Hill Cipher Encoder/Decoder

How it is calculated

The tool encodes letters in pairs using a 2×2 key matrix [a b; c d] with A=0 to Z=25. Each pair (x, y) becomes (a·x + b·y) mod 26 and (c·x + d·y) mod 26. The text is made upper case and everything that is not A to Z is dropped (spaces, digits, punctuation and accented letters). An odd number of letters gets an X added at the end. Decoding uses the inverse key, det⁻¹ × [d −b; −c a] mod 26, where det = ad − bc mod 26. The key only works if det has an inverse mod 26, meaning det is odd and not a multiple of 13; otherwise the tool returns nothing. The determinant is displayed with the result.

Common mistakes

• Choosing a key with an even determinant or one divisible by 13 (for example 2, 4, 1, 2, which gives det 0); there is no inverse, so the tool returns nothing.
• Expecting spaces, digits and punctuation to survive; they are removed, so decoding gives back letters only, with no word breaks.
• Forgetting that an odd-length message gets a trailing X when encrypted, so the decoded text may end in an extra X.
• Decoding with a different key than was used to encode, or leaving the mode on encode when you want the plaintext back.
• Reading the key matrix by columns instead of rows: the tool multiplies row (a b) by the pair for the first letter, and row (c d) for the second.