If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all n. Then apply your method to compute 1032 (mod 41).
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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![If m = 2* is a power of 2, explain how you could use repeated squaring
to compute am (mod n) for all n. Then apply your method to compute 102
(mod 41).
If m is not a power of 2, explain how you could use the results of the exercise
above to compute a™ (mod n) for any n. Then apply your method to compute
1726 (mod 44).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55f7e17a-bf33-4c09-981d-d6d267ec0a99%2F4429b174-416b-4f06-a5e8-05a9dcd0dbba%2Fu64ar7_processed.png&w=3840&q=75)
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a) We have m = . First we illustrate the method of repeated squaring to compute . Then we will compute .
The following steps compute the value of
1. Write m as a sum of powers of 2,
,
where each is either 0 or 1. (This is called the binary expansion of m.)
2.Make a table of powers of a modulo m using successive squaring.
Note that to compute each line of the table we only need to take the number at the end of
the previous line, square it, and then reduce it modulo m. Also note that the table has r + 1
lines, where r is the highest exponent of 2 appearing in the binary expansion of k in Step 1.
3. The product , will be congruent to . Note that all of the 's are either 0 or 1, so this number is really just the product of those for which u1 is 1.
We compute
Now, we compute the value of .
We can 32 as
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