| name | quantum-isogeny-sampling |
| description | Quantum polynomial-time sampling of secure supersingular elliptic curves with unknown endomorphism rings — foundational methodology for isogeny-based cryptography instantiation without trusted setup. |
Quantum Isogeny Sampling
Description
Quantum algorithm methodology for sampling random supersingular elliptic curves with unknown endomorphism rings. First provable quantum polynomial-time solution to a problem critical for isogeny-based cryptographic protocols. Two variants: (1) O~(log^4 p) quantum gate complexity under GRH, outputting provably secure curves based on endomorphism ring problem hardness; (2) uniform O-oriented curve sampling based on Vectorization Problem hardness. Enables secure instantiation of CGL hash function and related isogeny-based primitives when combined with interactive quantum verification.
Activation Keywords
- isogeny graph
- supersingular elliptic curve sampling
- quantum isogeny sampling
- CGL hash function
- vectorization problem
- endomorphism ring problem
- 同源图采样
- 超奇异椭圆曲线
- isogeny-based cryptography
- quantum number theory
Tools Used
- terminal: Run mathematical computations and quantum simulations
- web_search: Search for related cryptographic protocols
- browser: Access mathematical libraries
Usage Patterns
Pattern 1: Secure Curve Generation for Isogeny Cryptography
- Determine field characteristic p
- Choose imaginary quadratic order O (for oriented variant)
- Run quantum sampling algorithm:
- Variant 1: O~(log^4 p) gate complexity (GRH)
- Variant 2: O~(log^13 p) (unconditional)
- Verify curve security via endomorphism ring problem hardness
Pattern 2: CGL Hash Function Instantiation
- Sample secure supersingular curve using quantum algorithm
- Combine with interactive quantum computation verification
- Use as input to CGL hash function
- Achieve provably secure instantiation without trusted setup
Pattern 3: Spectral Analysis of Isogeny Graphs
- Model isogeny graph as expander
- Compute spectral gap via quantum walk
- Relate spectral properties to mixing time
- Use for security parameter selection
Instructions for Agents
Step 1: Understand the Problem
Supersingular elliptic curve sampling requires:
- Random curve from S_p (supersingular curves over F_{p^2})
- Unknown endomorphism ring (critical for security)
- Previous methods: trusted setup only
Step 2: Choose Algorithm Variant
| Variant | Complexity | Security Assumption | Output |
|---|
| Booher-based | O~(log^4 p) under GRH | Average-case ERP | Secure curve |
| Oriented | O~(log^13 p) | Vectorization Problem | Uniform O-oriented |
Step 3: Verify Security
- Endomorphism Ring Problem (ERP): Given curve E, find End(E)
- Average-case hardness ensures sampled curve is secure
- Interactive verification adds quantum proof of correctness
Step 4: Application to Cryptography
- CGL hash: Map curve to hash output via isogeny walks
- SIDH/SIKE: Use sampled curves as starting points
- SQIsign: Use for signature scheme instantiation
Error Handling
GRH Dependency
If Generalized Riemann Hypothesis cannot be assumed:
- Fall back to unconditional O~(log^13 p) variant
- Or use classical methods with trusted setup
Small Prime p
For small p, spectral methods may not provide sufficient security:
- Choose p >= 2^256 for cryptographic security
- Verify isogeny graph expansion properties
Resources
- arXiv: 2602.02263 — "On the Spectral theory of Isogeny Graphs and Quantum Sampling of Secure Supersingular Elliptic curves"
- Booher et al. algorithm for isogeny-based sampling
- CGL hash function (Charles-Goren-Lauter)
