inverse,multiplicative inverse and Congruence of a prime field

Question

I am dealing with ECC in these days which heavily based on finite fields. I want to how to find a inverse of a value in finite field and what is multiplicative inverse and also Congruence

F29- {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 28}

For any integer a, a mod p shall denote the unique integer remainder r , 0 ≤ r ≤ p − 1, obtained upon dividing a by p; this operation is called reduction modulo p.

(i) Addition: 17 + 20 = 8 since 37 mod 29 = 8

(ii) Subtraction: 17 − 20 = 26 since −3 mod 29 = 26

(iii) Multiplication: 17 · 20 = 21 since 340 mod 29 = 21

(iv) Inversion: 17−1 = 12 since 17 · 12 mod 29 = 1 //can't understand how this happen

17⋅12=1+7⋅29≡1 (mod 29) ⇒ 17−1≡12 (mod 29) explain this also

You'll get better answers if you say precisely why you don't understand that $$\rm 17\cdot 12 = 1 + 7\cdot 29\equiv 1,\ (mod\ 29)\ \Rightarrow\ 17^{-1}\equiv 12,\ (mod\ 29)$$ — Bill Dubuque, Sep 29 '12 at 19:26
oh,,,I can't understand this either, lets forget 12,,,then how we find inverse(17). (I am new to modular arithmatic in finite field maths)...can you give me a simple explanation please,,thanks — user827918, Sep 29 '12 at 19:33
You can compute modular inverses by the extended Euclidean algorithm. — Bill Dubuque, Sep 29 '12 at 19:34
thanks very much, I really appreciate your help, but I can't understand these complex things, really sorry about that. could you please give me steps to find inverse of 17 in ff(29). I mean mathematical steps — user827918, Sep 29 '12 at 19:41

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

Below are a couple of ways to compute $\rm\: 17^{-1}\pmod{29}.\ $ First, twiddling fractions

$$\rm mod\ 29\!:\ \ \ {{\ \ \, 1\,\equiv\ \ \ 30}\atop{17\,\equiv\, -12}}\ \ \ \Rightarrow\ \ \ \frac{1}{17}\ \equiv\ \frac{\ 30\ }{-12\ }\ \equiv\ \frac{5}{-2\ }\,\equiv\,\frac{-5\,}2\,\equiv\,\frac{24}2\,\equiv\,12$$

Alternatively, mechanically applying the Extended Euclidean Algorithm

$$\begin{eqnarray} 29 &\ =\ &\ \ \ 1&\cdot& 29 &+&\ 0&\cdot& 17\\ 17 &=&\ \ \ 0&\cdot& 29 &+&\ 1&\cdot& 17\\ 12 &=&\ \ \ 1&\cdot& 29 &-&\ 1&\cdot& 17\\ 5 &=& -1&\cdot &29 &+&\ 2&\cdot& 17\\ 2 &=&\ \ \ 3&\cdot& 29 &-&\ 5&\cdot& 17\\ 1 &=& -7&\cdot& 29 &+& 12&\cdot& 17\end{eqnarray}$$

Hence the final equation implies that $\rm\: 12\cdot 17\equiv 1\pmod{29}$

See here for more on twiddling and related methods (and see this list) — Bill Dubuque, Dec 06 '22 at 17:50

inverse,multiplicative inverse and Congruence of a prime field

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