FHE Mathematical Foundations

2025-07-12   blogpage sketch fhe mathdependencygraph

FHE = Fully Homomorphic Encryption

Prompting style: "You're Timothy Gowers. A brilliant and most elegant math explainer"

Watch these for friendly intros

• Craig B. Gentry Homomorphic Encryption
  

• Fully Homomorphic Encryption from the Ground Up
  

FHE Math Dependency Graph

x.com/i/grok/share/5tgjOQGzzRmZmKfULzNSCKHvY

FHE schemes: CKKS (Approximate Reals for ML); BGV (Exact Integers for Precise Ops)

FHE*
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├── Ring-Learning With Errors (Ring-LWE)
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│   ├── Learning With Errors (LWE)
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│   │   ├── Lattices
|   |   |   |
│   │   │   ├── Linear Algebra
│   │   │   │   └── Basic Algebra
|   |   |   |
│   │   │   └── Number Theory
│   │   │       └── Modular Arithmetic
|   |   |
│   │   ├── Shortest Vector Problem (SVP)
│   │   │   └── Closest Vector Problem (CVP)
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│   │   └── Probability and Statistics
│   │       ├── Gaussian Distributions
│   │       └── Statistical Distance
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│   └── Polynomial Rings
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│       ├── Rings
│       │   ├── Groups
│       │   │   └── Abelian Groups
│       │   └── Fields
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│       └── Ideals and Quotient Rings
│           └── Ring Homomorphisms
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├── Homomorphic Operations
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│   ├── Noise Management and Growth
│   │   └── Error Analysis
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│   └── Chinese Remainder Theorem (CRT)
│       └── Euler's Totient Function
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├── Encoding/Decoding
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│   ├── Plaintext Representations
│   │   └── Complex Numbers and Floating-Point (CKKS-specific)
│   │       └── Real Analysis
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│   └── Polynomial Representations
│       └── Fast Fourier Transform (FFT) / Number Theoretic Transform (NTT)
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├── Scheme-Specific Techniques
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│   ├── Rescaling (CKKS)
│   │   └── Modulus Switching
│   │       └── Modular Arithmetic (dependency reinforced)
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│   ├── Modulus Switching (BGV)
│   │   └── Chain of Moduli
│   │       └── Modular Arithmetic (dependency reinforced)
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│   └── Key Switching / Relinearization (BGV)
│       └── Gadget Decomposition
│           └── Linear Algebra (dependency reinforced)
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└── Bootstrapping
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    ├── Advanced Homomorphic Evaluation
    │   └── Ring Homomorphisms (dependency reinforced)
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    └── Lattice Reductions
        └── Linear Algebra (dependency reinforced)

[*] Note that this is a tree-view of an actual graph (more accurately, a non-tree graph). You may realize some nodes are duplicated on the leaves, they supposed to be a single node. It's shown as a tree since showing a (non-tree) graph on text is harder.

More detailed

https://claude.ai/chat/5b2bb8fb-114d-4ae1-a051-21f5007e7dc1

CKKS-specific

https://claude.ai/public/artifacts/035ef45e-00bd-4bec-a13f-e5acf69393bf

BFV-specific

https://claude.ai/public/artifacts/df731299-300c-43a0-ae13-04cb4dcac37c

A Study Structure for all concepts

https://claude.ai/public/artifacts/aa61c440-61e0-472c-b556-dd3ad7dd6801