后量子密码学（PQC）：GL(8,F251) 上的广义 ElGamal 密码
Post-Quantum Cryptography(PQC): Generalized ElGamal Cipher over GL(8,F251)
AI总结：
本文提出一种基于非交换一般线性群和广义 ElGamal 协议的后量子非对称密码，通过递归更新参数并将安全性建立在广义对称分解问题困难变体上，在 GL(8,F251) 下实现 64 位安全，且无需大数库，适用于智能卡等资源受限平台。
摘要：
后量子密码学（PQC）试图寻找能够抵抗攻击的密码协议，例如抵抗 Shor 多项式时间算法对整数分解（IFP）或离散对数（DLP）等数域问题的攻击。其他方面还包括确定性随机生成器中发现的后门，以及求解某些 DLP 实例的最新进展。使用替代代数结构，如非交换或非结合的部分群胚、原群、幺半群、半群、拟群或群，是这些新协议的有效选择。本文聚焦于一种基于广义 ElGamal 非仲裁协议的非对称密码，该协议使用非交换一般线性群。所开发的协议将一个困难的子群成员搜索问题强制嵌入非交换结构中。该协议首先涉及广义 Diffie-Hellman 密钥交换，并且在每次启动新的密码会话时，私钥和公开参数都会被递归更新。安全性基于广义对称分解问题（GSDP）的一个困难变体。使用 GL(8, F251) 可实现 64 位安全性，如果选择 GL(16, F251)，安全性则提升至 127 位。一个吸引人的特点是无需大数库，因为所有算术都在 Z_251 中进行。因此，该新协议特别适用于计算能力非常有限的平台，如智能手机或智能卡。
英文摘要：
Post-quantum cryptography (PQC) attempts to find cryptographic protocols resistant to attacks using for instance Shor's polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. Using alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new protocols. This paper focuses on an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with GL(8, F251) 64-bit security is achieved, and if GL(16, F251) is chosen, the security rises to 127-bit. An appealing feature is that there is no need for big number libraries as all arithmetic is performed in Z_251. Therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.
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