Wouldn't it be possible to block-wise xor the random data onto the key?

Maybe use a windowing mechanism, where the window is moved forward depending on the random data (e.g. for a 16 bit key, xor random bits 0 to 15, then move forward 4 to 16 bits, depending on the current window; iterate until the maximum number of window movements necessary to go over the whole random data is reached [leak less bits via timing]; if the end of the data is reached, again start at the beginning, but with some offset to avoid result_bit_0 = secret_bit_0 xor random_bit_0 xor random_bit_0).

   -  /* $OpenBSD: authfd.c,v 1.113 2018/12/27 23:02:11 djm Exp $ */ 
   +  /* $OpenBSD: authfd.c,v 1.114 2019/06/21 04:21:04 djm Exp $ */ 

I actually liked these identifiers. There were tools like ident (http://manpages.org/ident/1) to deal with them.