26、向量函数覆盖序列与有限域的uq - 尖锐子集

向量函数覆盖序列与有限域的uq - 尖锐子集

1. 向量函数覆盖序列相关内容

在向量函数的研究中，覆盖序列是一个重要的概念。对于任意的 (R - 1) 元组 (k1, …, kR - 1)，函数 Fk1, …, FkR - 1 允许 (ϕ, ψ) 和 (ψ, ϕ) 作为覆盖序列。基于此，这些轮函数满足命题 13 的假设，并且按方程 (27) 定义的函数 Gk1, …, kR - 1 也允许 (ϕ, ψ) 和 (ψ, ϕ) 作为覆盖序列。

接下来的推论给出了满足上述备注第二部分假设的函数示例。
- 推论 5 ：设 F 是 Fn₂ = Fr₂ × Fs₂ 上的一个置换，定义为 F(x, y) = (F ′(x, y), P(y))，其中 F ′ 是属于 Mr + s, r 的一个 (r + s, r) 函数，P 是 Fs₂ 上的一个置换。如果 K 包含于 Fn₂，并且 R 轮迭代分组密码的轮函数 Fk 定义为 Fk(x, y) = F(x + k(L), y + k(R))，其中 k = (k(L), k(R)) ∈ K，那么简化密码 Gk1, ···, kR - 1 允许 (δFr₂×{0}, δFr₂×{0}) 作为覆盖序列。
- 证明 ：根据推论 4，(δFr₂×{0}, δFr₂×{0}) 是 F 的一个覆盖序列。由于轮密钥 k 是通过加法引入的，即 F(k(L), k(R))(x, y) = F(x + k(L), y + k(R))，所以对于每一个轮密钥 k = (k(L), k(R)) 的选择，(δFr₂×{0}, δFr₂×{0}) 都是所有函数 Fk 的覆盖序列。对轮函数 Fk1, …, FkR - 1（其中 (k1, …, kR - 1) 是 K 的任意 (R - 1) 元组）应用命题 13，我们可以推断出简化密码 Gk1, ···, kR - 1 允许 (δFr₂×{0}, δFr₂×{0}) 作为覆盖序列。

还有一个重要的备注：
- 备注 18 ：从覆盖序列导出的区分器的存在意味着可以定义一个从高阶结构导出的新区分器。实际上，如果一个 (n, m) 函数 G 允许 (ϕ, ψ) 作为覆盖序列，那么它允许一个高阶结构 (V, c)，使得 V = {a ∈ Fn₂; ϕ(a) ≡ 1 mod 2} 且 c = ∑b∈Fm₂ (ψ(b) mod 2) × b，因为总是可以对 (23) 中的求和进行模 2 约简以获得高阶结构。然而，由于覆盖序列比从它们定义的高阶结构提供了更多关于函数的信息，所以使用覆盖序列的区分器比从相应高阶结构导出的区分器更有效。

为了攻击那些不能用高阶结构攻击的简化密码，需要对覆盖序列进行推广：
- 定义 8 ：设 F 是从 Fn₂ 到 Fm₂ 的一个函数。我们称 F 的广义覆盖序列是任意一对函数 (Φ, Ψ)，它们分别从 (Fn₂)² 和 Im F × Fm₂ 到 R，使得：
∀x ∈ Fn₂, ∀b ∈ Fm₂, ∑a∈Fn₂, DaF(x)=b ϕx(a) = ψF(x)(b)
其中 ϕx(a) 和 ψF(x)(b) 分别表示 Φ(x, a) 和 Ψ(F(x), b)。
- 备注 19 ：任何单射的 (n, m) 函数 F 都允许广义覆盖序列 (Φ, Ψ)，其中 Φ 是常数函数 (x, a) ∈ Fn₂² → 1，Ψ(F(x), b) 表示集合 {a ∈ Fn₂; DaF(x) = b} 的基数。
- 命题 15 ：设 F : Fn₂ → Fk₂ 和 G : Fk₂ → Fm₂ 是两个分别允许 (Φ, Θ) 和 (Θ, Ψ) 作为广义覆盖序列的函数。那么，(Φ, Ψ) 是 G ∘ F 的广义覆盖序列。
- 证明 ：证明过程与命题 13 类似。

