Cryptanalysis of the RSA Cryptosystem Based on $n$ Prime Numbers

As digital platforms become essential in daily life, the demand for robust information security has grown. Cryptography, particularly RSA, plays a central role in safeguarding sensitive data. In response to vulnerabilities in RSA, the modified RSA cryptosystem (MRSA), based on multiple prime numbers, was introduced. This study analyzes MRSA's resilience against attacks, employing techniques like Pollard's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> - 1 factorization, the common modulus attack, and the cycling attack. Our results have shown that Pollard's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> - 1 attack is faster when applied to factorize the integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> in MRSA compared to RSA. Additionally, common modulus attacks, multiplicative properties attacks, and cycling attacks can be used to cryptanalyze MRSA in polynomial time. This research highlights MRSA's limitations, suggesting that simply increasing the number of prime factors does not guarantee stronger security.