Survey on Applications of List Decoding to Cryptography
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摘要: 列表译码自上世纪50年代提出以来，不仅在通信与编码等方面得到了广泛应用，也在计算复杂性理论和密码学领域有着广泛的应用。近年来，随着量子计算的发展，基于整数分解等传统困难问题设计的密码方案受到了巨大的威胁。由于编码理论中一些计算问题的NP困难性被广泛认为是量子概率多项式时间不可攻克的，建立在其上的基于纠错码的密码体制得到了越来越多的重视，列表译码也越来越引起人们的关注。该文系统梳理了列表译码在密码学中的应用，包括早期在证明任何单向函数都存在硬核谓词、设计叛徒追踪方案、以多项式重建作为密码原语设计公钥方案、改进传统基于纠错码的密码方案和求解离散对数问题(DLP)等方面的应用，以及近期，列表译码在设计安全通信协议、求解椭圆曲线离散对数问题、设计新的基于纠错码的密码方案等方面的应用。该文对列表译码的算法改进及其在密码协议设计和密码分析中的应用、新应用场景探索等方面的发展趋势进行了探讨。Abstract: Since the conception of list decoding is proposed in the 1950s, list decoding not only is applied to communication and coding theory, but also plays a significant role in computational complexity and cryptography. In recent years, with the rapid development of quantum computing, the traditional cryptographic schemes based on factorization and other difficult problems are greatly threatened. The code-based cryptosystems, whose security relies on the NP-hard problems in coding theory, are attracting more and more attention as a candidate of the post-quantum cryptography, and so does the list decoding algorithm. This paper systematically reviews the applications of list decoding to cryptography, including early applications in proving that any one-way function has hard-core bits, designing traitor tracing schemes, designing public key schemes using polynomial reconstruction as cryptographic primitives, improving the traditional code-based cryptosystems and solving Discrete Logarithm Problems (DLP), and recent applications to designing secure communication interactive protocols, solving the elliptic curve discrete logarithm problem, and designs new cryptographic schemes based on error correction codes. Finally, the new research issues of the algorithm improvement of list decoding, its application to the design of cryptographic protocol and cryptoanalysis, and the exploration of new application scenarios are discussed.
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Key words:
- Public key cryptography /
- List decoding /
- Discrete logarithm /
- Post-quantum cryptography
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表 1 Guruswami-Sudan列表译码算法
${\rm{ListDecode}}({\cal{C}},{{r}},t)$ 输入：有限域${\mathbb{F}_q}$，曲线${\cal{X}}$，除子$G = \alpha Q$和$D$，接受向量${{r} } = ({r_1},{r_2}, ··· ,{r_n})$ 以及错误重量上界$t$。 初始化： (1) 设置表单${\Omega _r}: = \varnothing $； (2) 由$n,k,t$计算译码参数$l$，要求$l > \alpha $；一般地，设
$r = 1 + \dfrac{{(2g + \alpha )n - 2gt + \sqrt {{{((2g + \alpha )n - 2gt)}^2} - 4({g^2} - 1)({{(n - t)}^2} - \alpha n)} }}{{2{{(n - t)}^2} - \alpha n}}$, $l = r(n - t) - 1$；(3) 固定${\cal{L}}(lQ)$的一组极基$\{ {\phi _{ {j_1} } }:1 \le {j_1} \le l - g + 1\} $，使得Q最多为${\phi _{{j_1}}}$的${j_1} + g - 1$次极点； (4) 对任意${P_i}$, $1 \le i \le n$，找${\cal{L}}(lQ)$的一组零基$\{ {\psi _{ {j_3} } }:1 \le {j_3} \le l - g + 1\} $，使得${P_i}$为${\psi _{{j_3},{P_i}}}$重数(至少)为${j_3} - 1$的零点； (5) 计算集合$\{ {a_{ {P_i},{j_1},{j_3} } } \in {\mathbb{F}_q}:1 \le i \le n,1 \le {j_1},{j_3} \le l - g + 1\} $，使得对任意i和${j_1}$，都有${\phi _{{j_1}}} = {\Sigma _{{j_3}}}{a_{{P_i},{j_1},{j_3}}}{\psi _{{j_3},{P_i}}}$。 插值：
令$s = \dfrac{{l - g}}{\alpha }$，找非零多项式$H \in {\cal{L}}(lQ)[T]$，它具有以下形式：$H[T] = \displaystyle\sum\limits_{ {j_2} = 0}^s {\displaystyle\sum\limits_{ {j_1} = 1}^{l - g + 1 - \alpha {j_2} } { {h_{ {j_1},{j_2} } }{\phi _{ {j_1} } }{T^{ {j_2} } } } } $；其中，系数${h_{{j_1},{j_2}}} \in {\mathbb{F}_q}$满足：至少有一个${h_{{j_1},{j_2}}}$是非零的，且对任意$i \in [n]$，和满足${j_3} + {j_4} \le r$的${j_3} \ge 1,{j_4} \ge 0$，有
$h_{{j_3},{j_4}}^{(i)} = \displaystyle\sum\limits_{{j_2} = {j_4}}^s {\displaystyle\sum\limits_{{j_1} = 1}^{l - g + 1 - \alpha {j_2}} {\left( \begin{array}{l} {j_2} \\ {j_4} \\ \end{array} \right)r_i^{{j_2} - {j_4}} \cdot {h_{{j_1},{j_2}}}{\alpha _{{x_i},{j_1},{j_3}}} = 0} } $求根： 找到$H[T]$的所有根$h \in {\cal{L}}(\alpha Q) \subseteq {\cal{L}}(lQ)$。对每一个$h$，检查是否对至少$n - t$个$i \in \{ 1,2, ··· ,n\} $有$h({P_i}) = {r_i}$，即$d({{r} },{{c} }) \le t$。如果成
立，将$h$加入${\Omega _r}$。输出：码字列表${\Omega _r}$。 
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