A public-key cryptosystem based on Diophantine equations

Till now many public-key cryptosystems, for example the RSA cryptosystem, have been proposed, but few cryptosystems are used because of their computational complexity. Enciphering and deciphering operations require too much time and space. A public-key cryptosystem based on Diophantine equations is proposed in this paper. The operations do not require so much computational complexity in this cryptosystem as compared with RSA cryptosystem. A digital signature in ths cryptosystem can be generated by extending the region of plaintext xn. The cryptosystem is constructed as follows. With n+1 prime numbers Pi (i=0,...,n), n positive integers ki (i=1,...,n), we get bi such that bi=kiP0,...,PiÅ-1 (i=1,...,n). We find positive integers ai<bi (i=1,...,n), R and β such that R>b1P1+ëëë+bnPn, ai•biβmodR (i=1,...,n), where β is relatively prime to R, and R>a1P1+ëëë+anPn. We construct a cryptosystem based on Diophantine equations such that C=a1x1+ëëë+anxn, where 0•xi<Pi (i=1,...,n)ëxi are plain texts and C is ciphertex in this cryptosystem. [In Japanese.]