Multivariate quadratic (MQ
) cryptography requires the use of long public and private keys to ensure a sufficient security level, but this is not favorable to embedded systems, which have limited system resources. Recently, various approaches to MQ
cryptography using reduced public keys have been studied. As a result of this, at CHES2011 (Cryptographic Hardware and Embedded Systems, 2011), a small public key MQ
scheme, was proposed, and its feasible implementation on an embedded microprocessor was reported at CHES2012. However, the implementation of a small private key MQ
scheme was not reported. For efficient implementation, random number generators can contribute to reduce the key size, but the cost of using a random number generator is much more complex than computing MQ
on modern microprocessors. Therefore, no feasible results have been reported on embedded microprocessors. In this paper, we propose a feasible implementation on embedded microprocessors for a small private key MQ
scheme using a pseudo-random number generator and hash function based on a block-cipher exploiting a hardware Advanced Encryption Standard (AES) accelerator. To speed up the performance, we apply various implementation methods, including parallel computation, on-the-fly computation, optimized logarithm representation, vinegar monomials and assembly programming. The proposed method reduces the private key size by about 99.9% and boosts signature generation and verification by 5.78% and 12.19% than previous results in CHES2012.