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Additional info for Cryptography in C and C++ (2nd Edition)
Schönhage and V. Strassen for multiplying large numbers by application of fast Fourier transforms over finite fields. 3). These techniques encompass the fastest known multiplication algorithms, but their advantage in speed over the classical O n2 methods comes into play only when the number of binary digits is in the range 8,000–10,000. Based on the demands of cryptographic systems, such numbers, at least for the present, are far beyond the range envisioned in the application domain of our functions.
Set t ← ai + bi + c, si ← t mod B , and c ← ⌊t/B⌋. 3. Set i ← i + 1; if i ≤ n − 1, go to step 2. 4. Set t ← ai + c, si ← t mod B , and c ← ⌊t/B⌋. 5. Set i ← i + 1; if i ≤ m − 1, go to step 4. 6. Set sm ← c. 7. Output s = (sm sm−1 . . s0 )B . The digits of the summands, together with the carry, are added in step 2, with the less-significant part stored as a digit of the sum, while the more-significant part is carried to the next digit. If the most-significant digit of one of the summands is reached, then in step 4 any remaining digits of the other summand are added to any remaining carries one after the other.
7. Set pi+n ← c. 8. Set j ← i + 1; if i = n − 1, go to step 7. 9. Output p = (p2n−1 p2n−2 . . p0 )B . In selecting the necessary data types for the representation of the variables we must note that t can assume the value (B − 1) + 2(B − 1)2 + (B − 1) = 2B 2 − 2B (in step 5 of the algorithm). But this means that for representing t to base B more than two digits to base B will be needed, since we also have B 2 −1 < 2B 2 −2B < 41 Chapter 4 2B 2 − 1, and so a ULONG will not suffice for representing t (the inequality above is derived from the fact that one additional binary digit is needed).