[Cryptography] How do I tell if a byte sequence repeats?

[Pierre Abbat]

I'm doing a chosen-plaintext attack on my stream cipher Daphne with reduced 
key length. (The shift register is as long as the key.) I take a Daphne, make 
two copies of it, and feed them the same byte sequence except that two 
different bytes are exchanged at or near the beginning. The attack stops when 
each Daphne has been fed 2^32 bytes (I may increase this in the future) 
without noticing a statistical anomaly, or when it's been fed twice as many 
bytes as when a statistical anomaly is noticed (the number of times a byte 
pair has occured is too far from i/65536, where i is the iteration number). I 
then plot a heat map of the byte pairs and a density plot (smoothed histogram) 
of the numbers of times the byte pairs occur.

I've seen different appearances of the graphs:

* The heat map looks like snow, the density looks normal, and it ran all the 
way to 2^32. This is good.

* The heat map looks like a diagonal line, and the density is bimodal with one 
bump much smaller than the other. I haven't seen this plotted; I was still 
looking at the console when it happened. This was when a 4-byte Daphne was fed 
random bytes, and the two Daphnes fell together after a while.

* The heat map looks like tartan, and the density looks normalish but is too 
wide. The input was all zeros except for a single nonzero byte at the 
beginning. I think the two Daphnes fell into different loops with period a few 
thousand.

* The heat map looks like snow, and the density looks normalish but is too 
wide. I think the Daphnes fell into two different loops with period between 
2^16 and 2^24.

The last case is what I'm asking about, and with longer keys there may be 
longer loops. I don't want to store the entire 8 GiB output of the Daphnes to 
measure the period. I know how to measure the period of a sequence where each 
number in the period occurs only once, but this is a sequence of bytes that 
repeats with a period that may be in the millions. How can I measure the 
period without storing the whole sequence?

Pierre
-- 
The Black Garden on the Mountain is not on the Black Mountain.