Non-commutitive and nonassociative algebraic structures in cryptography
A self-distributive algebra is an algebra $(X,*)$ that satisfies the identity
$x*(y*z)=(x*y)*(x*z)$. There are several cryptosystems that use self-distributive algebras as platforms and these cryptosystems should be thought of as non-associative versions of the more well-known non-abelian group based cryptosystems (such as the Anshel-Anshel-Goldfeld and Ko-Lee key exchanges). Since the current quantum algorithms have nothing to do with self-distributivity, I would suspect that cryptosystems based on self-distributive algebras would be no less secure against quantum attacks than they are against classical attacks.

Cryptosystem 1: This paper by Kalka and Teicher presents a self-distributive algebra based key exchange which is a modifification the Anshel-Anshel-Goldfeld key exchange for groups.

Cryptosystem 2: In this paper, Patrick Dehornoy proposed an authentication protocol which (with some modification) could use any self-distributive algebra as a platform.

Cryptosystem 3: The Ko-Lee key exchange for non-abelian groups extends to a self-distributive algebra based key exchange. Define terms $t_{n}$ for $n\geq 1$ by letting
$$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y).$$
Then $x*(y*z)=t_{n+1}(x,y)*(t_{n}(x,y)*z)$ for all $x,y,z$ and $n\geq 1$. Consider the following key exchange.


  A self-distributive structure $(X,*)$ and an element $x\in X$ are
  public.
  
  
  Alice selects some $a\in X$ and $n\geq 1$ and sends $r_{1}=t_{n+1}(a,x),r_{2}=t_{n}(a,x)$ to Bob.
  Bob selects some $b\in X$ and sends $s=x*b$ to Alice.
  
  
  Let $K=a*(x*b)$.
  
  
  Alice computes $K$ using the fact that $K=a*s$.
  Bob computes $K$ using the fact that $K=r_{1}*(r_{2}*b)$.
  


Now these cryptosystems just need a suitable self-distributive algebra as a platform.

Platform type 1: racks and quandles. Racks and quandles are among the most well-known self-distributive algebras. Racks and quandles arise from groups, so I would consider these cryptosystems a part of non-abelian group based cryptography instead of non-associative cryptography.

Platform type 2: Shifted conjugacy on braids. Braid based cryptography does not appear to be secure and Dehornoy's authentication scheme for shifted conjugacy has been attacked in this paper.

Platform type 3: Algebras of elementary embeddings. These are the algebras which I have been working on, and there is currently no one else studying these structures. The algebras of elementary embeddings are self-distributive structures that arise from the rank-into-rank cardinals at the top of the large cardinal hierarchy. It is too early to comment whether these algebras of elementary embeddings could be made into secure cryptosystems.

Note: The action of braid groups on quandles could be used to greatly improve the security and efficiency of braid group based cryptosystems.