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# Geometric Cipher (Original Algorithm)

[GitHub](https://github.com/HectorTablero/geometric-cipher) | [Live Demo](https://geometric-cipher.experiments.tablerus.es)

**Date:** April 2024
**Technologies:** Python

---

## Project Overview

The Geometric Cipher is a novel encryption algorithm that arranges text characters on regular polygons (triangles, squares, pentagons, etc.) in a spiral pattern. The cipher combines geometric positioning with optional character substitution, creating a two-layer encryption system. Characters are placed on concentric polygons and then read in a specific order to produce the ciphertext.

The final ciphertext is generated by reading the positioned characters from top to bottom, left to right - as if reading the geometric arrangement as normal text.

## Core Concept

The cipher works by:

1. Splitting plaintext into blocks of size `shape` (number of polygon sides)
2. Positioning each block's characters on a regular polygon
3. Applying rotation and direction transformations
4. Optionally applying a character substitution cipher
5. Reading the arranged characters in geometric order to produce ciphertext

### Example: Basic Cipher

```python
geometric_cipher("Ciphertext", shape=5, starting_pos=0, clockwise=True, turn_left=True)
# Output: 'tCireepthx'
```

![Basic geometric cipher visualization - "Ciphertext" with pentagon (shape=5).](../../assets//projects/geometric-cipher/ciphertext-base-pentagon.webp)

The input "Ciphertext" is arranged on a pentagon, with characters positioned at vertices. Reading from top to bottom produces the encrypted result.

### Example: With Character Substitution

```python
geometric_cipher("Ciphertext", shape=5, starting_pos=0, clockwise=True, turn_left=True, cipher=caesar_cipher)
# Output: 'uCisfepuhy'
```

![Modified cipher with Caesar shift - "Ciphertext" with pentagon and caesar_cipher.](../../assets//projects/geometric-cipher/ciphertext-caesar-pentagon.webp)

Adding `caesar` shifts each character by its layer number (the innermost is 0) before geometric positioning, creating a double-encrypted result that is extremely hard to break without knowing the shape.

## Algorithm Details

### Text Processing Pipeline

The `process_text` function handles the core transformation:

1. **Character Substitution**: Apply optional cipher to the block
2. **Padding**: Extend block to `shape` length if needed
3. **Direction Reversal**: Reverse text if `clockwise=False`
4. **Rotation**: Rotate the block based on `starting_pos`, block index, and `turn_left` parameter
5. **Pairing**: Split characters into pairs for symmetric polygon placement
6. **Positioning**: Calculate geometric coordinates using trigonometric functions

### Coordinate Calculation

Characters are positioned using polar coordinates converted to Cartesian:

```python
angle = π/2 + π * ((n + 1) % 2 + i * 2) / shape
radius = cos(π/shape)^(-n)
y = radius * sin(angle)
x = ±√(radius² - y²)
```

- `n`: Block index (determines polygon size)
- `shape`: Number of polygon sides
- `i`: Pair index within the block

The radius grows exponentially with each block, creating concentric polygons.

### Character Ordering

The cipher sorts positioned characters by their y-coordinate (descending), then reads left to right for each y-level. This creates the final ciphertext by reading the geometric arrangement as if it were normal text.

## Configuration Parameters

| Parameter      | Type     | Default  | Description                                                      |
| -------------- | -------- | -------- | ---------------------------------------------------------------- |
| `shape`        | int      | 4        | Number of polygon sides (3=triangle, 4=square, 5=pentagon, etc.) |
| `starting_pos` | int      | 0        | Initial rotation offset (0 to shape-1)                           |
| `clockwise`    | bool     | True     | Direction of character placement                                 |
| `turn_left`    | bool     | True     | Rotation direction for subsequent blocks                         |
| `cipher`       | function | identity | Optional character substitution function                         |

## Character Substitution Functions

### Caesar Cipher Variant (`caesar_cipher`)

```python
def caesar_cipher(txt, i):
    def shift_ascii(c):
        if 'A' <= c <= 'Z':
            return chr((ord(c) - ord('A') + i) % 26 + ord('A'))
        elif 'a' <= c <= 'z':
            return chr((ord(c) - ord('a') + i) % 26 + ord('a'))
        elif c.isdigit():
            return str((int(c) + i) % 10)
        return c

    return [shift_ascii(c) for c in txt]
```

