Optimized Digital Signature Algorithm with Multi-Prime RSA

Abstract

Digital signatures are an essential cryptographic technique used to ensure the authenticity, integrity, and non-repudiation of digital messages. This paper presents on optimizing RSA-based digital signatures by integrating Multi-Prime RSA, Chinese Remainder Theorem (CRT)-based signing, Sliding Window Exponentiation, and SHA-3 hashing to enhance performance and security. Multi-Prime RSA improves efficiency by using multiple primes, reducing computational complexity while maintaining strong encryption. The CRT-based optimization accelerates signing by breaking computations into smaller modular exponentiations, and Sliding Window Exponentiation speeds up verification by reducing multiplication steps. SHA-3 hashing ensures message integrity and protection against cryptographic attacks. These techniques improve key generation, signing, and verification, making RSA more efficient for real-time applications such as secure transactions, encrypted communication, and blockchain security.

Keywords

RSA Digital Signature, Multi-Prime RSA, Chinese Remainder Theorem (CRT), Sliding Window Exponentiation, SHA-3 Hashing, Signature Verification, Public-Key Cryptography, Optimized Key Generation

Introduction

Digital signatures are an essential cryptographic technique used to ensure the authenticity, integrity, and non-repudiation of digital messages or documents. Among various digital signature schemes, the RSA-based digital signature is one of the most widely used due to its strong security foundation in public-key cryptography. However, traditional RSA implementations face challenges such as computational inefficiency and slow key generation, signing, and verification processes. To overcome these limitations, this project explores optimized RSA-based digital signatures by integrating Multi-Prime RSA, Chinese Remainder Theorem (CRT)-based signing, Sliding Window Exponentiation for faster verification, and SHA-3 hashing for improved security. These techniques collectively enhance performance while maintaining robust cryptographic security, making RSA signatures more efficient for real-time applications such as secure communication, digital certificates, and blockchain technology. This study aims to optimize key generation and verification without replacing RSA with alternative cryptographic methods, ensuring compatibility with existing RSA-based security infrastructures.  The study integrates multiple advanced algorithms to optimize RSA-based digital signatures. Multi-Prime RSA enhances efficiency by using three or more primes, reducing computation time while maintaining security. The Chinese Remainder Theorem (CRT) accelerates signing by breaking complex calculations into smaller modular exponentiations. Sliding Window Exponentiation further optimizes RSA verification by reducing multiplication steps. SHA-3 hashing ensures message integrity and enhanced security against cryptographic attacks. The combination of these techniques significantly improves key generation, signing, and verification. Multi-Prime RSA accelerates encryption and decryption, while CRT and Sliding Window Exponentiation boost signing and verification speeds. SHA-3 hashing strengthens protection against forgery. These enhancements make RSA signatures more practical for real-time applications, including secure transactions and blockchain security. Overall, the improved system ensures faster performance without compromising RSA's strong security foundation.

LITERATURE SURVEY

Digital signatures, particularly RSA-based, are essential in ensuring the authenticity, integrity, and non-repudiation of digital messages [1]. RSA, a widely used public-key cryptosystem, faces inefficiencies in key generation, signing, and verification, which have led to efforts to optimize these processes [2]. Multi-Prime RSA, which uses multiple primes instead of two, enhances encryption and decryption speed by reducing the size of the private exponent [4]. The Chinese Remainder Theorem (CRT) accelerates RSA operations by dividing them into smaller, parallel computations [5]. Sliding Window Exponentiation optimizes signature verification by reducing multiplication steps [9], while SHA-3 hashing improves message integrity and security [6]. Optimized key generation techniques, such as dynamic public exponent selection, enhance both efficiency and security [7]. These improvements make RSA more suitable for large-scale, real-time applications like blockchain, secure communications, and digital transactions, ensuring both speed and robustness in modern cryptography [3].

RSA Digital Signature

RSA digital signatures are a crucial component of public-key cryptography, ensuring authentication, integrity, and non-repudiation of digital messages or documents. The process involves generating a unique digital signature by encrypting a message hash with a private key. The recipient verifies the signature by decrypting it with the sender’s public key and comparing it with the independently computed hash of the received message. If both hashes match, the signature is valid, confirming that the message was sent by the claimed sender and was not altered in transit. RSA digital signatures are widely used in secure communication, software distribution, and blockchain applications.

