feat: temp stage

paper/scripts/example/shamir_example.py

@@ -0,0 +1,173 @@
+# type: ignore
+# ruff: noqa
+import numpy as np
+from fractions import Fraction
+
+
+def generate_shamir_example(secret, t, n, a_coeffs=None):
+    """
+    生成一个Shamir秘密共享的例子并输出LaTeX代码
+
+    参数:
+        secret: 需要共享的秘密整数
+        t: 恢复秘密所需的最小分片数
+        n: 总分片数
+        a_coeffs: 多项式系数（可选）
+    """
+    # 如果未提供系数，则随机生成
+    if a_coeffs is None:
+        a_coeffs = [np.random.randint(1, 100) for _ in range(t - 1)]
+
+    # 确保a_coeffs长度正确
+    assert len(a_coeffs) == t - 1, f"系数长度应为 {t - 1}"
+
+    # 构建多项式 f(x) = secret + a_1*x + a_2*x^2 + ... + a_{t-1}*x^{t-1}
+    all_coeffs = [secret] + a_coeffs
+
+    # 计算每个参与者的分片
+    shares = []
+    for i in range(1, n + 1):
+        y = sum(all_coeffs[j] * (i**j) for j in range(t))
+        shares.append((i, y))
+
+    # 选择t个分片进行恢复
+    selected_shares = shares[:t]
+
+    # LaTeX输出
+    latex_output = generate_latex(secret, t, n, all_coeffs, shares, selected_shares)
+
+    return {
+        "secret": secret,
+        "polynomial": all_coeffs,
+        "shares": shares,
+        "selected_shares": selected_shares,
+        "latex": latex_output,
+    }
+
+
+def lagrange_interpolation(points, x_value=0):
+    """使用Lagrange插值法恢复秘密"""
+    result = 0
+    x_points, y_points = zip(*points)
+
+    # 计算每个点的拉格朗日基本多项式值并相加
+    for i in range(len(points)):
+        term = y_points[i]
+        for j in range(len(points)):
+            if i != j:
+                term *= Fraction(x_value - x_points[j], x_points[i] - x_points[j])
+        result += term
+
+    return result
+
+
+def generate_latex(secret, t, n, coeffs, shares, selected_shares):
+    """生成完整的LaTeX代码"""
+    # 构建多项式字符串
+    poly_str = f"{coeffs[0]}"
+    for i in range(1, len(coeffs)):
+        if coeffs[i] > 0:
+            poly_str += f" + {coeffs[i]}"
+        else:
+            poly_str += f" - {abs(coeffs[i])}"
+
+        if i == 1:
+            poly_str += "x"
+        else:
+            poly_str += f"x^{i}"
+
+    # 计算拉格朗日插值步骤
+    lagrange_steps = []
+    result = 0
+    for i, (x_i, y_i) in enumerate(selected_shares):
+        term = y_i
+        term_str = f"{y_i}"
+
+        for j, (x_j, _) in enumerate(selected_shares):
+            if i != j:
+                fraction = Fraction(0 - x_j, x_i - x_j)
+                term *= fraction
+
+                # 构建拉格朗日基本多项式的LaTeX表示
+                term_str += f"\\cdot\\frac{{0-{x_j}}}{{{x_i}-{x_j}}}"
+
+        # 添加简化步骤
+        simplified_term = int(term) if term.denominator == 1 else term
+        lagrange_steps.append((term_str, simplified_term))
+        result += simplified_term
+
+    # 生成最终的LaTeX代码
+    latex = f"""
+举例说明：假设我们有一个秘密数字$S={secret}$，我们希望将其分割成{n}个片段，
+但只需要{t}个片段就可以恢复它（即$t={t},n={n}$）：
+选择一个{t - 1}次多项式（因为$t-1={t - 1}$）:
+\\[f(x) = """
+
+    # 添加多项式表达式
+    for i in range(t):
+        if i == 0:
+            latex += f"a_{i}"
+        else:
+            latex += f" + a_{i}x^{i}"
+    latex += "\\]\n"
+
+    # 添加具体的多项式
+    latex += f"假设我们选择"
+    for i in range(1, t):
+        if i < t - 1:
+            latex += f"$a_{i}={coeffs[i]}$, "
+        else:
+            latex += f"$a_{i}={coeffs[i]}$"
+    latex += f"，那么多项式就是：\n\\[f(x) = {poly_str}\\]\n"
+
+    # 添加分片计算
+    latex += "为每个参与者选择一个$x$值并计算$f(x)$：\n"
+    for x, y in shares:
+        latex += f"\\[f({x}) = {' + '.join([f'{coeffs[i]}\\cdot{x}^{i}' if i > 0 else f'{coeffs[i]}' for i in range(t)])} = {y}\\]\n"
+
+    # 添加分片分配
+    share_str = ", ".join([f"$({x},{y})$" for x, y in shares])
+    latex += f"每个参与者分别得到一个片段：{share_str}。\n"
+
+    # 添加恢复秘密的说明
+    latex += f"要恢复秘密，我们可以选择其中任意{t}个片段。\n"
+    selected_str = "和".join([f"$({x},{y})$" for x, y in selected_shares])
+    latex += f"假设我们选择{selected_str}，使用Lagrange插值可以计算出$f(0)={secret}$，从而恢复秘密。\n"
+
+    # 添加拉格朗日插值公式
+    latex += "具体的Lagrange插值公式为：\n"
+    latex += "\\[f(x) = \\sum_{i=1}^t y_i \\prod_{j=1,j\\neq i}^t \\frac{x-x_j}{x_i-x_j}\\]\n"
+
+    # 添加具体的拉格朗日插值计算
+    latex += f"在这个例子中，使用{selected_str}进行插值：\n"
+    terms = []
+    calculations = []
+
+    for term_str, simplified_term in lagrange_steps:
+        terms.append(term_str)
+        calculations.append(str(simplified_term))
+
+    latex += (
+        f"\\[f(0) = {' + '.join(terms)} = {' + '.join(calculations)} = {result}\\]\n"
+    )
+    latex += f"这就恢复出了原始的秘密值{secret}。"
+
+    return latex
+
+
+if __name__ == "__main__":
+    # 设置的参数
+    my_secret = 89  # 秘密数字
+    threshold = 2  # 恢复秘密所需的最小分片数
+    total_shares = 4  # 总分片数
+    coefficients = [23]  # 多项式系数 (除了常数项外，t-1个系数)
+
+    # 生成例子
+    example = generate_shamir_example(my_secret, threshold, total_shares, coefficients)
+
+    # 打印LaTeX代码
+    print(example["latex"])
+
+    # 验证秘密恢复是否正确
+    recovered_secret = lagrange_interpolation(example["selected_shares"])
+    assert recovered_secret == my_secret, "恢复的秘密不正确！"