基于Richelot同构的超特殊亏格-2曲线的Hash函数
Hash functions from superspecial genus-2 curves using Richelot isogenies
摘要：
去年，Takashima提出了一个版本的Charles，Goren和Lauter的哈希函数，使用Richelot isogenies，从亏格2曲线开始，允许所有后续的算术在二次有限域Fp 2上执行。在最近的一篇论文中，Flynn和Ti指出Takashima的哈希函数是不安全的，因为它存在小的iscurrencies。我们重新审视的建设，并表明，它可以通过施加一个简单的限制，而且澄清了安全分析。所得到的散列函数的运行时间由对消息的每个3位块提取3个平方根来控制，与椭圆曲线情况下的每个位一个平方根相比;然而，在我们的设置中，提取可以并行化，并且在有限域中完成，其位大小减少了3倍。一路上，我们认为完整的超奇异同源图是研究Charles、Goren和Lauter哈希函数的高维类似物的错误背景，并主张使用超特殊子图，这是查看高岛的自然框架Fp 2友好的开始曲线。
英文摘要：
Last year Takashima proposed a version of Charles, Goren and Lauter's hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field Fp2. In a very recent paper Flynn and Ti point out that Takashima's hash function is insecure due to the existence of small isogeny cycles. We revisit the construction and show that it can be repaired by imposing a simple restriction, which moreover clarifies the security analysis. The runtime of the resulting hash function is dominated by the extraction of 3 square roots for every block of 3 bits of the message, as compared to one square root per bit in the elliptic curve case; however in our setting the extractions can be parallelized and are done in a finite field whose bit size is reduced by a factor 3. Along the way we argue that the full supersingular isogeny graph is the wrong context in which to study higher-dimensional analogues of Charles, Goren and Lauter's hash function, and advocate the use of the superspecial subgraph, which is the natural framework in which to view Takashima's Fp2-friendly starting curve.
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