GATE CS 2015 Set 3 Q.59 (CS Q.49): Combinatorics, Counting, Recurrence Relations and Generating Functions

Suppose $c = (c[0], \dots, c[k - 1])$ is an array of length $k$ , where all the entries are from the set $\{0, 1\}$ . For any positive integers $a$ and $n$ , consider the following pseudocode:

DOSOMETHING (c, a, n)
z = 1
for i = 0 to k - 1
    do z = z^2 mod n
    if c[i] == 1
        then z = (z * a) mod n
return z

If $k = 4, c = (1, 0, 1, 1), a = 2$ and $n = 8$ , then the output of DOSOMETHING(c, a, n) is _____.

Your answer

Official answer and solution

Official answer: 0
From: Official GATE 2015 answer key

Solution

Trace the loop iterations of the modular exponentiation algorithm.

Given inputs:

$k = 4$
Array $c = (1, 0, 1, 1)$ so $c[0] = 1, c[1] = 0, c[2] = 1, c[3] = 1$
Base $a = 2$
Modulus $n = 8$

Trace execution:

Initial: $z = 1$
$i = 0$ : $z \leftarrow (1^2) \bmod 8 = 1$ .
Since $c[0] = 1$ , $z \leftarrow (1 \times 2) \bmod 8 = 2$ .
$i = 1$ : $z \leftarrow (2^2) \bmod 8 = 4$ .
Since $c[1] = 0$ , $z$ remains 4.
$i = 2$ : $z \leftarrow (4^2) \bmod 8 = 16 \bmod 8 = 0$ .
Since $c[2] = 1$ , $z \leftarrow (0 \times 2) \bmod 8 = 0$ .
$i = 3$ : $z \leftarrow (0^2) \bmod 8 = 0$ .
Since $c[3] = 1$ , $z \leftarrow (0 \times 2) \bmod 8 = 0$ .

The function returns $z = 0$ .

Final answer

Common trap

Overlooking that 4^2 = 16 is divisible by 8 (16 mod 8 = 0), collapsing all subsequent multiplications to 0.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 3
Question: 59 of 65, printed as CS Q.49
Section: Computer Science and Information Technology

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