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Paper IPM / M / 16321  


Abstract:  
Many cryptographic protocols which are based on elliptic curves require to efficiently encode bitstrings into the points of a given elliptic curve such that the encoding function satisfies computability, regularity, and samplability, or generally admissibility. All the admissible encoding functions from the finite field Fq are restricted to the class of elliptic curves with a nontrivial ltorsion Fqrational point, where lâ2,3. Therefore, there is no admissible encoding function to the class of many cryptographically interesting elliptic curves of prime order. In this paper, we present an admissible 2:1 encoding function from the set 0,1,â¦,qâ12 to the Fqrational points of arbitrary elliptic curves. We also propose an injective encoding function to elliptic curves with a nontrivial Fq
rational point of order two, that acts the same as the Bernstein et al.âs injective encoding function. Conversely, occasionally we have to transmit points of a known curve through an insecure channel. Traditional methods for transferring points enable an adversary to recognize patterns in the transmitted data. Consequently, one finds valuable information to attack the cryptosystem using the network traffic. By the help of the inverse of the injective encoding functions, Bernstein et al. introduced an interesting solution to this problem, namely Elligator. In this paper, we present an indistinguishable elliptic curve point representation using our given encoding function, which unlike the previous wellknown encoding functions is not injective but covers almost all of elliptic curves over odd characteristic finite fields. Indeed, since we proposed a 2:1 encoding function to elliptic curves in short Weierstrass form, we have to select one preimage randomly and transmit its corresponding bitstring instead of the point.
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