AI 中文总结
本文提出一种针对任意有效除子的椭圆码McEliece密码系统的多项式时间结构性攻击，利用三个已知点恢复整个除子，并优化为无需额外信息的版本，从而破坏该密码系统。
AI 中文摘要
基于代数几何码的McEliece密码系统曾被提出用于减小基于码的密码学的密钥大小，但若干结构性攻击已经证明了特定代数几何码族的脆弱性。尽管如此，直到最近，仍有一些方案和参数集不受任何已知攻击的影响。我们提出了一种带有“提示”的新型结构性攻击，适用于具有任意有效除子的椭圆码。特别地，我们证明，给定椭圆曲线、公开生成矩阵以及求值除子中的三个点，可以在多项式时间内恢复整个除子，且与密码系统中使用的错误数量无关。该攻击在$\mathbb{F}_q$上需要$\mathcal{O}(k^2n^2+|\mathcal{E}(\mathbb{F}_q)|+n)$次运算，并以压倒性概率成功，之后第二个除子在$\mathcal{O}\left(k^2n^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$次运算内被恢复。我们进一步提出了一种优化版本的攻击，该版本完全不需要额外信息。利用曲线自同构的作用，三个已知点被替换为单对域元素的枚举，这将在给定公开曲线上平均$\mathcal{O}\left(k^2n^2 + q^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$次运算内产生一个等价密钥。
英文摘要
The McEliece cryptosystem based on algebraic geometry codes has been proposed as a way to reduce the key size of code-based cryptography, but several structural attacks have demonstrated the vulnerability of particular families of algebraic geometry codes. Despite this, until recently, there remained schemes and parameter sets that were not vulnerable to any known attack. We propose a new structural attack with ``hints'' that applies to elliptic codes with arbitrary effective divisors. In particular, we prove that, given the elliptic curve, the public generator matrix, and three points from the evaluation divisor, the entire divisor can be recovered in polynomial time, independently of the number of errors used in the cryptosystem. The attack requires $\mathcal{O}(k^2n^2+|\mathcal{E}(\mathbb{F}_q)|+n)$ operations in $\mathbb{F}_q$ and succeeds with overwhelming probability, after which the second divisor is recovered in $\mathcal{O}\!\left(k^2n^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$ operations. We further propose an optimized version of the attack that requires no additional information at all. Exploiting the action of the automorphisms of the curve, the three known points are replaced by the enumeration of a single pair of field elements, which yields an equivalent key on the given public curve in $\mathcal{O}\!\left(k^2n^2 + q^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$ operations on average.
