# Is simple EC arithmetics enough to construct security proofs?

Using the simple rules of modular arithmetics for Elliptic Curves, including associative and commutative rules for $$+$$ and $$*$$, where uppercase letters are points and lowercase are scalars on the order of G.

1. $$P = a.G$$ as the generator trapdoor.
2. $$P + R = D \Leftrightarrow P = D - R$$
3. $$P + 0 = P$$ , identity for $$+$$ using points
4. $$a + b = c \Leftrightarrow a = c - b$$
5. $$a + 0 = a$$ , identity for $$+$$ using scalars
6. $$a*a^{-1} = 1$$ , identity for $$*$$ using scalars

Assuming any scheme I build always use those arithmetic rules and trapdoors. Can one use these to construct security proofs without messing around with polynomials?

Good crypto libraries (like bouncycastle) already provide such operations. Am I missing something, can the polynomials bring anything important to the proofs?

As an example, one can observe the discussion in this other thread: How do I create a cryptographic signature and commitment scheme with accountable evidence?

• It's not a trapdoor unless you have a secret that allows you to invert it. I suspect you mean "one-way function" instead of "trapdoor". – Ella Rose Feb 8 at 18:30