Multikey homomorphic encryption (MKHE) supports arbitrary homomorphic evaluation on the ciphertext of different users and thus can be applied to scenarios involving multiusers (e.g., cloud computing and artificial intelligence) to protect user privacy. CDKS19 is the current most efficient MKHE scheme, and its relinearization process consumes most of the time of homomorphic evaluation. In this study, an optimized relinearization algorithm of CDKS19 is proposed. This algorithm reorganizes the evaluation key during the key generation process, decreases the complexity of relinearization, and reduces the error growth rate during homomorphic evaluation. First, we reduce the scale of the evaluation key by increasing its modulus instead of using a gadget vector to decompose the user's public key and extend the ciphertext of homomorphic multiplication. Second, we use rescaling technology to optimize the relinearization algorithm; thus, the error bound of the ciphertext is reduced, and the homomorphic operation efficiency is improved. Lastly, the average-case error estimation on the variances of polynomial coefficients and the upper bound of the canonical embedding map are provided. Results show that our scheme reduces the scale of the evaluation key, the error variance, and the computational cost of the relinearization process. Our scheme can effectively perform the homomorphic multiplication of ciphertexts.

The Multi-Key Fully Homomorphic Encryption (MKFHE) based on the NTRU cryptosystem is an important alternative to the post-quantum cryptography due to its simple scheme form, high efficiency, and fewer ciphertexts and keys. In 2012, López-Alt et al. proposed the first NTRU-type MKFHE scheme, the LTV12 scheme, using the key-switching and modulus-reduction techniques, whose security relies on two assumptions: the Ring Learning With Error (RLWE) assumption and the Decisional Small Polynomial Ratio (DSPR) assumption. However, the LTV12 and subsequent NTRU-type schemes are restricted to the family of power-of-2 cyclotomic rings, which may affect the security in the case of subfield attacks. Moreover, the key-switching technique of the LTV12 scheme requires a circular application of evaluation keys, which causes rapid growth of the error and thus affects the circuit depth. In this paper, an NTRU-type MKFHE scheme over prime cyclotomic rings without key-switching is proposed, which has the potential to resist the subfield attack and decrease the error exponentially during the homomorphic evaluating process. First, based on the RLWE and DSPR assumptions over the prime cyclotomic rings, a detailed analysis of the factors affecting the error during the homomorphic evaluations in the LTV12 scheme is provided. Next, a Low Bit Discarded & Dimension Expansion of Ciphertexts (LBD&DEC) technique is proposed, and the inherent homomorphic multiplication decryption structure of the NTRU is proposed, which can eliminate the key-switching operation in the LTV12 scheme. Finally, a leveled NTRU-type MKFHE scheme is developed using the LBD&DEC and modulus-reduction techniques. The analysis shows that the proposed scheme compared to the LTV12 scheme can decrease the magnitude of the error exponentially and minimize the dimension of ciphertexts.

To address the increasing labor cost of resin tapping, more efficient methods for resin tapping need to be developed. This study aimed to evaluate the features of resinosis as affected by stimulant pastes in Pinus elliottii × P. caribaea, which is also one of the predominant resin-producing species hybrids in South China. The resin yields and resin compositions were assessed in 33 P. elliottii × P. caribaea F1 families, with the application of four kinds of chemical stimulants, potassium (K2SO4) paste, naphthalene acetic acid (NAA) paste, benzoic acid (BA) paste and 2-chloroethylphosphonic acid (CEPA) paste. Our results showed that all four pastes significantly increased the resin yield by at least 20% at each tapping, and 3- to fivefold increases were detected at the beginning of each year. The correlations between resin yield and growth at each tapping ranged from uncorrelated to moderately positively correlated, indicating that resin yield was mostly but not always determined by tree size. The concentration of each resin component did not change with the stimulant applications. In P. elliottii × P. caribaea, selecting a larger tree diameter at breast height and employing the chemical stimulants at the first several tapping rounds are efficient tapping procedures. Moreover, the K2SO4-based stimulant can be recommended considering its promoting effects on resin yield and the low cost of the chemicals required to produce it.

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