Shifts depend on block index `i`, creating variable encryption strength across blocks. Supports:

- Uppercase letters (A-Z)
- Lowercase letters (a-z)
- Digits (0-9)

### Decryption (`caesar_decipher`)

```python
def caesar_decipher(txt, i):
    def shift_ascii(c):
        if 'A' <= c <= 'Z':
            return chr((ord(c) - ord('A') - i) % 26 + ord('A'))
        elif 'a' <= c <= 'z':
            return chr((ord(c) - ord('a') - i) % 26 + ord('a'))
        elif c.isdigit():
            return str((int(c) - i) % 10)
        return c

    return [shift_ascii(c) for c in txt]
```

## Decryption Process

The `geometric_decipher` function reverses the encryption:

1. **Order Reconstruction**: Create dummy indices to track character positions
2. **Geometric Positioning**: Apply same transformations as encryption
3. **Index Mapping**: Determine original position of each ciphertext character
4. **Reordering**: Place characters back in original sequence
5. **Character Reversal**: Apply decipher function to reverse substitution

Returns a tuple: `(reordered_text, text_blocks, final_plaintext)`

### Example: Decryption

```python
geometric_decipher("tCireepthx", shape=5, starting_pos=0, clockwise=True, turn_left=True)
# Output: ('Ciphertext', [['C', 'i', 'p', 'h', 'e'], ['r', 't', 'e', 'x', 't']], 'Ciphertext')

geometric_decipher("uCisfepuhy", shape=5, starting_pos=0, clockwise=True, turn_left=True, decipher=caesar_decipher)
# Output: ('Ciphesufyu', [['C', 'i', 'p', 'h', 'e'], ['s', 'u', 'f', 'y', 'u']], 'Ciphertext')
```

## Visualization

The `visualize_geometric_cipher` function generates matplotlib plots showing:

- Character positions on polygon vertices
- Concentric polygons for multiple blocks (color-coded by depth)
- Geometric arrangement before reading

![Multiple block visualization showing nested polygons.](../../assets//projects/geometric-cipher/alphabet-base-hexagon.webp)

**Color Encoding:**

- Blue (inner): First block (n=0)
- Gradient to red: Subsequent blocks
- Polygon transparency: 50% for edge visibility

## Security Analysis

**Strengths:**

- Unconventional character ordering resistant to frequency analysis
- Variable encryption with block-dependent transformations
- Customizable parameters increase keyspace
- Optional layered encryption (geometric + substitution)

**Weaknesses:**

- Deterministic algorithm vulnerable to known-plaintext attacks
- Geometric positioning patterns may be recognizable with multiple samples

## Use Cases

- Educational cryptography demonstrations
- Artistic text encoding
- Low-security obfuscation for puzzles/games
- Steganographic applications (embed in geometric art)

## Example Applications

### Short Message

```python
geometric_cipher("Hello", shape=5, starting_pos=2, clockwise=False, turn_left=False)
# Output: 'lloeH'
```

![Single pentagon example - "Hello".](../../assets//projects/geometric-cipher/hello-base-pentagon.webp)

### Long Text

```python
geometric_cipher("The quick brown fox jumps over the lazy dog", shape=6, starting_pos=1, clockwise=True, turn_left=True, cipher=caesar_cipher)
# Output: 'j uzxinpl jqhempd  syT hvgjc suqm qsq aweyf'
```

![Multi-polygon example showing text spanning multiple layers.](../../assets//projects/geometric-cipher/fox-caesar-hexagon.webp)

### Custom Shape

```python
geometric_cipher("Octagon", shape=8, starting_pos=0, clockwise=True, turn_left=True)
# Output: 'Octnaog'
```

![Octagon (8-sided) cipher visualization.](../../assets//projects/geometric-cipher/octagon-base-octagon.webp)

## Implementation Notes

- Uses Python's `math` library for trigonometric calculations
- Matplotlib required for visualization (`matplotlib.pyplot`, `matplotlib.patches`)
- Floating-point rounding to 10 decimal places prevents coordinate drift
- Handles odd-length blocks by placing center character at polygon center

## Conclusion

The Geometric Cipher demonstrates how spatial transformations can create encryption beyond traditional character substitution. While not cryptographically secure for sensitive data, it provides an intuitive visualization of how geometric mathematics can encode information. The customizable parameters and optional substitution layer make it a flexible tool for educational purposes and creative text obfuscation.