By some basic transforms and invariant theory, we give two results: 1) an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are equivalent. This is useful in studying Boolean functions and in engineering. For example, we classify all 8-variable homogeneous bent functions of degree 3 into two classes; 2) Reed-Muller codes R(4,6)/R(1,6), R(3,7)/R(1,7) are classified efficiently.

Advances in quantum computers pose great threats on the currently used public key cryptographic algorithms such as RSA and ECC. As a promising candidate secure against attackers equipped with quantum computational power, multivariate public key cryptosystems (MPKCs) have attracted increasing attention in recently years. Unfortunately, the existing MPKCs can only be used as a multivariate signature scheme, and it remains unknown how to construct an efficient MPKC enabling secure encryption. Furthermore, some multivariate signature schemes have been shown insecure in recent years, and it is also not trivial to build MPKC which can serve as a secure signature scheme. By employing the basic MQ-trapdoors, this paper proposes a novel MPKC and shows how it can be used as a multivariate signature scheme and a multivariate encryption scheme, respectively. The goal is achieved by incorporating our new hash authentication techniques and some modification methods such as the Shamir's minus method. Thorough analysis shows that our schemes are secure and efficient. Our MPKC gives a positive response to the challenges in multivariate public key cryptography.

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Background The significance of spiritual care needs among chronic diseases patients has been emphasized across countries and cultures in many studies. However, there were few studies on spiritual care needs among elderly patients with moderate-to-severe chronic heart failure (CHF) in China. Objective To investigate spiritual care needs and associated influencing factors among elderly patients with moderate-to-severe CHF, and to examine the relationships among spiritual care needs, self-perceived burden, symptom management self-efficacy, and perceived social support. Methods A cross-sectional design was implemented, and the STROBE Checklist was used to report the study. A convenience sample of 474 elderly patients with moderate-to-severe CHF were selected from seven hospitals in Tianjin, China. The sociodemographic characteristics questionnaire, the Spiritual Needs Questionnaire Scale, the Self-Perceived Burden Scale, the Self-efficacy for Symptom Management Scale, and the Perceived Social Support Scale were used. Descriptive statistics, univariate, multiple linear regression, and Pearson's correlation analysis were used to analyze data. Results The total score of spiritual care needs among 474 elderly patients with moderate-to-severe CHF was 37.95 ± 14.71, which was moderate. Religious belief, educational background, self-perceived burden, symptom management self-efficacy, and perceived social support were the main factors affecting spiritual care needs, and spiritual care needs were negatively correlated with self-perceived burden (r = −0.637, p < 0.01) and positively correlated with symptom management self-efficacy (r = 0.802, p < 0.01) and social support (r = 0.717, p < 0.01). Significance of results The spiritual care needs of elderly patients with moderate-to-severe CHF were moderate, which were influenced by five factors. It is suggested that clinical nurses, families, and society should take targeted spiritual care measures to improve patients’ symptom management self-efficacy and perceived social support from many aspects, and reduce self-perceived burden to meet their spiritual care needs and improve the quality and satisfaction of spiritual care in nursing practice.

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