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Fixed argument pairing inversion on elliptic curves

机译：固定椭圆曲线上的参数配对反转

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摘要

Let be an elliptic curve over a finite field with a power of prime a prime dividing , and the smallest positive integer satisfying , called embedding degree. Then a bilinear map is defined, called the Tate pairing. The Ate pairing and other variants are obtained by reducing the domain for each argument and raising it to some power. In this paper we consider the Fixed Argument Pairing Inversion (FAPI) problem for the Tate pairing and its variants. In 2012, considering FAPI for the Ate pairing, Kanayama and Okamoto formulated the Exponentiation Inversion (EI) problem. However the definition gives a somewhat inaccurate description of the hardness of EI. We point out that the described EI can be easily solved, and hence give a repaired definition of EI so that the problem does contain the actual hardness in connection with the prescribed domain for given pairings. Next we show that inverting the Ate pairing (including other variants of the Tate pairing) defined on the smaller domain is neither easier nor harder than inverting the Tate pairing defined on the larger domain. This is interesting because the structure of the Ate pairing is so simple and good (that is, the Miller length is short, the solution domain is small and has an algebraic structure induced from the Frobenius map) that it looks more probable that attackers find further approach to solve FAPI for the Ate pairing, differently from the Tate pairing.
机译：设椭圆形曲线在有限域上以素数除以质数，且最小的正整数满足，称为嵌入度。然后定义了一个双线性图，称为泰特配对。通过减少每个自变量的域并将其提高到一定的幂，可以获得Ate配对和其他变体。在本文中，我们考虑了泰特（Tate）配对及其变体的固定参数配对反转（FAPI）问题。在2012年，考虑到Ate配对的FAPI，金山和冈本提出了指数倒置（EI）问题。但是，该定义对EI硬度的描述有些不准确。我们指出，所描述的EI可以很容易地解决，因此可以给出EI的修正定义，从而使问题的确包含与给定配对的规定范围相关的实际硬度。接下来，我们证明反转在较小域上定义的Ate配对（包括Tate配对的其他变体）与反转在较大域上定义的Tate配对既不容易也不困难。这很有趣，因为Ate配对的结构是如此简单和良好（也就是说，米勒长度短，解域小并且具有从Frobenius映射得出的代数结构），从而使攻击者更容易发现更多解决Ate配对FAPI的方法，与Tate配对不同。

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