+.ciphertext[_ciphertext_letter_] =.plaintext[_plaintext_letter_] × a + b
* Convert letters to numbers
* Take the total modulus 26
* Convert letters to numbers
* Take the total modulus 26
@@ -83,11+95,11 @@ This is not always defined in modular arithmetic. For instance, 7 × 4 = 28 = 2
Result from number theory: only numbers coprime with _n_ have multiplicative inverses in arithmetic mod _n_.
Result from number theory: only numbers coprime with _n_ have multiplicative inverses in arithmetic mod _n_.
-Another result from number theory: for non-negative integers _a_ and _n_, and there exist unique integers _x_ and _y_ such that _ax_ + _ny_ = gcd(_a_, _b_)
+Another result from number theory: for non-negative integers _m_ and _n_, and there exist unique integers _x_ and _y_ such that _mx_ + _ny_ = gcd(_m_, _n_)
Coprime numbers have gcd of 1.
Coprime numbers have gcd of 1.
-_ax_ + _ny_ = 1 mod _n_. But _ny_ mod _n_ = 0, so _ax_ = 1 mod _n_, so _a_ = _x_<sup>-1</sup>.
+_mx_ + _ny_ = 1 mod _n_. But _ny_ mod _n_ = 0, so _mx_ = 1 mod _m_, so _m_ = _x_<sup>-1</sup>.
Perhaps the algorithm for finding gcds could be useful?
Perhaps the algorithm for finding gcds could be useful?