Multi-Prime RSA

Multi-Prime RSA is an advanced variant of the traditional RSA cryptosystem that enhances security and computational efficiency. Instead of using two prime numbers to generate the modulus n, Multi-Prime RSA utilizes three or more primes. This approach results in a smaller private exponent d due to a larger totient function \phi(n), reducing decryption and signing time while maintaining security levels. The key advantage is that it allows for faster encryption and decryption operations, making RSA signatures more efficient, particularly for large-scale systems that require high-speed cryptographic operations.

 Pseudocode:

1. Generate three large prime numbers: p, q, r.
2. Compute modulus: n = p * q * r.
3. Compute Euler’s totient function:

             φ(n)=(p-1)*(q-1)*(r-1).

4. Select a random e such that 1 < e < φ(n)
   and gcd(e,φ(n))                                                  = 1.
5. Compute private key d as the modular inverse of

             d=e mod φ(n).

6. Return the public key (e, n) and
   private key (d, p, q, r).

 Advantages of Multi-Prime RSA Over Traditional RSA

1. Improved Efficiency:

Multi-Prime RSA uses three or more primes instead of just two, which reduces the size of the private exponent. This leads to faster decryption and signing processes.

2. Enhanced Performance with CRT:

The integration of the Chinese Remainder Theorem (CRT) allows RSA calculations to be divided into smaller, parallel computations, significantly accelerating encryption, decryption, and signing operations. This method can improve RSA speed by nearly four times.

3. Faster Verification with Sliding Window Exponentiation:

This technique reduces the number of multiplication steps in modular exponentiation, optimizing RSA signature verification for real-time applications.

4. Improved Security with SHA-3 Hashing:

SHA-3 provides enhanced protection against collision and preimage attacks, ensuring message integrity and resistance to forgery.

5. Optimized Key Generation:

Using multiple primes in the modulus generation process results in faster key generation without compromising security.

6. Greater Suitability for Real-Time Applications:

Due to reduced computational overhead, this                optimized RSA approach is better suited for secure transactions, blockchain security, and other applications requiring fast cryptographic operations.

Chinese  Remainder Theorem(CRT)

The Chinese Remainder Theorem (CRT) is a fundamental mathematical principle used in modular arithmetic to solve systems of congruences. In RSA, CRT is leveraged to speed up modular exponentiation computations, which are a core part of both encryption and digital signature generation. By breaking a large modular exponentiation problem into smaller, parallel computations using the prime factors of n, CRT significantly reduces the time complexity of operations. This optimization is particularly beneficial for decryption and signing, where the private key operations are computationally expensive. Using CRT, RSA operations become nearly four times faster than standard methods.

     Pseudocode:

1.Extract private key components (d, p, q,r).
2.Compute:  d_p = d mod (p-1),  d_q = d mod(q-1),    d_r=dmod(r-1).
3.Compute signature parts: s_p = message^d_p mod p,  s_q = message^d_q mod q,

   s_r = message^d_r mod r.                                        
4.Reconstruct signature using CRT.
5.Return the final signature.

Sliding Window Exponentiation

Sliding Window Exponentiation is an optimization technique used to accelerate modular exponentiation, which is a critical operation in RSA signature verification. Instead of computing powers iteratively, this method precomputes a set of small exponentiations and then processes the exponent in blocks (windows) of bits. By reducing the number of multiplications required, this technique enhances the efficiency of RSA verification, making it practical for real-time cryptographic applications. Sliding Window Exponentiation is especially useful when dealing with large exponents, as it balances computation speed with memory usage to achieve optimal performance.

Pseudocode:
1.Convert exponent to binary.
2.Choose a window size (e.g., 4-bit window).
3.Precompute small powers of the base modulo n.
4.Process the exponent in k-bit windows.
5.Multiply result by precomputed values and perform        necessary squaring.    
6.Return the final result.

