Key distribution system and attribute-based encryption on non-commutative ring

I propose the new key distribution system and attribute-based encryption scheme on noncommutative ring where the complexity required for enciphering and deciphering is small. As in this system encryption keys and decryption keys involve the attributes of each user, the system is adaptive for cloud computing systems. The security of this system is based on the complexity for solving the multivariate algebraic equations of high degree over finite field, that is, one of NP complete problems. So this system is immune from the Gröbner basis attacks. The key size of this system becomes to be small enough to handle. key words: key distribution, attribute-based encryption ,multivariate polynomial, Gröbner basis, NP complete problems