现在，可以用类似于覆盖序列的方式从轮函数 F 的广义覆盖序列导出一个区分器。显然，这个新的区分器不一定意味着涉及高阶结构的第二个区分器。如果不同的函数 ψx 的数量和集合 ∪x∈Fn₂ Supp ϕx 的基数较小，那么这样的区分器将是有效的。

下面用 mermaid 格式给出一个简单的流程图，展示从覆盖序列到广义覆盖序列及区分器的推导过程：

graph LR
    A[覆盖序列] --> B[满足命题13假设的轮函数]
    B --> C[简化密码的覆盖序列]
    A --> D[高阶结构区分器]
    A --> E[广义覆盖序列]
    E --> F[广义覆盖序列区分器]

2. 有限域的 uq - 尖锐子集相关内容

在有限域的研究中，uq - 尖锐子集是一个新的概念。我们先从一些基本定义开始：
- 定义 1 ：设 f(x1, · · ·, xn) ∈ Fq[x1, · · ·, xn]。f 的值集 V(f) 定义为 V(f) = {f(α1, · · ·, αn) : (α1, · · ·, αn) ∈ Fnq}，即 f 的值域。我们还定义 uq(f) 为最小的正整数 i，使得 ∑(α1,···,αn)∈Fnq f(α1, · · ·, αn)i ≠ 0。如果这样的 i 不存在，则设置 uq(f) = ∞。
- 定理 1 ：设 f ∈ Fq[x1, x2, · · ·, xn]，V(f) 是 f 的值集。如果 uq(f) < ∞，那么 |V(f)| ≥ uq(f) + 1。

还有经典的 Cauchy - Davenport 定理：
- 定义 2 ：设 A 和 B 是域 F 的非空子集。和集 A + B 定义为 A + B = {a + b : a ∈ A 且 b ∈ B}。
- 定理 2 ：设 A, B 是 Fp 的非空子集。那么 |A + B| ≥ min{p, |A| + |B| - 1}。

为了给出 Cauchy - Davenport 定理的另一个证明，引入了 uq - 尖锐子集的概念：
- 定义 3 ：设 f ∈ Fq[x]。如果 V(f) = uq(f) + 1，我们定义 f 为 uq - 尖锐的。如果 A 是 Fq 的非空子集，并且存在一个多项式 f ∈ Fq[x] 使得 f 是 uq - 尖锐的且 V(f) = A，我们定义 A 为 Fq 的 uq - 尖锐子集。

有以下一些关于 uq - 尖锐子集的性质：
- 命题 1 ：
1. Fq 是 uq - 尖锐的。
2. 如果 A ⊂ Fq 且 |A| = 2，那么 A 是 uq - 尖锐的。
3. 如果 A ⊂ Fq，q ≠ 3 且 |A| = q - 1，那么 A 不是 uq - 尖锐的。
- 定义 4 ：设 A = {a1, · · ·, aK} ⊂ Fq 且 |A| = K。设 f ∈ Fq[x] 使得 V(f) = A。对于 1 ≤ i ≤ K，我们设 mi = |f - 1(ai)| = |{α ∈ Fq : f(α) = ai}|。我们称 M(f) = {mi : i = 1, · · ·, K} 为 f 的重数集。
- 命题 2 ：设 A ⊂ Fq 且 |A| = K > 1。那么 A 是 Fq 的 uq - 尖锐子集当且仅当存在正整数 m1, · · ·, mK 且 ∑mi = q，使得 mi 满足以下在 Fq 中的线性方程组：
- m1 + m2 + · · · + mK = 0
- a1m1 + a2m2 + · · · + aKmK = 0
- …
- aK - 1₁m1 + aK - 1₂m2 + · · · + aK - 1ₖmK = β，其中 β ≠ 0。

