A Fast RSA-Type Public-Key Primitive Modulo p^kq Using Hensel Lifting

We propose a public-key primitive modulo p^kq based on the RSA primitive. The decryption process of the proposed scheme is faster than those of two variants of PKCS #1 version 2.1, namely the RSA cryptosystem using Chinese remainder theorem (CRT) and the Multi-Prime RSA. The message M of the proposed scheme is decrypted from M mod p^k and M mod q using the CRT, where we apply the Hensel lifting to calculate M mod p^k from M mod p that requires only quadratic complexity O((log₂ p)²). Moreover, we propose a trick that avoids modular inversions used for the Hensel lifting, and thus the proposed algorithm can be computed without modular inversion. We implemented in software both the proposed scheme with 1024-bit modulus p²g and the 1024-bit Multi-Prime RSA for modulus p ₁p₂p₃, where p, q, p₁, p ₂, p₃ are 342bits. The improvements of the proposed scheme over the Multi-Prime RSA are as follows: The key generation is about 49% faster, the decryption time is about 42% faster, and the total secret key size is 33% smaller.