What is the affine cipher formula?

Encryption is E(x) = (a times x plus b) mod 26, where x is the position of a letter from 0 for A to 25 for Z. You pick two key numbers, a and b. For example with a = 5 and b = 8, the letter A at position 0 becomes (5 times 0 plus 8) mod 26 = 8, which is I, and B at position 1 becomes (5 plus 8) mod 26 = 13, which is N. Decryption reverses it with D(y) = inverse-of-a times (y minus b) mod 26.

Why must a be coprime with 26?

The decrypt step divides by a, which in modular arithmetic means multiplying by the modular inverse of a. That inverse only exists when a and 26 share no common factor, that is when gcd(a, 26) = 1. If a is even or a multiple of 13, two different plaintext letters collapse onto the same cipher letter and the message can no longer be recovered. The twelve valid values are 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25, and this tool rejects any other a.

What is a modular inverse and where does it appear here?

The modular inverse of a mod 26 is the number that multiplies with a to give 1 mod 26. For a = 5 it is 21, because 5 times 21 = 105 and 105 mod 26 = 1. Decryption needs it: D(y) = 21 times (y minus 8) mod 26 undoes the encryption keyed with a = 5, b = 8. The tool computes and displays this inverse for whatever a you enter, so you can verify the decryption by hand.

Is the Caesar cipher a special case of the affine cipher?

Yes. Set a = 1 and the formula becomes 1 times x plus b, which is just x plus b, the plain Caesar shift by b. So encrypting with a = 1, b = 3 gives identical output to a Caesar shift of 3, where A becomes D and B becomes E. The affine cipher is the next step up: it adds a multiply by a on top of the shift, expanding the key space from 26 shifts to 312 combinations.

How do I decrypt an affine cipher?

If you know a and b, enter both, press Decrypt, and the tool applies D(y) = inverse-of-a times (y minus b) mod 26 to recover the plaintext. For instance ciphertext "IHHWVC" with a = 5, b = 8 decrypts to "AFFINE". If you do not know the keys, there are only 312 possibilities, so try each coprime a with a known crib such as a flag prefix and read the combination that yields readable text.

Solve an affine cipher in a CTF crypto challenge

A challenge hands you ciphertext and the hint that the key is "ax+b". Set a and b, switch to decrypt, and the flag inside CTF{...} resolves instantly. If only the ciphertext is given, the two unknowns a and b mean there are 12 valid choices for a times 26 for b, so paste a known crib like the flag prefix, try a few coprime a values, and read the line where the braces frame plain text.

Teach modular inverses in a number-theory class

The affine cipher is the cleanest classroom example of why a number needs a multiplicative inverse mod 26. Type a value of a that shares a factor with 26, such as 13, and the tool refuses with a clear error; switch to a coprime a like 5 and it shows the inverse 21, then decrypts correctly. Students see in real time why gcd(a, 26) = 1 is not optional.

Show that Caesar is a special case of affine

Set a to 1 and the formula 1 times x plus b collapses to plain addition, which is exactly the Caesar shift by b. Encode a sentence at a = 1, b = 3 here and at shift 3 in the Caesar tool and the outputs match character for character. It makes the family relationship between shift, affine and substitution ciphers concrete instead of abstract.

Build a puzzle for an escape room or a treasure hunt

You want a clue that looks scrambled but decodes with a small printed key card. Pick a = 7, b = 11, encrypt the riddle, and print the cipher plus the line "key a equals 7, b equals 11" on the card. Players who find the card paste the text, enter the two numbers, hit decrypt and get the next location. The shareable link reopens the same puzzle.

Affine Cipher Encoder & Decoder (ax+b)

Encrypt and decrypt the ax+b affine cipher with live modular-inverse check, browser-only

Runs locally
Category Encoding & Crypto
Best for Checking small payloads, tokens, hashes, and encoded values quickly.

Affine cipher. Each letter maps with E(x) = (a·x + b) mod 26 (A=0 … Z=25); decrypt reverses it with D(y) = a⁻¹·(y − b) mod 26. Case is kept; digits, spaces, punctuation and non-Latin text pass through untouched. The key a must be coprime with 26, one of 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25.

a (multiply)b (shift)

E(x) = (a·x + b) mod 26Modular inverse a⁻¹ mod 26 = 21.

Input0 chars

Output0 chars

Transformed text appears here.

