HFHE

HFHE (Hypergraph Fully Homomorphic Encryption) is an approach to implementing a bootstrap FHE scheme using hypergraphs.

The implementation of logical gates through hypergraphs enables efficient binary operations:

1. AND The intersection of two hyperedges, creating a new hyperedge that is active only when both original hyperedges are active. $e_{\text{AND}}(H) = e_1(H) \cap e_2(H)$

2. OR A union of hyperedges, where a new hyperedge is active if at least one of the original hyperedges is active. $e_{\text{OR}}(H) = e_1(H) \cup e_2(H)$

3. XOR The combination of two hyperedges, AND and OR, is activated only when only one of the original hyperedges is active. $e_{\text{XOR}}(H) = (e_1(H) \cup e_2(H)) \cap \overline{(e_1(H) \cap e_2(H))}$

4. NOT Inverting a hyperedge: a new hyperedge becomes active when the original one is inactive. $e_{\text{NOT}}(H) = \overline{e(H)}$

5. NAND A mix of AND and NOT operations, with the NAND hyperedge active when the AND hyperedge is inactive. $e_{\text{NAND}}(H) = \overline{e_1(H) \cap e_2(H)}$

6. NOR The union of OR and NOT activates the NOR hyperedge when the OR hyperedge is inactive. $e_{\text{NOR}}(H) = \overline{e_1(H) \cup e_2(H)}$

7. XNOR Integration of XOR and NOT operations, where the XNOR hyperedge is active when the XOR hyperedge becomes inactive. $e_{\text{XNOR}}(H) = \overline{(e_1(H) \cup e_2(H)) \cap \overline{(e_1(H) \cap e_2(H))}}$

These hypergraph-based implementations provide an approach to solving full homomorphic encryption problems.

Hypergraphs naturally support parallel computation because different nodes and hyperedges are processed independently.