Hello smithhall

I have a solution - but I'm not confident that the coded letters are correct. I say this because there are clearly errors in the problem description. For example:This is obviously wrong: DI is 86, not 80. Such an elementary mistake in work where numerical accuracy is vital doesn't inspire confidence!For example, to convert DISCRETE into numbers we would do the following: DI–SC–RE–TE becomes

D is 3, I is 8: so DI is 3 × 26 + 8 = 80.

S is 18, C is 2: so SC is 18 × 26 + 2 = 470.

R is 17, E is 4: so RE is 17 × 26 + 4 = 446.

T is 19, E is 4: so TE is 19 × 26 + 4 = 498.

Thus, DISCRETE would be written 80–470–446–498.

Alsoshould read:Now calculate Ci = Mei mod pq for each i.

andshould again readTo decrypt, convert the encrypted message into numbers C1,C2, . . ., and calculate Mi = Cdi mod pq,where d is an integer satisfying de ≡ 1 mod (p − 1)(q − 1).

It took me some time to discover this - having never seen this encryption before. These formulae are to be found, for example, at RSA - Wikipedia, the free encyclopedia.

So I have to ask: is this sequence correct?I'm not sure.We have received an important encrypted message – it reads FRWHVWNUTSHX.

Anyway, here are my answers.I used the Euclidean algorithm and found . Can you do this OK?(a) Determine the d we need to use to decrypt this message. (HINT: use the Euclidean algorithm to first solve 257x+(p−1)(q −1)y = 1 – you should then be able to determine d; think about it!)

This is very straightforward: 147, 579, 568, 358, 512, 205. To save time and (hopefully) avoid the sort of arithmetic error above I used a look-up table in an Excel spreadsheet.(b) Convert the encrypted message into a sequence of numbers less than 676.

This is the part where the arithmetic gets very difficult. Again I used the Excel spreadsheet, but even so it was very tricky because finding can't be done in a single operation, because of overflow errors. In the end, I found I could calculate without causing errors. Then by careful manipulation of powers (I used ), I obtained the following numbers: 147, 108, 92, 358, 512, 676, which being interpreted is: FREEODUNSTAZ.(c) Decrypt the message using your answers to the first two parts and convert into English.

Not sure what that means, though!

Spreadsheet attached.

Grandad