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I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation of determinant.

Formally we have square matrix of order n under the finite ring of order r. There is the trivial algorithm which always gives the answer, but it's quite impractical. Gaussian elimination method is completely clear for me only in the case of matrix under real numbers. For example, we have $$ A= \begin{bmatrix} 6 & 1 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 5 \\ \end{bmatrix} $$ According to Gaussian method I should multiple the first row by -2/6 and add to the second row. In result I will have 0 instead of 2 in the second row, and so on. But how does Gaussian method work in the following case: $$ B= \begin{bmatrix} 6 & 1 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 5 \\ \end{bmatrix} mod r$$

I can multiple the first row by -2/6 only when r is prime number, in this case we have finite field, so multiplicative inverse exists for all nonzero elements. But the determinant of this matrix certainly exists in the all cases even if modulus r is composite number and we don't have finite field.

How can I efficiently compute the determinant of matrix under finite ring of arbitrary order r?

Pequenito
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    In a field multiplicative inverses always exist, so the ring $\mathbb{Z}/30\mathbb{Z}$ is not a field. – Jair Taylor May 05 '15 at 07:53
  • When you work modulo $30$, you are not working over a field. May be you could extend your alphabet to size $31$ or shrink it to $29$? These are both primes, so the residue class ring is a field, and things work smoothly. Anyway, Gaussian method works over any field. For an example with $29$ see this walkthrough answer. In general you can only divide by elements that have a multiplicative inverse (w.r.t whatever modulus you are working with). The extended Euclid's algorithm also gives you modular inverses whenever they exist (gcd =1). – Jyrki Lahtonen May 05 '15 at 10:36
  • I have used incorrect terminology in my question. Actually I mean finite ring, that is we have square matrix under finite ring, where order of ring is arbitrary number (prime or composite). Gaussian method is clear only in the case when order is prime. I have edited my question. – Pequenito May 07 '15 at 07:00

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