Abstract
Debris-Alazard, Sendrier, and Tillich proposed SURF, which is a code-based signature scheme and enjoys efficient signature generation and verification (eprint in 2017). The security of this scheme is based on two problems: one is DOOM (Decoding One Out of Many), and the other is the plain (U,U+V)-code problem over \(\mathbb {F}_2\). There are many studies on the former one but few studies on the latter one. Later the security of SURF was broken because the hardness of the plain (U,U+V)-code problem does not hold with considering a notion of the hull.
Then Debris-Alazard et al. proposed Wave as a successor of SURF, which is known as one of the most promising quantum-resistant signature schemes (ASIACRYPT 2019). Wave is based on similar problems used in SURF. Wave uses DOOM and the normalized generalized (U,U+V)-code problem over \(\mathbb {F}_3\).
In this paper, we utilize a notion of the Gramian (the determinant of the Gram matrices) of public keys and analyze the plain (U,U+V)-code problem over \(\mathbb {F}_2\). For this purpose, we compute the asymptotic probability distribution of Gramians of random matrices. Furthermore, we also show a way to analyze the normalized generalized (U,U+V)-code problem over \(\mathbb {F}_2\). Finally, we apply our analysis to the normalized generalized (U,U+V)-code problem over \(\mathbb {F}_3\). By our analysis with Gramian, SURF is completely broken, however, Wave is not directly threatened.
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A part of this work was supported by JST CREST JP-MJCR2113 and JSPS KAKENHI JP21H04879.
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Iwata, I., Yoshida, Y., Tanaka, K. (2023). Analysis of (U,U+V)-code Problem with Gramian over Binary and Ternary Fields. In: Seo, SH., Seo, H. (eds) Information Security and Cryptology – ICISC 2022. ICISC 2022. Lecture Notes in Computer Science, vol 13849. Springer, Cham. https://doi.org/10.1007/978-3-031-29371-9_21
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