Rational Number Representation over Elliptic Curves via Equivalence Relations: Extending ECHC to Fractional Scalar Multiplication

We propose a novel framework for representing rational numbers over elliptic curves through equivalence relations, extending the classical scalar multiplication paradigm. Building upon the Elliptic Curve Homomorphic Cryptography (ECHC) foundation, we construct a systematic method to represent not only integers but also rational numbers as points on elliptic curves. Starting from the standard form aP, where a ∈ [0, n−1] on an elliptic curve E of order n, we first extend to signed integers using pairs (aP, s) with s ∈ {0,1} encoding the sign. Following the historical development from integers to rationals, we then construct an equivalence relation on pairs of elliptic curve points (aP, bP) such that (a,b)P ∼ (c,d)P if and only if adP = bcP in E. This construction leads to a well-defined "elliptic rational" number system ℚ_E that generalizes scalar multiplication to rational coefficients. We establish the algebraic structure of ℚ_E, investigate its field-like properties using formal multiplication derived from ECHC, and analyze potential cryptographic applications including fractional homomorphic operations.