# Security of an Elliptic Curve Public Key with a “Small” x-coordinate

Consider an elliptic curve over a finite field $$F_p$$ with $$p$$ prime and order $$n$$. Let $$Q$$ be a generator for the field. Given a public key point $$P = aQ$$, suppose we have an algorithm that finds an integer $$b$$ such that $$P + bQ = R = (x, y)$$ with $$x \leqslant M$$ for some fixed $$M << n$$. Is this new point $$R$$ any less secure than the original public key i.e., could we find $$c$$ such that $$R = cQ$$ yielding $$a \equiv c - b \ (\!\!\!\!\mod n)$$.