下面给出一个非 uq - 尖锐子集的例子：
- 例子 ：设 p ≥ 5。使用最小正整数来表示 Fp 中的剩余类。设 c 是任意一个满足 2 ≤ c ≤ (p - 1) / 2 的正整数。考虑 A = {1, c, p - c, p - 1} ⊂ Fp。可以验证 (m1, m2, m3, m4) = (c, p - 1, 1, p - c) 给出了以下在 Fp 中的方程组的唯一解：
- m1 + m2 + m3 + m4 = 0
- m1 + cm2 + (p - c)m3 + (p - 1)m4 = 0
- m1 + c²m2 + (p - c)²m3 + (p - 1)²m4 = 0
- m1 + c³m2 + (p - c)³m3 + (p - 1)³m4 = β，其中 β = 2c(1 - c²) ≠ 0 (mod p)。注意在整数环 Z 中 ∑mi = 2p。对于所有非零的 β，在整数环中都有 ∑mi = 2p。因此，根据命题 2，A 不是 Fp 的 up - 尖锐子集。

还有一个重要的定理：
- 定理 3 ：设 A ⊂ Fp 使得 |A| > 1 且 q = pi > p。那么 A 是 Fq 的 uq - 尖锐子集。
- 证明 ：设 A = {a1, · · ·, aK}。设 (m1, · · ·, mk) 是以下在 Fp 中的方程组的唯一解：
- m1 + m2 + · · · + mK = 0
- a1m1 + a2m2 + · · · + aKmK = 0
- …
- aK - 1₁m1 + aK - 1₂m2 + · · · + aK - 1ₖmK = 1。
首先观察到所有的 mi 都必须是非零的。否则，不失一般性，假设 m1 = 0。那么前 K - 1 个方程构成了一个关于剩余 K - 1 个变量 m2, · · ·, mK 的齐次方程组。这个方程组的系数矩阵是一个范德蒙德矩阵，其行列式为 ∏2≤i<j≤K(aj - ai)。由于 ai 是不同的，所以行列式不为零。但是，齐次方程组必须只有零解，这与最后一个方程矛盾。我们使用 mi 的最小正整数代表。如果在整数环 Z 中 ∑mi = rp < q，其中 r 是一个满足 p ≤ rp ≤ K(p - 1) 的正整数，我们将代表剩余类 m1 的正整数替换为正整数 q - rp + m1。那么 ∑i mi = q，根据命题 2，证明完成。

利用定理 3 可以给出经典的 Cauchy - Davenport 定理的一个新证明：
- 证明 ：我们可以假设 |A| > 1 且 |B| > 1。还需要注意的是，如果 A 和 B 是 Fp 的子集，使得 |A| + |B| > p，那么对于每个 α ∈ Fp，A ∩ (α - B) ≠ ∅。因此，在这种情况下 |A + B| = p。所以我们可以假设 |A| + |B| ≤ p。设 q = p²。根据定理 3，我们可以找到 f, g ∈ Fq[x] 使得 V(f) = A 且 V(g) = B，其中 |A| = uq(f) + 1 且 |B| = uq(g) + 1。考虑多项式 h(x, y) = f(x) + g(y) ∈ Fq[x, y]。注意到 V(h) = A + B。使用二项式定理，我们得到：
∑(x,y)∈F₂q (h(x, y))N = ∑i + j = N N! / (i!j!) ∑x∈Fq (f(x))i ∑y∈Fq (g(y))j。
从不变量 uq 的定义可以得出，对于 i < uq(f)，∑x∈Fq (f(x))i = 0，对于 j < uq(g)，∑y∈Fq (g(y))j = 0。由于 uq(f) + uq(g) = |A| + |B| - 2 ≤ p - 2，所以 (uq(f) + uq(g))! / (uq(f)!uq(g)!) ≠ 0 (mod p)。由此可得 uq(h) = uq(f) + uq(g)。根据定理 1，我们有 |A + B| = V(h) ≥ uq(h) + 1 = uq(f) + uq(g) + 1 = |A| + |B| - 1。这就证明了定理。