SHA-3 Hashing

SHA-3 (Secure Hash Algorithm 3) is a modern cryptographic hash function designed to provide high security and resistance against collision and preimage attacks. In digital signatures, SHA-3 is used to hash the original message before it is signed with the private key. Hashing ensures that the digital signature remains compact and prevents attackers from deriving the original message from its signature. Unlike its predecessors (SHA-1 and SHA-2), SHA-3 is based on the Keccak sponge construction, offering enhanced security against vulnerabilities. Its integration into RSA digital signatures strengthens data integrity and protection against forgery.

Signature Verification

 Signature verification is the process of ensuring that a received digital signature is authentic and that the signed message has not been tampered with. It involves decrypting the signature using the sender’s public key, which recovers the hash value originally computed before signing. The verifier then independently hashes the received message and compares it with the decrypted hash. If the values match, the signature is considered valid, confirming that the document originated from a trusted source and was not altered. This process is critical in secure digital transactions, electronic documents, and blockchain-based authentication. Input for verification the digital signature provided document is  named as: Signature_document.txt

“Signature document                                                 

This is a test document for digital signature verification.”

Pseudocode:
1.Extract public key components (e, n).
2.Compute message hash using SHA-3.
3.Decrypt the signature using modular exponentiation      (Sliding Window method).
4.Compare the decrypted hash with the computed message hash.
5.If they match, the signature is valid

Public-Key Cryptography

Public-key cryptography, also known as asymmetric cryptography, operates with a pair of keys: a public key that is available to everyone and a private key that remains confidential. One of the most commonly utilized public-key cryptosystems is RSA. Unlike symmetric encryption, where both the sender and receiver use the same key, public-key cryptography allows for secure communication and verification even over insecure networks. RSA digital signatures, which are based on this cryptographic approach, ensure the integrity and authenticity of messages, playing a crucial role in secure email, SSL/TLS protocols, and blockchain technology..

Proposed  Digital Signature Schema Using  Multi_Prime RSA          

Key generation is a fundamental step in RSA, where a secure pair of public and private keys is created. Optimized key generation techniques focus on improving the efficiency and security of this process. Multi-Prime RSA enhances key generation by using additional prime factors, reducing computational overhead while maintaining strong cryptographic properties. Additionally, dynamically selecting the public exponent e rather than using a fixed value (such as 65537) ensures greater randomness and resistance to specific attacks. Optimized key generation is essential for large-scale cryptographic applications, ensuring that keys are both secure and computationally efficient.

Step 1: Key Generation

1. Choose Large Prime Numbers:

Select three large prime numbers: p, q, r

2. Compute Modulus (n):

Compute n = p × q × r

3. Compute Euler’s Totient Function (φ(n)):

Compute φ(n) = (p-1) × (q-1) × (r-1)

4. Choose Public Exponent (e):

Select e such that gcd(e, φ(n)) = 1

5. Compute Private Key (d):

Compute d = e^(-1) mod φ(n)

6. Return Public and Private Keys:

Public Key: (e, n)

Private Key: (d, p, q, r)

Step 2: Signing a Message Using CRT

1. Choose a Message (M):

Input message M

2. Compute Reduced Private Exponents:

d_p = d mod (p-1)

d_q = d mod (q-1)

d_r = d mod (r-1)

3. Compute Partial Signature

s_p = M^(d_p) mod p

s_q = M^(d_q) mod q

s_r = M^(d_r) mod r

         4.   Compute Final Signature Using CRT:

       Solve the following system of congruences:

S ≡ s_p mod p

S ≡ s_q mod q

S ≡ s_r mod r

5. Return Signature:

Digital Signature: S

Step 3: SHA-3 Hashing of the Message

1. Hash the Message:

Compute H = SHA3(M)

Step 4: Verifying the Signature Using Sliding Window Exponentiation

1. Convert e to Binary:

Convert e to binary format

2. Compute Base Powers:

Compute base^1 mod n

Compute base^2 mod n

Compute base^4 mod n

3. Compute Final Exponentiation Using Sliding Window:

Compute result = (base^8 × base^2 × base^1) mod n

4. Verify Hash Values:

If result == SHA3(M), signature is valid

Final Results:

Public Key: (e, n)

Private Key: (d, p, q, r)

Message: M

SHA-3 Hash: H

Signature: S

Verified Hash: H

Multi -prime RSA Algorithm for signing digitally

Example1:

Enhanced RSA with Larger Prime Numbers Example

Step 1: Key Generation

1.1 Choose Large Prime Numbers

Let's pick three large prime numbers:

p = 47,q = 59,r = 71

1.2 Compute Modulus  (n)

n= p × q × r n = 47 × 59 × 71 = 197243

1.3 Compute Euler’s Totient Function (φ(n))

φ(n) = (p - 1) × (q - 1) × (r - 1)

φ(n) = (47 - 1) × (59 - 1) × (71 - 1)

φ(n)   =    46 × 58 × 70 = 186320

1.4 Choose Public Exponent (e)

We choose e such that it is coprime with φ(n).
e = 11 gcd(11, 186320) = 1 (Valid choice)

1.5 Compute Private Key (d) using Modular Inverse

d = e^(-1) mod φ(n) d = 11^(-1) mod 186320 = 169471

Public Key: (e, n) = (11, 197243)

Private Key: (d, p, q, r) = (169471, 47, 59, 71)

Step 2: Signing a Message Using CRT

2.1 Choose a Message

M = 45

2.2 Compute Reduced Private Exponents

d_p = d mod (p - 1) = 169471 mod 46 = 25
d_q = d mod (q - 1) = 169471 mod 58 = 7
d_r = d mod (r - 1) = 169471 mod 70 = 21

2.3 Compute Partial Signatures

s_p = M^(d_p) mod p = 45^25 mod 47 = 13
s_q = M^(d_q) mod q = 45^7 mod 59 = 35
s_r = M^(d_r) mod r = 45^21 mod 71 = 18

2.4 Compute Final Signature (S) using CRT

S ≡ 13 mod 47

S ≡ 35 mod 59

S ≡ 18 mod 71

Using CRT, S = 85679

Digital Signature: S = 85679

Step 3: SHA-3 Hashing of the Message

To enhance security, we hash the message before signing.
For simplicity, assume

SHA3(45) = 93

Step 4: Verifying the Signature Using Sliding Window Exponentiation

4.1 Convert e to Binary

11 = (1011)_2

4.2 Apply Sliding Window Exponentiation

Compute Base Powers:

79^1 mod 197243 = 85679

79^2 mod 197243 = 54221

79^4 mod 197243 = 158282

Compute Final Exponentiation:

85679^11 = 85679^(8+2+1)

= (85679^8 × 85679^2 × 85679^1) mod 197243

= (158282 × 54221 × 85679) mod 197243
= 93

Since the recovered hash matches the original SHA-3 hash, the signature is valid.

Final Results

Public Key: (11, 197243)

Private Key: (169471, 47, 59, 71)

Message: 45

SHA-3 Hash: 93

Signature: 85679

Verified Hash: 93

Signature is Valid

RESULTS AND ANALYSIS

The optimized RSA system effectively improves security and performance by combining Multi-Prime RSA, CRT, Sliding Window Exponentiation, and SHA-3 hashing. Multi-Prime RSA reduces the size of the private exponent while ensuring strong encryption, and CRT divides modular calculations into smaller steps, improving signing efficiency. Sliding Window Exponentiation accelerates signature verification by minimizing multiplication steps, and SHA-3 hashing enhances resistance against collision attacks, strengthening data integrity. These optimizations collectively reduce computational overhead while retaining robust cryptographic security, making the method suitable for secure transactions, blockchain applications, and other real-time cryptographic systems demanding speed and reliability.

CONCLUSIONS

Optimizing RSA-based digital signatures enhances cryptographic efficiency and security for real-time applications. Incorporating Multi-Prime RSA, CRT-based signing, Sliding Window Exponentiation, and SHA-3 hashing improves key generation, signing, verification, and data integrity. These optimizations reduce computational overhead while maintaining RSA's security foundation, making it a practical choice for modern, high-security environments demanding speed and reliability. Overall, this study demonstrates that RSA digital signatures can be optimized without replacing the fundamental RSA framework, ensuring compatibility with existing security infrastructures while achieving improved performance and cryptographic strength. These enhancements make RSA a more viable choice for modern, high-security environments that demand both speed and reliability.

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