Why convert from hex strings to a number when in the end you compare anyway? Unless you use fancy bitmask arithmetic, just stay with strings:

use constant XXX => 'cccc';

$self->{auth_mode} = lc $val;

given ($self->{auth_mode}) {
    when (XXX) {  ... }
};
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Note that the change from numeric to string comparisons might prove problematic if you have multiple locations where you check for equality with == (and no test suite that smokes out such situations).