1. 椭圆曲线上的 Abel 群结构设 $\mathbb{F}_q$ 为有限域， $q = p$ 为奇素数（密码学中最常见情形）。一条椭圆曲线 $E$ 的短 Weierstrass 形式为$$ E : \; y^2 = x^3 + A x + B, \quad A, B \in \mathbb{F}_p, $$其中判别式$$ \Delta = -16(4A^3 + 27B^2) \not\equi

SM4 分组密码算法推导SM4是一种分组密码算法，其分组长度和密钥长度均为128位。它采用32轮迭代的非平衡Feistel结构。1. 初始设定明文分组长度为128位，密钥长度为128位。首先，将128位的输入明文 $P$ 分为4个32位的字 $(X_0, X_1, X_2, X_3)$。$$ P = (X_0, X_1, X_2, X_3) \quad \text{其中 } X_i \in \{0

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Advances in Machine Learning and Their Application to Materials ScienceMachine learning is rapidly transforming materials science, shifting the field from a traditional trial-and-error approach to a m