, is kept secret and not shared with anyone. Once data is encrypted with one of the keys, it can only be decrypted and recovered using the other key. A number of mathematical concepts from number theory are essential in the design of cryptographic algorithms.

Various theorems are clarified which are further applied to the RSA algorithm in our work plan.

Euclidean Algorithm

Prime Numbers

Many cryptographic algorithms require the random selection of one or more very large prime numbers. We are thus faced with the task of determining whether a given large number is prime. It is based on the conclusion that if n is prime, either the first element in the list of residues, or the residues (aq, a2q,.., a2k-1q, a2kq) modulo n is equal to 1, or an element in the list is equal to (n-1); otherwise, n is composite (that is, not prime).

On the other hand, if the condition is satisfied, it does not necessarily mean that n is prime.

Fermat’s And Euler’s Theorems

The Chinese Remainder Theorem

Let m and n be integers with gcd (m, n) =1, M=mn and let b and c be any integers

As stated in the proof, we write the solutions of the first congruence in the form x=11y + 8 and substitute it into the second congruence, which yields what it is. Finally, we want to check that our answer is correct, so substitute 41 for x and see this. We write the solutions of the first congruence as x= 3y+2 and substitute it into the second and.

If we now substitute the solutions for the first congruence into the third, we get , which is equivalent to 3(5z + 3) when y is substituted. So note that the statement can be extended to polynomials as long as the moduli are relatively prime to each other. Proof: Let g(x) denote the polynomial obtained by multiplying all the, since it is a monic polynomial of degree one, we can write it as (x- ), where one of the x- coordinates of our points is.

For each point ( ), we can write the expression , for which only and is a search algorithm. A note of extreme importance is the fact that the algorithm for P(x) is the well-known Lagrange interpolation formula found in numerical analysis.


Applications of Chinese Remainder Theorem

The conventional Chinese remainder theorem (CRT) is to determine a single integer from its remainders from a set of modulo. The two most important considerations in designing remainder number systems are the choices of the module sets and the conversion from the remainder to the weighted binary system. A new general conversion algorithm has been applied to the newly proposed conjugate moduli sets, resulting in a more efficient design for residue to binary conversion of the given moduli sets.

This more efficient design for the converter will make the conjugate moduli sets more attractive compared to other moduli sets. Another generalization of CRT has recently been proposed in which (instead of a single integer in CRT) multiple integers are to be determined from (not a sequence of residues but) a sequence of sets, residue sets, of residues. A remainder consists of the remainder of multiple integers modulo a modulus integers, and the remainder is unordered, i.e. the correspondence between the elements of the remainder and the multiple integers is not specified.

The general CRT was motivated by determining multiple frequencies in a superpositioned multisine signal from its multiple subsampled waveforms. This has applications in a sensor network where many sensors have low power and low bit rates, and their sampling rates can be low and much lower than the Nyquist rate of the signal of interest in the field. A generic CRT has been used in synthetic aperture radar (SAR) imaging of moving targets and polynomial phase signal detection.

It is found that the multi-frequency error rates are significantly reduced with the proposed algorithm considering residual errors compared to the one without. Chinese remainder theorem has been used for hundreds of years and has been applied to many domains such as integers and polynomials as explained in the last chapter.

Hill Cipher and its Applications


  • Public Key Issues
  • The RSA Algorithm
  • b Exponentiation in Modular Arithmetic
  • c Efficient Operation Using the Public Key
  • d Key Generation
  • RSA using Chinese Remainder Theorem
  • Fermat’s Little Theorem) Let p be a prime any integer a not divisible by p satisfies
    • RSA decryption using the CRT
    • a Algorithm for RSA using CRT
    • b Algorithm for RSA using Variable Radix Number System

Public key algorithms are symmetric, that is, the key used to encrypt the message is different from the key used to decrypt the message. The encryption key, known as the public key, is used to encrypt a message, but the message can only be decrypted by the person who has the decryption key, also known as the private key. The advantage of public key algorithms is that they are more computationally intensive than symmetric algorithms, and therefore take longer to encrypt and decrypt.

