  We study short sums of algebraic trace functions via the $q$-analogue of van
der Corput method, and develop methods of arithmetic exponent pairs that
coincide with the classical case while the moduli has sufficiently good
factorizations. As an application, we prove a quadratic analogue of
Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant
X$ with $p^2+1\equiv0\pmod q$. The other two applications include a larger
level of distribution of divisor functions in arithmetic progressions and a
sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by
Irving.
