Since there are many time polynomial algorithms (e.g., Berlekamp , Ben-Or , and Cantor-Zassenhaus ) for factoring a polynomial f (x) [member of] [Z.sub.p][x] into the product of irreducible polynomials, the RSA cryptosystem on the quotient ring of polynomials can be easily broken using these algorithms.
For the original RSA cryptosystem, because a plaintext m [member of] [Z.sub.n] has a length of bits, the modulus n must have the same length as m.
For the RSA cryptosystem on the quotient ring of Gaussian integers, a plaintext m = a + bi has a length of bits, and therefore, both a and b have a length of bits 512.
The equality [m.sup.ed] = m plays an important role in a RSA cryptosystem, it ensures encryption and decryption phases in the cryptosystem.
For the first trend, the RSA cryptosystems are developed on the ring [Z.sub.n].
Over years, numerous attacks on RSA illustrating RSA's present and potential vulnerability have brought our attention to the security issues of RSA cryptosystem. Although twenty years of research have led to several fascinating attacks, none of them is devastating.
In fact, the implementation of RSA Cryptosystem is heavily based on modular arithmetic and exponentiation involving large prime numbers.
A Tiny RSA Cryptosystem Based on Arduino Microcontroller Useful for Small Scale Networks.
Because only the buyer has the [sk.sub.N] key to decrypt the encrypted digital content, we insert the watermark s into the digital content M' along with a privacy homomorphism with respect to the multiplication operation under the RSA cryptosystem. To protect the buyer's rights, the new buyer' s watermark is encrypted by a one-time public key [pk.sup.i.sub.N] before sending it to the seller, as shown below:
Prior to inserting the watermark s into the digital content [M.sub.R]', the original seller must encrypt this digital content with the buyer's one-time public key [pk.sub.N]'; then he or she must insert the watermark into the encrypted digital content along with a privacy homomorphism into the RSA cryptosystem as discussed in subsection 3.1.