There is a paper by D. Biss and S. Dasgupta (A Presentation for the Unipotent Group over Rings
with Identity, J. Algebra, vol. 237, 691-707 (2001)) which gives a presentation of the group of upper triangular matrices over $R$ with diagonal entries equal to 1, where $R$ is any ring with identity. 

(Moreover, in some cases they show that their presentations are minimal both in the sense of having no redundant relations, and in the stronger sense of having the fewest possible number of generators and relations.) 

In the case $R=\mathbb{Z}$, their generators are the same as the ones in the question above, and the relations are $$[a_{i,i+1},a_{j,j+1}] =1\ \ (i&lt;j-1\leq n-2)$$ 

$$[a_{i,i+1},[a_{i,i+1},a_{i+1,i+2}]]=[a_{i+1,i+2},[a_{i,i+1},a_{i+1,i+2}]]=1$$ $$[[a_{i,i+1},a_{i+1,i+2}],[a_{i+1,i+2},a_{i+2,i+3}]]=1.$$

Presentation of discrete upper triangular group
Earlier this year, Yitang Zhang published a proof of a weakened form of the Twin Prime Conjecture. I'm wondering if any of the new mathematical machinery he developed has uses in cryptography or could be adapted for future use in the field. 