If the public key is used to encrypt something, then it can only be decrypted using the private key. And similarly, if the private key is used to encrypt something, then it can only be decrypted using the public key. With public key cryptography it is thus possible for two people who have never met to exchange secure messages.

Another major advantage of public key systems is that they can provide a method for digital signatures. Public key authentication, on the other hand, prevents this kind of rejection; each user is solely responsible for protecting his or her private key. One disadvantage of using public key cryptography for encryption is speed; there are popular secret key encryption methods that are significantly faster than any currently available public key encryption method.

However, public key cryptography can be used with secret key cryptography to get the best of both worlds. A public key system can be used to encrypt a secret key that is used to encrypt a larger portion of a file or message. Public-key cryptography can be vulnerable to spoofing even when users' private keys are not available.

In some situations, public key cryptography is not necessary and secret key cryptography alone is sufficient. Public key cryptography is not intended to replace secret key cryptography, but rather to complement it, to make it more secure. The first use of public key techniques was for secure to complement it, to make it more secure.

The first use of public key techniques was for secure key exchange in an otherwise secret key system; this is still one of its primary functions. Before using the public key cryptosystem, each participant must generate a pair of keys.

What is MAPLE?


MAPLE and the symbolic Math Toolbox:-

MATLAB to MAPLE code translation:-

MATLAB Code Generation:-



To verify that this value of a is a valid RSA encryption exponent, we enter the following Maple igcd command, which returns the greatest common divisor of the integers a and m. Next, we convert this message to a list of 2-digit integers and combines these integers into a single block. If this procedure is stored as the text file for number in the directory from which we run Maple, then we can include the for number procedure in this Maple session by entering the following command.

The message can then be converted to its numeric equivalent as a single block by entering the following command. Since this plaintext integer is less than n, we can encrypt this message as a single block. This means that we can encrypt this message by raising the plaintext to the power of a and reducing it modulo n.

Like the preceding igcd command, the following igcdex command returns the greatest common divisor of the integers a and m. However, the following igcdex command also takes two additional user-defined variable inputs, leaving them as integers b and y satisfying ab + my = (a,m). To see the decryption exponent b defined by the previous command, and to express this value as a positive number less than m, we enter the following command.

Then, by entering the following command, we verify that this value of b satisfies ab = 1 mod m. To see the original plaintext letters, we need to split this single block into a list of 2-digit integers and convert these integers to letters. We can then turn the plain text into a list of letters by entering the following command.

Recall that the security of the RSA cryptosystem is based on the apparent difficulty of factoring the value of n. Therefore, for the RSA system used in this section to be secure, it should be very difficult for an intruder to factor the 42-digit value of n used in this section. Although this value of n is one digit shorter than the integer used in the previous command, because the prime factors of n are both very large, it will take ifactor much longer to return these prime factors.


RSA without Chinese Remainder Theorem Outputs

RSA with Chinese Remainder Theorem Outputs

RSA in Variable Radix Number System



1] W.Stallings; “Cryptography and Network Security” 2 nd Edition, Prentice Hall, 1999

4] Michael Welschenbach “Cryptography in C and C++”

6] Applications of Abstract Algebra with MAPLE – Richard E.Kilma,Neil Sigmon,Ernest Stitzinger

7] A Generalized CRT for residue sets with errors and its application in

8]Applications of New Chinese Remainder Theorem to RNS with two pairs of conjugate moduli – Alex Skavantzos,Yuke Wang

9]A New cryptosystem using matrix transformation – Yi Shiung Yeh,Tzong-Chen Wu,Chin-Chen Chang

10]Modular Matrix Cipher and its application in authentication protocol – Bao Ngoc TRAN,Thuc Dinh NGUYEN


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