**Edited 3/10/2017**

Very large cardinals around the rank-into-rank area potentially have applications in cryptography. Rank-into-rank cardinals produce self-distributive algebras which may be used as platforms or to produce platforms for authentication schemes and key exchange protocols. Therefore, if these self-distributive algebras are seriously considered as platforms for new cryptosystems, then one will need all of the large cardinal hierarchy in order to research applied mathematics.

Recall that a rank-into-rank embedding is an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$. Let $\mathcal{E}_{\lambda}$ denote the collection of all rank-into-rank embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Define a binary operation $*$ on $\mathcal{E}_{\lambda}$ by letting $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Then $(\mathcal{E}_{\lambda},*)$ satisfies the self-distributivity identity $j*(k*l)=(j*k)*(j*l)$. If $\gamma$ is a limit ordinal with $\gamma<\lambda$, then define a congruence $\equiv^{\gamma}$ on $(\mathcal{E}_{\lambda},*)$ by letting
$j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ whenever $x\in V_{\gamma}$. Then my question and answer shows that the algebra $(\mathcal{E}_{\lambda}/\equiv^{\gamma},*)$ is locally finite and hence locally computable.

The non-abelian group based cryptosystems can generally be modified to produce cryptosystems that could use any left-distributive algebra as a platform. For example, in this paper, the authors have modified the Anshel-Anshel-Goldfeld key exchange to produce a cryptosystem that could use any left-distributive algebra as a platform. Furthermore, the Ko-Lee key exchange also could be modified to produce a cryptosystem for left-distributive algebras. In this paper, Using shifted conjugacy in braid-based cryptography, Dehornoy proposes an authentication scheme that can use any self-distributive algebra as a platform (though you need to tweak this cryptosystem if the self-distributive algebra is not left-cancellative) rather than just conjugacy on groups (Dehornoy had the platform shifted conjugacy on braid groups in mind, but this platform was later shown to be insecure).

Dehornoy in 4 remarks that the combinatorial complexity of the Laver tables and the fact that the classical Laver tables produce extremely fast growing functions suggests that the classical Laver tables or similar structures may be a good platform for his authentication scheme or some other cryptosystem. However, the classical Laver tables are currently a very insecure platform for all cryptosystems. First of all, $A_{48}$ is the largest classical Laver table which has even been computed. Therefore, the classical Laver table based cryptosystems only provide at most 48 bits of security. Furthermore, the homomorphisms between the classical Laver tables allow one to easily write computer programs that break all cryptosystems based on the classical Laver tables. My generalized Laver tables which you can compute online here are not a secure platform for self-distributive algebra based cryptosystems either since it is easy to factorize elements in generalized Laver tables. **Fortunately, unless I have simply overlooked something, the ternary Laver tables so far appear to be plausible platforms for these self-distributive algebra based cryptosystems.** Click here for a ternary Laver table calculator on my website.

Of course, it is way to soon to comment on the security or the insecurity of these ternary Laver table based cryptosystems, and much more research needs to be done on cryptography based on Laver-like algebra. These line of investigation have barely been researched, but I have recently proposed a polymath project to promote an investigation into these directions of inquiry.

Is the axiom of choice relevant for applications of mathematics to empirical sciences ?It is often used, through theorems of Functional Analysis, but thus might be due to our lazyness. $\endgroup$ – Denis Serre Feb 9 '11 at 7:16