## Question

(a) Use this function to put your name into code.

(b) Now put your coded name into code (so that you are using the coding function twice).

(c) Show that using the coding function twice is equivalent to using the coding function d = 23x + 10 mod 26 once.

(d) What is the moral of this story?

4.

(a) Use the Euclidean algorithm to find the inverse of 17 mod 26.

(b) Consider the coding function c = 17x+3. Find the decoding function.

(c) The following message was coded using c = 17x + 3. Decode the message.

Z E N K J J G T I T W S S T L N G Z R I E.

(For this question, use A = 1, B = 2, ..., Y = 25, and Z = 0.)

(d) Why?

## Solution Preview

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4. (a) Since gcd(17, 26) = 1, thus its multiplicative inverse existsNow 26 = 17*1 + 9 => 9 = 26 – 17*1 …… (a)

17 = 9*1 + 8 => 8 = 17 – 9.1 …….. (b)

9 = 8*1 + 1 => 1 = 9 – 8*1 ……. (c)

From (b), put value in (a) we have 1 = 9 –(17-9*1)*1

1 = 9 – 17*1 + 9*1 = 9*2 -17*1 … (d)

Now substitute value of 9 form (a) in (d)

1 = (26-17*1)*2 – 17*1 = 26*2 – 17*3

=> 26*2 + 17*(-3) = 1 mod 26

Thus -3 is inverse of 17 but – 3 = 23 mod 26

This shows 23 is inverse of 17....

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