Suppose that strings are composed only by 0 and 1.. If the text is s with length(s) = n and the pattern is t with length(t) = m, reverse the pattern first, and then for each string build the polynomial P(w) = w[0] + w[1] × x + ... + w[length(w) - 1] × x^{length(w) - 1}. The coefficient of the term i + m - 1 in P(s + s) × P(rev(t)) give you the matches of the i - th cyclic shift of s with t.. For solving the original problem calculate the answer for each different letters and add up the result for obtain the answer for the original.

• Consider masks for each letter (e.g. for 'A': AATGCA -> 110001).
• Treat mask as a polynomial: x⁵ + x⁴ + 1.
• Multiply poly(S₁) and poly(reversed(S₂)) modulo xⁿ - 1. Modulo means setting the relation xⁿ = 1 which is saying that rotating left by n changes nothing. To implement: the result of multiplication has size 2n, just add the high half to the low and remove the high half). Reverse is needed because normal convolution will pair first letters "asymmetrically".
• The result will contain number of matches for each position.
• Sum such results for all letters and choose the maximum.