= Solution
\b[No.] The procedure exposes a square-root oracle for the <Rabin cryptosystem>. The verifier chooses $m$, sends $r=m^2\bmod N$, and receives another square root $m_1$. If
$$
m_1\not\equiv\pm m\pmod N,
$$
then the <factorization from distinct square roots modulo a semiprime> gives
$$
\boxed{\gcd(m-m_1,N)\text{ is a nontrivial factor of }N.}
$$
For a product of two odd primes, a square normally has four roots, and a returned root lies outside the pair $\{\pm m\}$ with probability about $1/2$. Repeating chosen-square challenges therefore factors $N$ with overwhelming probability. The attacker can then compute square roots and impersonate Alice. This is the <chosen-square attack on Rabin authentication>, so the proposed protocol is insecure.
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