Mistake in my simple RSA encryption

Question

Could someone point out where I am going wrong I cannot seem to do the following question and get the correct encryption value for my chosen values below.

Your alphabet consists of the $10$ digits, $0,1,\ldots,9$.

Pick up any two unique odd primes $a, b$ and the encryption exponent $e$ defining the RSA system, and encrypt $1715171.$

Ensure that your choice of $a, b$ allows you to encrypt in a way in which the encrypted block size is twice the integer part of the average of the digits of the number.

E.g. $1234565$ gives a block size of $2\left(\frac{26}{7}\right) = 2·3 = 6.$

My solution;

Worked out block size = $4$

chose $a = 7, b = 11$ as my primes, so $n = 77$ and $e= 7$

then got Cipher $= (1715171)^{7} \bmod 77 = 52$

to get a final solution of $0052$ for $4$ letter block encryption, checking this using an online calculator gives me $0044.$

Answer 1

I don't know what method you used to evaluate $1413071^7 \pmod {77}$ but it is congruent to $44$, not $52$.

If you did it by hand, you probably made a mistake somewhere.

If you used some special algorithm, it might have failed because $1413071$ is divisible by $11,$ so that choice of $b$ wasn't very good.

Answer 2

$1413071 \equiv 44\pmod{77}\Rightarrow (1413071)^7\equiv (44^7)\equiv (44^2)^3\cdot44\equiv 11^3\cdot44\equiv22\cdot 44\equiv 44\pmod {77}$