Question

2. (8 points) Show how the modular exponentiation algorithm computes $33^{27} \mod 50$. Show your work by recording the values $i$, $a_i$, $x$, and power at the beginning and end of each iteration in the table. proc mod exponent($b \in \mathbb{Z}$, $n = (a_{k-1}a_{k-2} \cdots a_1a_0)_2$, $m \in \mathbb{Z}^+$) $x := 1$ power $:= b \mod m$ for $i = 0: k - 1$ if $a_i = 1$ then $x := (x \cdot \text{power}) \mod m$ power $:= (\text{power} \cdot \text{power}) \mod m$ endfor return($x$) // $x = b^n \mod m$ Beginning of Iteration End of Iteration $i$ $a_i$ $x$ power $x$ power 0 1 1 33 33 39 1 1 33 39 37 21 2 3 4

          2. (8 points) Show how the modular exponentiation algorithm computes $33^{27} \mod 50$. Show
your work by recording the values $i$, $a_i$, $x$, and power at the beginning and end of each iteration
in the table.
proc mod exponent($b \in \mathbb{Z}$, $n = (a_{k-1}a_{k-2} \cdots a_1a_0)_2$, $m \in \mathbb{Z}^+$)
$x := 1$
power $:= b \mod m$
for $i = 0: k - 1$
if $a_i = 1$ then $x := (x \cdot \text{power}) \mod m$
power $:= (\text{power} \cdot \text{power}) \mod m$
endfor
return($x$)  // $x = b^n \mod m$
Beginning
of Iteration
End of
Iteration
$i$  $a_i$  $x$  power  $x$  power
0  1  1  33  33  39
1  1  33  39  37  21
2
3
4

2. (8 points) Show how the modular exponentiation algorithm computes 33^27 50. Show
your work by recording the values i, ai, x, and power at the beginning and end of each iteration
in the table.
proc mod exponent(b ∈ℤ, n = (ak-1ak-2⋯ a1a0)2, m ∈ℤ^+)
x := 1
power := b m
for i = 0: k - 1
if ai = 1 then x := (x ·power) m
power := (power·power) m
endfor
return(x) // x = b^n m
Beginning
of Iteration
End of
Iteration
i ai x power x power
0 1 1 33 33 39
1 1 33 39 37 21
2
3
4

Added by Juan Jos- B.

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