In rank-metric cryptography, a vector from a finite dimensional linear space over a finite field is viewed as the linear space spanned by its entries. The rank decoding problem which is the analogue of the problem of decoding a random linear code consists in recovering a basis of a random noise vector that was used to perturb a set of random linear equations sharing a secret solution. Assuming the intractability of this problem, we introduce a new construction of injective one-way trapdoor functions. Our solution departs from the frequent way of building public key primitives from error-correcting codes where, to establish the security, ad hoc assumptions about a hidden structure are made. Our method produces a hard-to-distinguish linear code together with low weight vectors which constitute the secret that helps recover the inputs.The key idea is to focus on trapdoor functions that take sufficiently enough input vectors sharing the same support. Applying then the error correcting algorithm designed for Low Rank Parity Check (LRPC) codes, we obtain an inverting algorithm that recovers the inputs with overwhelming probability.

翻译：在秩度量密码学中，有限域上有限维线性空间中的向量被视为由其分量张成的线性空间。秩译码问题是随机线性码译码问题的类比，其目标在于恢复用于扰动一组共享秘密解的随机线性方程的随机噪声向量的基。基于该问题的难解性，我们提出了一种新的单射单向陷门函数构造方案。我们的方案不同于传统上通过纠错码构建公钥原语的方式——后者通常需要对隐藏结构提出特殊假设以建立安全性。本方法能够生成一个难以区分的线性码，同时附带构成恢复输入关键秘密的低权重向量。核心思想在于聚焦于接收足够多共享相同支撑输入向量的陷门函数。通过应用为低秩奇偶校验（LRPC）码设计的纠错算法，我们获得了能以压倒性概率恢复输入的求逆算法。