Code-Based Public-Key Cryptosystems and Their Applications

Abstract

Code-based public-key cryptosystems are based on the hardness of a decoding problem. Their advantages include: 1) quantum tolerant, i.e. no polynomial time algorithm is known even on quantum computers whereas number theoretic public-key cryptosystems, such as RSA, Elliptic Curve Cryptosystems, DH, DSA, are vulnerable against them. 2) arithmetic unit is small for encryption and signature verification since they consists mostly of exclusive-ors that are highly parallelizable. The drawback is, however, that the public-key size is large, which is around some hundreds KB to some MB for typical parameters. Several attempts have been conducted to reduce the public-key size. Most of them, however, failed except one, which is Quasi-Dyadic (QD) public-key (for large extention degrees). While an attack has been proposed on QD public-key (for small extension degrees), it can be prevented by making the extension degree m larger, specifically by making q ⁽ m (m − 1)) large enough where q is the base filed and q = 2 for a binary code. QD approach can be improved further by using the method proposed in this paper. We call it “Flexible” Quasi-Dyadic (FQD) since it is flexible in its parameter choice, i.e. FQD can even achieve the maximum code length n = 2^m − t with one shot for given error correction capability t whereas QD must hold n < < 2^m − t (at least n ≤ 2^m − 1) and the key generation is performed by trial and error. Achieving n = 2^m − t or more loosely $n = 2^m - 2^{\lceil \log_2 t \rceil}$) is crucial for code-based digital signatures since they must make $2^{mt}/{n \choose t}$ small enough and without making n close to 2^m − t it cannot be satisfied. FQD can also be applied to code-based digital signatures.