Primary audience: Cryptographers, security engineers, quantum networking researchers
ASA for Cryptographers
Portfolio D applies the non-associative geometry of the ASA to cryptographic protocol design. The central insight is that non-associativity — normally treated as an obstacle in algebra — is a cryptographic resource: a sequence of non-associative operations is order-dependent in a way that cannot be compressed or reordered without destroying the result.
The Post-Quantum Opportunity
Standard public-key cryptography (RSA, ECC) relies on algebraic hardness over associative structures — integer multiplication, elliptic curve group addition. These are broken by Shor’s algorithm on a sufficiently large quantum computer, because Shor exploits the Fourier analysis of abelian groups. The Magmoidal Cipher is immune to this attack: octonion multiplication forms a magma, not a group. There is no Fourier transform over a magma, and no quantum algorithm analogous to Shor’s is known.
The Magmoidal Cipher (Paper 208)
Sequence-Dependent Cryptography introduces the Magmoidal Cipher, grounded in the non-associativity of octonion multiplication. A secret is encoded as a sequence of Fano-line selections: because the octonion product $(e_{i_1} e_{i_2}) e_{i_3}$ depends irreducibly on the order of operations — not just the set ${i_1, i_2, i_3}$ — an adversary who intercepts the output but not the sequence has no algebraic shortcut to the key. The associator $\mathcal{A}(e_i, e_j, e_k) = (e_i e_j)e_k - e_i(e_j e_k)$ vanishes on Fano lines and equals $\pm 2$ elsewhere, giving a natural binary hard/easy distinction that the cipher exploits.
Three constructions are developed:
Verifiable Delay Functions (VDFs). A sequence of $n$ non-associative Fano multiplications takes exactly $n$ steps — there is no parallel shortcut because each step depends on the result of the previous one. This gives a cryptographic time-lock with a publicly verifiable proof of elapsed computation, immune to parallelisation even on quantum hardware.
V31-QKD (Fano Quantum Key Distribution). A quantum key distribution protocol using the 31 non-Fano triples of $\mathrm{PG}(2,2)$ as its alphabet. Eavesdropping disturbs the associator signature in a detectable way: any measurement collapsing a superposition of Fano/non-Fano states leaves a geometric fingerprint $|\mathcal{A}| \in {0, 2}$. For quantum networking engineers: V31-QKD is designed for integration with quantum repeater networks and is compatible with the $G_2$ fibre bundle structure of the Fibrational Tensor Codes (Paper 206, Portfolio C), suggesting a unified framework for quantum communication and quantum error correction over the same geometric substrate.
Asymmetric Sequential Locks. A public key is a Fano-compatible sequence (associator = 0 at every step); the private key is the full non-associative generating sequence. Verification is $O(n)$; forgery requires finding a sequence with the correct Fano shadow, which is conjectured hard (no polynomial-time algorithm is known classically or quantumly).
Connections to Portfolio F
The Fano Monogamy Paradox (Paper 270, Portfolio F) provides a geometric security foundation for V31-QKD: it proves that an eavesdropper cannot be simultaneously Fano-entangled with both Alice and Bob without leaving a detectable associator signature, giving a no-cloning-style security proof grounded in octonion geometry rather than Hilbert space axioms.
Papers
| # | Paper |
|---|---|
| 208 | Sequence-Dependent Cryptography (The Magmoidal Cipher): Verifiable Delay Functions, V31-QKD, and Asymmetric Sequential Locks via Non-Associative Fano-Geometry |
Key Glossary Terms
Fano Plane · Associator · $G_2$ · Division Algebra Ladder · Steane Code / QEC