Multivariate Public Key Cryptosystems (MPKC) are candidates for post-quantum cryptography. Rainbow is a digital signature scheme in MPKC, whose encryption and decryption are relatively efficient. However, the security of MPKC depends on the difficulty in solving a system of multivariate polynomials, and the key length of MPKC becomes substantially large compared with that of RSA cryptosystems for the same level of security. The size of the public key in MPKC has been reduced in previous research, but to the best of our knowledge, there are no algorithms to reduce the size of a private key . In this paper, we propose NC-Rainbow, a variation of Rainbow using non-commutative rings and we describe the ability of the proposed scheme to reduce the size of a private key in comparison with the ordinary Rainbow while maintaining the same level of security. In particular, using the proposed NC-Rainbow, the size of a private key is reduced by about 75% at the 80 bit security level. Moreover, the speed of signature generation is accelerated by about 34% at the 80 bit security level.