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. eAND(H)=e1(H)e2(H)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. eOR(H)=e1(H)e2(H)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. eXOR(H)=(e1(H)e2(H))(e1(H)e2(H))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. eNOT(H)=e(H)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. eNAND(H)=e1(H)e2(H)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. eNOR(H)=e1(H)e2(H)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. eXNOR(H)=(e1(H)e2(H))(e1(H)e2(H))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.

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