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DOI：http://dx.doi.org/10.26855/jamc.2022.12.001

# A Factorization Attack Algorithm on RSA Cryptosystem Using Fast Searching Algorithm

Date: October 25,2022 |Hits: 2616 Download PDF How to cite this paper

Yuh-Sien Sun1, Che Wun Chiou1,*, Wei-Cheng Sun2

1Chien Hsin University of Science and Technology, Taoyuan City, Taiwan, China.

2Business Unit. III, Himax Technologies, Inc., Taipei City, Taiwan, China.

*Corresponding author: Che Wun Chiou

### Abstract

The RSA cryptosystem is the first public key cryptosystem and has widely applied in privacy and ensure authenticity of digital data and included in many standards such as FIPS PUB 186-4. The security of RSA cryptosystem is heavily relied on the practical difficulty of factoring the product of p and q (N=p×q). Many studies are reported for attacking RSA cryptosystem, but no direct solving algorithms for finding p and q of the RSA cryptosystem by mathematical derivative are proposed. This paper uses strictly mathematical steps to derive out important parameters such as K=p+q, B = p-q, and   for breaking the RSA cryptosystem. Two fast searching methods are proposed to find out the correct K value, and then two primary values p =   and q =   can be computed directly. Proposed algorithms need division operations with complexity  in average to solve the q solution and suggest some important up-down search limits to attack RSA-2048 for helping cryptanalysis on such unbroken cryptosystem. The proposed algorithms can find out the precision K value with computation complexity  , and thus unbroken RSA cryptosystems with huge numbers such as the RSA-2048 cryptosystem will be solved immediately because exhaustive search is no more needed.

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### How to cite this paper

A Factorization Attack Algorithm on RSA Cryptosystem Using Fast Searching Algorithm

How to cite this paper:  Yuh-Sien Sun, Che Wun Chiou, Wei-Cheng Sun. (2022) A Factorization Attack Algorithm on RSA Cryptosystem Using Fast Searching Algorithm. Journal of Applied Mathematics and Computation6(4), 390-404.

DOI: http://dx.doi.org/10.26855/jamc.2022.12.001