The Piece in Hand method is a security enhancement technique for Multivariate Public Key Cryptosystems (MPKCs). Since 2004, many types of this method have been proposed. In this paper, we consider the 2-layer nonlinear Piece in Hand method as proposed by Tsuji et al. in 2009. The key point of this method is to introduce an invertible quadratic polynomial map on the plaintext variables to add perturbation to the original MPKC. An additional quadratic map allows the owner of the secret key to remove this perturbation from the system. By our analysis, we find that the security of the enhanced scheme depends mainly on the structure of the quadratic polynomials of this auxiliary map. The two examples proposed by Tsuji et al. for this map can not resist the Linearization Equations attack. Given a valid ciphertext, we can easily get a public key which is equivalent to that of the underlying MPKC. If there exists an algorithm that can recover the plaintext corresponding to a valid ciphertext of the underlying MPKC, we can construct an algorithm that can recover the plaintext corresponding to a valid ciphertext of the enhanced MPKC.

ZHFE, proposed by Porras et al. at PQCrypto'14, is one of the very few existing multivariate encryption schemes and a very promising candidate for post-quantum cryptosystems. The only one drawback is its slow key generation. At PQCrypto'16, Baena et al. proposed an algorithm to construct the private ZHFE keys, which is much faster than the original algorithm, but still inefficient for practical parameters. Recently, Zhang and Tan proposed another private key generation algorithm, which is very fast but not necessarily able to generate all the private ZHFE keys. In this paper we propose a new efficient algorithm for the private key generation and estimate the number of possible keys generated by all existing private key generation algorithms for the ZHFE scheme. Our algorithm generates as many private ZHFE keys as the original and Baena et al.'s ones and reduces the complexity from O(n2ω+1) by Baena et al. to O(nω+3), where n is the number of variables and ω is a linear algebra constant. Moreover, we also analyze when the decryption of the ZHFE scheme does not work.

Multivariate Public Key Cryptography (MPKC) is one of the main candidates for secure communication in a post-quantum era. Recently, Yasuda and Sakurai proposed at ICICS 2015 a new multivariate encryption scheme called SRP, which offers efficient decryption, a small blow up factor between plaintext and ciphertext and resists all known attacks against multivariate schemes. However, similar to other MPKC schemes, the key sizes of SRP are quite large. In this paper we propose a technique to reduce the key size of the SRP scheme, which enables us to reduce the size of the public key by up to 54%. Furthermore, we can use the additional structure in the public key polynomials to speed up the encryption process of the scheme by up to 50%. We show by experiments that our modifications do not weaken the security of the scheme.