# The following message was coded using c = 17x + 3. How to decode the message?

Q V E B A A X K Z K N J D K C E X Q I Z V.

A = 1, B = 2, . . . , Y = 25, and Z = 0.

Please show me how to decode this message. Where does c = 17x + 3 come to play? Thanks

Also, how do I find the decoding function of c = 17x+3? Thanks.

### 1 Answer

- MorewoodLv 74 years agoFavorite Answer
You have left out an important part of the coding formula, the modulus! Notice that you only have 26 defined values for "x" (whole numbers from 0 to 25), and you also have only those same 26 possible values for "c". This implies that your domain is not all integers, but integers modulus 26:

c = (17x + 3) mod 26 = 17x+3 - 26k, where "k" is the integer which makes 0 ≤ c < 26.

Then the decoding procedure is just the inverse:

x = (17⁻¹·(c - 3)) mod 26

where 17⁻¹ is the inverse "mod 26", that is any integer for which

(17⁻¹)·(17) mod 26 = 1.

This inverse is easily found with the "Euclidean Algorithm" which gives the greatest common divisor of two numbers as a linear combination of those numbers. Since the greatest common divisor of 17 and 26 is 1, that algorithm gives:

17A + 26B = 1 where that "A" is the desired 17⁻¹ mod 26.

Then start calculating: x = A(c-3) mod 26.

A spreadsheet is nice to do the actual computation.

Put your coded message in row 1, place in cell A2 the following formula

(replacing the first "A" with the 17⁻¹ mod 26 number from above):

=char(64+mod(A*(code(A$1)-64 - 3),26))

and fill right.

(Actually, that will put "@" in place of "Z" - but I think the correct decoding has no "Z" - see below.)

***BUT***, you will get nonsense! It appears that the message was actually coded using c=17x-6. I also think that someone made a copy error (missed coding one character). I suspect the coded message should actually have been:

Q V E B A A X K Z K N J J K C E X Q I Z V

If you don t know the Euclidean algorithm, check the source below. With the correct coding formula:

c = 17x - 6 mod 26

the decoding should not be a hard day's (or night's) work!

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