What this tool does

Free online affine cipher tool that encrypts and decrypts the classic ax+b substitution cipher right in your browser. The affine cipher maps each letter x to a times x plus b, all taken modulo 26, so it generalises the Caesar shift by adding a multiply step before the add. Enter your two keys, a and b, choose encrypt or decrypt, and every letter transforms live while case is preserved and digits, spaces, punctuation, accented letters and CJK text pass through untouched, because the cipher is defined only on A to Z and a to z. The key a must be coprime with 26, meaning one of 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 or 25; any other a has no modular inverse and the tool flags it instead of producing a cipher that cannot be decoded. Decryption multiplies by the modular inverse of a, which is shown on screen so you can follow the maths. Everything is 100% client-side with a one-click copy button and a shareable link that reopens the same text and the same a and b. No upload, no server round-trip.

Tool details

Input: Files + Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Encoding & Crypto · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Affine Cipher Encoder & Decoder fits into your work

Use it for quick browser-side encoding, decoding, hashing, token checks, and share-safe transformations.

Encoding jobs

Checking small payloads, tokens, hashes, and encoded values quickly.
Preparing values for APIs, URLs, docs, or support tickets.
Avoiding account-based tools when the input might be sensitive.

Encoding checks

Do not paste live secrets unless you are comfortable with local browser handling.
Confirm whether the operation is reversible before sharing the result.
For hashes, compare the exact algorithm and casing expected by the receiver.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Solve an affine cipher in a CTF crypto challenge
A challenge hands you ciphertext and the hint that the key is "ax+b". Set a and b, switch to decrypt, and the flag inside CTF{...} resolves instantly. If only the ciphertext is given, the two unknowns a and b mean there are 12 valid choices for a times 26 for b, so paste a known crib like the flag prefix, try a few coprime a values, and read the line where the braces frame plain text.
Teach modular inverses in a number-theory class
The affine cipher is the cleanest classroom example of why a number needs a multiplicative inverse mod 26. Type a value of a that shares a factor with 26, such as 13, and the tool refuses with a clear error; switch to a coprime a like 5 and it shows the inverse 21, then decrypts correctly. Students see in real time why gcd(a, 26) = 1 is not optional.
Show that Caesar is a special case of affine
Set a to 1 and the formula 1 times x plus b collapses to plain addition, which is exactly the Caesar shift by b. Encode a sentence at a = 1, b = 3 here and at shift 3 in the Caesar tool and the outputs match character for character. It makes the family relationship between shift, affine and substitution ciphers concrete instead of abstract.
Build a puzzle for an escape room or a treasure hunt
You want a clue that looks scrambled but decodes with a small printed key card. Pick a = 7, b = 11, encrypt the riddle, and print the cipher plus the line "key a equals 7, b equals 11" on the card. Players who find the card paste the text, enter the two numbers, hit decrypt and get the next location. The shareable link reopens the same puzzle.

Common pitfalls

Choosing an a that is not coprime with 26. If gcd(a, 26) is greater than 1, such as a = 2, 13 or 26, no modular inverse exists, two different letters can map to the same cipher letter, and decryption is impossible. The valid a values are exactly 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25; the tool blocks every other a with an error.
Mixing up the encrypt and decrypt formulas. Encryption is a times x plus b, but decryption is not a times y minus b. You must multiply by the modular inverse of a, giving the inverse of a times the quantity y minus b. Use the Decrypt button rather than swapping a and b by hand, or the result stays scrambled.
Expecting digits and punctuation to be enciphered. The affine cipher is defined only on A to Z. A string like "Day 9!" leaves the 9, the space and the exclamation mark untouched and only transforms the letters. If a "decoded" message still has its original numbers, that is correct behaviour, not a bug.

Privacy

Every step runs in plain JavaScript inside your browser tab. The affine formula, the modular inverse and the encrypt or decrypt pass all happen on the page, and the text you type never leaves it, with no logging of what you encode. State this plainly though: the affine cipher is not real encryption. With only 12 valid choices for a and 26 for b there are 312 keys in total, breakable in a blink, so never use it to protect a password or a private message. The shareable URL encodes your text, a and b in the query string, so a link pasted into chat lands those characters in the recipient server access log. Use it for puzzles and lessons, not secrets.

FAQ

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Affine Cipher Encoder & Decoder (ax+b)

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Affine Cipher Encoder & Decoder fits into your work

Encoding jobs

Encoding checks

Good next steps

Real-world use cases

Solve an affine cipher in a CTF crypto challenge

Teach modular inverses in a number-theory class

Show that Caesar is a special case of affine

Build a puzzle for an escape room or a treasure hunt

Common pitfalls

Privacy

FAQ

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