下面用表格总结一下 uq - 尖锐子集的相关性质：
|性质|描述|
| ---- | ---- |
|Fq|是 uq - 尖锐的|
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |

向量函数覆盖序列与有限域的 uq - 尖锐子集

3. 研究展望与问题探讨

虽然我们已经对向量函数的覆盖序列和有限域的 uq - 尖锐子集有了一定的了解，但仍有许多问题值得进一步研究。

对于向量函数覆盖序列，尽管广义覆盖序列的定义为攻击一些难以用高阶结构攻击的简化密码提供了思路，但目前对于广义覆盖序列的性质和应用还缺乏深入的研究。例如，如何高效地计算广义覆盖序列，以及如何找到更多满足广义覆盖序列条件的函数，都是需要解决的问题。

在有限域的 uq - 尖锐子集方面，虽然我们已经证明了一些集合是否为 uq - 尖锐子集的性质，但对于 uq - 尖锐子集的结构还不清楚。我们可以提出以下几个研究问题：
1. uq - 尖锐子集的结构研究 ：找出 uq - 尖锐子集的一般结构特征，例如它们在有限域中的分布规律，以及与其他数学结构（如子域、理想等）的关系。
2. uq - 尖锐多项式的寻找 ：寻找新的多项式族，使得它们是 uq - 尖锐的。这有助于我们更好地理解 uq - 尖锐子集的形成机制。
3. uq - 尖锐子集的判定条件 ：确定一个集合成为 uq - 尖锐子集的更具体的条件，以便更方便地判断一个给定的集合是否为 uq - 尖锐子集。

下面用一个 mermaid 格式的流程图展示研究展望的方向：

graph LR
    A[向量函数覆盖序列] --> B[广义覆盖序列性质研究]
    A --> C[广义覆盖序列计算方法研究]
    A --> D[更多满足条件的函数寻找]
    E[有限域的uq - 尖锐子集] --> F[结构特征研究]
    E --> G[新的uq - 尖锐多项式族寻找]
    E --> H[判定条件确定]

4. 总结

本文主要介绍了向量函数覆盖序列和有限域的 uq - 尖锐子集的相关内容。在向量函数覆盖序列方面，我们从基本的覆盖序列概念出发，引入了广义覆盖序列的定义，并说明了如何从广义覆盖序列导出区分器。在有限域的 uq - 尖锐子集方面，我们定义了 uq - 尖锐子集的概念，证明了一些集合的 uq - 尖锐性质，并利用 uq - 尖锐子集给出了经典的 Cauchy - Davenport 定理的一个新证明。

通过对这些内容的研究，我们不仅加深了对向量函数和有限域的理解，还为密码学中的攻击和证明提供了新的工具和方法。然而，正如前面所提到的，这两个领域还有许多问题有待进一步研究和解决，未来的研究有望在这些方面取得更多的成果。

下面用表格总结本文的主要内容：
|研究领域|主要概念|重要结论|
| ---- | ---- | ---- |
|向量函数覆盖序列|覆盖序列、广义覆盖序列、区分器|广义覆盖序列可用于攻击难以用高阶结构攻击的简化密码；使用覆盖序列的区分器比从高阶结构导出的区分器更有效|
|有限域的 uq - 尖锐子集|uq - 尖锐子集、Cauchy - Davenport 定理|可以利用 uq - 尖锐子集给出 Cauchy - Davenport 定理的新证明；部分集合的 uq - 尖锐性质已被证明|