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列表译码在密码中的应用综述

张卓然 张煌 张方国

张卓然, 张煌, 张方国. 列表译码在密码中的应用综述[J]. 电子与信息学报, 2020, 42(5): 1049-1060. doi: 10.11999/JEIT190851
引用本文: 张卓然, 张煌, 张方国. 列表译码在密码中的应用综述[J]. 电子与信息学报, 2020, 42(5): 1049-1060. doi: 10.11999/JEIT190851
Zhuoran ZHANG, Huang ZHANG, Fangguo ZHANG. Survey on Applications of List Decoding to Cryptography[J]. Journal of Electronics & Information Technology, 2020, 42(5): 1049-1060. doi: 10.11999/JEIT190851
Citation: Zhuoran ZHANG, Huang ZHANG, Fangguo ZHANG. Survey on Applications of List Decoding to Cryptography[J]. Journal of Electronics & Information Technology, 2020, 42(5): 1049-1060. doi: 10.11999/JEIT190851

列表译码在密码中的应用综述

doi: 10.11999/JEIT190851
基金项目: 国家自然科学基金(61672550, 61972429),国家重点研发计划(2017YFB0802503)
详细信息
    作者简介:

    张卓然:女,1995年生,博士生,研究方向为基于纠错码的密码学

    张煌:男,1988年生,博士生,研究方向为格密码和零知识

    张方国:男,1972年生,教授,研究方向为密码学理论及其应用,特别是椭圆曲线密码体制、安全多方计算、可证明安全性等

    通讯作者:

    张方国 isszhfg@mail.sysu.edu.cn

  • 中图分类号: TP393

Survey on Applications of List Decoding to Cryptography

Funds: The National Natural Science Foundation of China (61672550, 61972429), The National Key R & D Program of China (2017YFB0802503)
  • 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。
  • 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}$。
    下载: 导出CSV
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  • 收稿日期:  2019-11-01
  • 修回日期:  2020-02-25
  • 网络出版日期:  2020-03-19
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