  Choiceless Polynomial Time (CPT) is one of the few remaining candidate logics
for capturing PTIME. In this paper, we make progress towards separating CPT
from polynomial time by firstly establishing a connection between the
expressive power of CPT and the existence of certain symmetric circuit
families, and secondly, proving lower bounds against these circuits. We focus
on the isomorphism problem of unordered Cai-F\"urer-Immerman-graphs (the
CFI-query) as a potential candidate for separating CPT from P. Results by
Dawar, Richerby and Rossman, and subsequently by Pakusa, Schalth\"ofer and
Selman show that the CFI-query is CPT-definable on linearly ordered and
preordered base graphs with small colour classes. We define a class of
CPT-algorithms, that we call "CFI-symmetric algorithms", which generalises all
the known ones, and show that such algorithms can only define the CFI-query on
a given class of base graphs if there exists a family of symmetric XOR-circuits
with certain properties. These properties include that the circuits have the
same symmetries as the base graphs, are of polynomial size, and satisfy certain
fan-in restrictions. Then we prove that such circuits with slightly
strengthened requirements (i.e. stronger symmetry and fan-in and fan-out
restrictions) do not exist for the n-dimensional hypercubes as base graphs.
This almost separates the CFI-symmetric algorithms from polynomial time - up to
the gap that remains between the circuits whose existence we can currently
disprove and the circuits whose existence is necessary for the definability of
the CFI-query by a CFI-symmetric algorithm.
