388342-digit largest known twin prime: faster sum of squares or sqrt(-1) (mod p) ?

[hermann@stamm-wilbrandt.de]

This is summary diagram for 388342-digit largest known twin prime 
determination of sum of square or sqrt(-1) (mod p).
At top terminology used is described.

GraphvizFiddle Share link:
https://stamm-wilbrandt.de/388342-digit_prime_summary.html
Screenshot:
https://stamm-wilbrandt.de/images/388342-digits_prime_summary.png


The very efficient millisecond runtimes and used Pari/GP script for 
three edges on the right are discussed in this posting:
https://pari.math.u-bordeaux.fr/archives/pari-users-2306/msg00020.html

The efficient determination of smallest quadratic nonresidue mod p left 
bottom is from Bill Allombert:
https://pari.math.u-bordeaux.fr/archives/pari-users-2306/msg00042.html

The 3:20:18h time to compute sqrtm1 from sqnr was determined with this 
C++ with libgmpxx code:
https://github.com/Hermann-SW/RSA_numbers_factored/blob/main/c%2B%2B/sqrtm1.cc

Runtimes for seven primes in range 10000-digits..388342-digits are in 
this diagram:
https://github.com/Hermann-SW/QuadraticRegression#self-contained-example-with-multiple-curves


All black instructions on diagram edges are Pari/GP instructions.
The orange instruction "powm(sqnr, p/4, p)" is libgmp instruction.

Determining smallest quadratic non-residue fast first made determination 
of sqrtm1 with powm() deterministic.

Using "halfgcd()" and "lift(sqrt(Mod(-1, p)))" allows to determine sum 
of squares from sqrtm1 and vice versa very fast.

Best Pari/GP runtime for that is more than 10.5h, best C++ runtime is 
3.3h.

Are there any alternative ways to determine say sum of squares for 
largest twin prime with significantly faster than 3h runtime for that 
CPU?


Regards,

Hermann Stamm-Wilbrandt.