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 $a \equiv c - b \ (\!\!\!\!\mod n)$.


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