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Mohamed Mohamed Mourad Abdel W avatar Crypto

Galois field problem in Cryptography

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Mohamed Mohamed Mourad Abdel W

This problem is related to Fields in Cryptography, My Question is why there is no multiplicative inverse for 2, isn't it 0.5?? or matters are diffrent if it was related to galois field ? I don't quite understand.

This is the Addition Tabel... enter image description here

This Is the Multiplication Tabel... enter image description here

And this is the tabel with the additive and multiplicative inverse that i have a problem with.... enter image description here

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user93353
$\mathbb Z_8$ is not a field. It's a ring denoted as $\mathbb{Z}^*_8$. All elements in a ring need not have multiplicative inverses. Only the elements $\{1, 3, 5, 7\}$ are invertible. The elements of $\mathbb Z_8$ are $\{0, 1, 2, 3, 4, 5, 6, 7\}$ - $0.5$ is not part of $\mathbb Z_8$. $\mathbb F_{2^3}$ on the other hand is a field & $\mathbb F_{2^3}$ is not $\bmod 8$, it's modulo an irreducible polynomial like $t^3 + t + 1$ - i.e. $\bmod t^3 + t + 1$.
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Maarten Bodewes avatar
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Maarten Bodewes
I’m voting to close this question because this is a misunderstanding about discrete math specifically ring & field theory.
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kodlu avatar Crypto
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kodlu

As in the comment, $\mathbb{Z}_8$ is not a field, and thus not every element has a multiplicative inverse.

In general $\mathbb{Z}_m$ is a field if and only if $m$ is a prime.

If $m$ is not a prime, then $m=pq,$ where $p,q$ are integers satisfying $1<p,q<m.$ If you pick $a$ which is divisible by $p,$ say, then $a=pk,$ for some integer $k\geq 1,$ and thus $qa=kpq=km$ is divisible by $m,$ which means $qa\equiv 0 \pmod m.$

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