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2. This problem considers examples of the "double-and-add" algorithm for efficiently computing the multiple of a point on an elliptic curve. Suppose we have an elliptic curve $E$ and a point $P \in E$, and suppose we want to compute $nP$ for some positive integer $n$. Here is one way to do it efficiently. • Write $n$ in binary form: $n = n_0 + n_1 \cdot 2 + \dots + n_r \cdot 2^r$, with each $n_i \in \{0, 1\}$. We will assume that $n_r = 1$. • Compute the sequence of points, $Q_0 = P$, $Q_1 = 2Q_0$, $Q_2 = 2Q_1$, ..., $Q_r = 2Q_{r-1}$. • We can then compute $nP$ as $nP = n_0Q_0 + n_1Q_1 + n_2Q_2 + \dots + n_rQ_r$. Thus we can compute $nP$ by performing at most $r$ doublings and $r$ additions on $E$. Since $n \ge 2^r$, we see that $r \le \log_2(n)$. Therefore, the number of necessary computations is small with respect to $n$, and this makes it possible to compute $nP$ even for large values of $n$. You may find this reminiscent of the method of successive squaring for computing $a^n \mod m$ in modular arithmetic. Now for the actual problems: Use the double-and-add procedure to compute $nP$ on the elliptic curves over finite fields below.

          2. This problem considers examples of the "double-and-add" algorithm for efficiently computing
the multiple of a point on an elliptic curve.
Suppose we have an elliptic curve $E$ and a point $P \in E$, and suppose we want to compute
$nP$ for some positive integer $n$. Here is one way to do it efficiently.
• Write $n$ in binary form: $n = n_0 + n_1 \cdot 2 + \dots + n_r \cdot 2^r$, with each $n_i \in \{0, 1\}$. We will
assume that $n_r = 1$.
• Compute the sequence of points,
$Q_0 = P$, $Q_1 = 2Q_0$, $Q_2 = 2Q_1$, ..., $Q_r = 2Q_{r-1}$.
• We can then compute $nP$ as
$nP = n_0Q_0 + n_1Q_1 + n_2Q_2 + \dots + n_rQ_r$.
Thus we can compute $nP$ by performing at most $r$ doublings and $r$ additions on $E$. Since
$n \ge 2^r$, we see that $r \le \log_2(n)$. Therefore, the number of necessary computations is small
with respect to $n$, and this makes it possible to compute $nP$ even for large values of $n$.
You may find this reminiscent of the method of successive squaring for computing $a^n \mod m$
in modular arithmetic.
Now for the actual problems:
Use the double-and-add procedure to compute $nP$ on the elliptic curves over finite fields
below.

$2. This problem considers examples of the "double-and-add" algorithm for efficiently computing the multiple of a point on an elliptic curve. Suppose we have an elliptic curve E and a point P ∈ E, and suppose we want to compute nP for some positive integer n. Here is one way to do it efficiently. • Write n in binary form: n = n0 + n1 · 2 + … + nr · 2^r, with each ni ∈{0, 1}. We will assume that nr = 1. • Compute the sequence of points, Q0 = P, Q1 = 2Q0, Q2 = 2Q1, ..., Qr = 2Qr-1. • We can then compute nP as nP = n0Q0 + n1Q1 + n2Q2 + … + nrQr. Thus we can compute nP by performing at most r doublings and r additions on E. Since n ≥ 2^r, we see that r ≤log2(n). Therefore, the number of necessary computations is small with respect to n, and this makes it possible to compute nP even for large values of n. You may find this reminiscent of the method of successive squaring for computing a^n m in modular arithmetic. Now for the actual problems: Use the double-and-add procedure to compute nP on the elliptic curves over finite fields below.$

Added by Cassandra J.

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