# Secure Bio-Data Encryption Using Finite Fields and Group Theory

By [Automata.AI.Labs](https://paragraph.com/@automata-ai-labs) · 2024-08-28

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**1.** **Group Theory Foundations**

![](https://storage.googleapis.com/papyrus_images/8335d794f47db1c37225aa17c497fbda27caf14c8cd6065eb2d5749eb100ca07.png)

**2\. Finite Fields (Galois Fields)**

![](https://storage.googleapis.com/papyrus_images/97a9e78125b651d698c2a5ed98e3be2eadf76c19336277f81adc68dabd385610.png)

**3\. Data Representation in**

$$GF(p^n)$$

![](https://storage.googleapis.com/papyrus_images/e80c45538944fd9385e4a284cbf919464c775b41027fc04261ebd15f2b3d1b0f.png)

**4\. Encryption and Decryption Functions**

![](https://storage.googleapis.com/papyrus_images/e50fee1f1f36b3ec37d7fef185c8347488476e1e295ff678f7dcea1db65ed82f.png)

**5\. Security Considerations**

![](https://storage.googleapis.com/papyrus_images/abfd4938f6aa2f68d7ed542e416f3077aa89d49b989404db9985863b7f665670.png)

**6\. Logical Framework**

![](https://storage.googleapis.com/papyrus_images/8661296f12f8077e45c0fd328edec6bbae1224b52cd0619bdc26cc179cc379fd.png)

**7\. Symbolic Logic Mapping**

![](https://storage.googleapis.com/papyrus_images/aabb9a58f01ad95b75f084b3dfbf5ea782ecf9bce04d730a202a19f26b17f028.png)

    +---------------------------------------------+
    |                                             |
    |             System Boundary                 |
    |                                             |
    |  +-------------------+                      |
    |  | Bio-Data Input    |                      |
    |  | Module            |                      |
    |  +---------+---------+                      |
    |            |                                |
    |            v                                |
    |  +---------+---------+                      |
    |  | Data Representation|                     |
    |  | Module             |                     |
    |  +---------+---------+                      |
    |            |                                |
    |            v                                |
    |  +---------+---------+                      |
    |  | Finite Field      |                      |
    |  | Operations        |                      |
    |  |  +-------------+  |                      |
    |  |  | Encryption  |  |                      |
    |  |  | Function    |  |                      |
    |  |  +------+------|  |                      |
    |  |         |         |                      |
    |  |         v         |                      |
    |  |  +------+------|  |                      |
    |  |  | Decryption  |  |                      |
    |  |  | Function    |  |                      |
    |  |  +-------------+  |                      |
    |  +---------+---------+                      |
    |            |                                |
    |            v                                |
    |  +---------+---------+                      |
    |  | Decryption Output |                      |
    |  | Module            |                      |
    |  +-------------------+                      |
    |                                             |
    +-----------------+---------------------------+
                      |
                      v
    +-----------------+---------------------------+
    | External Components                          |
    |                                             |
    |  +-------------------+  +-----------------+ |
    |  | Key Management    |  | Encrypted Data  | |
    |  | System (KMS)      |  | Storage         | |
    |  +---------+---------+  +---------+-------+ |
    |            |                      |         |
    |            v                      v         |
    |  +-------------------+  +-----------------+ |
    |  | Communication     |  | Communication   | |
    |  | Channel           |  | Channel         | |
    |  +-------------------+  +-----------------+ |
    |                                             |
    +---------------------------------------------+

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*Originally published on [Automata.AI.Labs](https://paragraph.com/@automata-ai-labs/secure-bio-data-encryption-using-finite-fields-and-group-theory)*