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**2021/1604**

**Title:**

The most efficient indifferentiable hashing to elliptic curves of j-invariant 1728

**Authors:**
Dmitrii Koshelev

**Abstract:**

This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic finite field \mathbb{F}_{\!q}. More precisely, we construct a new indifferentiable hash function to any ordinary elliptic \mathbb{F}_{\!q}-curve E_a of j-invariant 1728 with the cost of extracting one quartic root in \mathbb{F}_{\!q}. As is known, the latter operation is equivalent to one exponentiation in finite fields with which we deal in practice. In comparison, the previous fastest random oracles to E_a require to perform two exponentiations in \mathbb{F}_{\!q}. Since it is highly unlikely that there is a hash function to an elliptic curve without exponentiations at all (even if it is supersingular), the new result seems to be unimprovable.

**ePrint:**
https://eprint.iacr.org/2021/1604

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