2

Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$.

I've always done matrix arithmetic in a field $\mathbb F$ and that is what every Linear Algebra book has taught me to do.

However, I've started looking into Cryptography and here the term residue matrix come up.

Can someone prove the above result or tell me where to look it up ?

Where can I find books about Cryptography-related Linear Algebra ? I've studied several books on Linear Algebra, but they are all equal in the sense that the ring of interest is a field.

Normally a matrix $A$ (with coefficients in a field $\mathbb F$) has an inverse if and only if $\det A \neq 0$, and in this case $\det A$ is neccesarily invertible, since we are in a field. So the above result is a more general result than this.

Shuzheng
  • 5,533

1 Answers1

3

Over a commutative ring $R$ a matrix $A$ is invertible iff $\det A$ is a unit: see Invertible matrices over a commutative ring and their determinants. In your case this means exactly $\gcd(\det A,n)=1$.

Community
  • 1
user26857
  • 52,094
  • So the proof is that simple, just by considering the adjoint matrix $\text {adj} A$ and seeing that $\det(AB) = \det(A) \det(B)$ implies $\det A$ is a unit, which in $\mathbb Z_n$ is true if and only if $\gcd (\det A, n) = 1$. I thought the results $\det(AB) = \det (A) \det (B)$ and so on were true only for a field ! – Shuzheng Dec 01 '14 at 13:23
  • @NicolasLykkeIversen There are a lot of nice proofs here. – user26857 Dec 01 '14 at 15:08