[Lagrange's four-squares theorem](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem) states that every natural number can be represented as the sum of four integer squares. [Rabin and Shallit](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem#Algorithms) gave a randomised algorithm that finds one of these solutions in quadratic time. My question is if anything is known about the *deterministic* time complexity of finding one of the solutions? Any pointers would be appreciated.


(It seems that *enumerating* all the solutions is hard as factoring in certain cases (via [Jacobi's four-square theorem](https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem)), but correct me if I am wrong.)



Lagrange four-squares theorem --- deterministic complexity
As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper [Finding the four squares in Lagrange's theorem](http://campus.lakeforest.edu/trevino/finding4squares.pdf) by Pollack and Treviño. They mention that there is a deterministic polynomial-time algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic polynomial-time algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time. 


