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**2010/037**

**Title:**

Estimating the Size of the Image of Deterministic Hash Functions to Elliptic Curves

**Authors:**
Pierre-Alain Fouque, Mehdi Tibouchi

**Abstract:**

Let E be a non-supersingular elliptic curve over a finite field F_q. At CRYPTO 2009, Icart introduced a deterministic function F_q->E(F_q) which can be computed efficiently, and allowed him and Coron to define well-behaved hash functions with values in E(F_q). Some properties of this function rely on a conjecture which was left as an open problem in Icartâ€™s paper. We prove this conjecture as well as analogues for other hash functions. See also Farahashi, Shparlinski and Voloch, *On Hashing into Elliptic Curves*, for independent results of a similar form.

**ePrint:**
https://eprint.iacr.org/2010/037

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