Mathematical Proceedings of the Cambridge Philosophical Society

Corrigendum

Pseudoprime Reductions of Elliptic Curves – CORRIGENDUM

ALINA CARMEN COJOCARU, FLORIAN LUCA and IGOR E. SHPARLINSKI

Unfortunately, there are two inaccuracies in the argument of [CLS]. First, the statements of Lemmas 3, 4, 6, and 7 of [CLS] hold only under the additional condition gcd(m, ME) = 1 for some integer ME ≥ 1 depending only on E. Second, the divisibility condition (3·6) in [CLS] implies that tb(ℓ) | nE(p)−1 (rather than tb(ℓ) | nE(p), as it was erroneously claimed on p. 519 in [CLS]). In particular, instead of the divisibility ℓtb(ℓ) | nE(p) (see the last displayed formula on p. 519 in [CLS]), we conclude that for every prime ℓ | L there is an integer a such that


\begin{equation}
\label{eq:Cong}
n_\E(p) \equiv a_\ell \pmod {\ell t_b(\ell)} .
\end{equation}

(0.1)

However, the final result is correct and can easily be recovered. To do so, we remark that under the condition gcd(m,ME) =1, we have full analogues of Lemmas 6, 7, 9, and 10 of [CLS] for the function Π(x;m,a) defined as the number of primes px with nE(p) ≡ a (mod m) (rather than just for Π(x;m) = Π(x;m,0) as in [CLS]). Define ρ*(n) as the largest square-free divisor of n which is relatively prime to ME. We then derive from (0.1) above that

\begin{linenomath}$$
n_\E(p) \equiv a_\ell \pmod {\ell \rho^*(t_b(\ell))} .
$$\end{linenomath}

Therefore

\begin{equation}
\label{eq:T Pi}
\# \cT \le \sum_{y < \ell \le z} \Pi(x; \ell \rho^*(t_b(\ell)), a_\ell).
\end{equation}

(0.2)

Since

\begin{linenomath}$$
\rho^*(n)\mid \rho(n) \mand \rho^*(n)\ge \rho(n) / M_E,
$$\end{linenomath}

we see that (0.2) above implies the bound (3·7) from [CLS], and the result now follows without any further changes.

Related Articles

    A. C. Cojocaru, F. Luca and I. E. Shparlinski Pseudoprime reductions of elliptic curves. Math. Proc. Camb. Phil. Soc. 146 (2009), 513–522, doi: 10.1017/S0305004108001758, Published by Cambridge University Press, 14 July 2008.