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By Zimmer H.G.
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Additional resources for Computational aspects of theory of elliptic curves
4. Set t ← pi+j + ai aj + c, pi+j ← t mod B , and c ← t/B . 5. Set j ← j + 1; if j ≤ n − 1, go to step 4. 6. Set pi+n ← c. 7. Set i ← i + 1; if i ≤ n − 2, go to step 3. 8. Multiplication of inner products by 2: Set i ← 1 and c ← 0. 9. Set t ← 2pi + c, pi ← t mod B , and c ← t/B . 10. Set i ← i + 1; if i ≤ 2n − 2, go to step 9. 11. Set p2n−1 ← c. 12. Addition of the inner squares: Set i ← 0 and c ← 0. 13. Set t ← p2i + a2i + c, p2i ← t mod B , and c ← t/B . 14. Set t ← p2i+1 + c, p2i+1 ← t mod B , and c ← t/B .
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).
We assume that the type USHORT is represented by 16 bits and that the type ULONG can fully accept results of arithmetic operations with USHORT types, which is to say that the informally formulated size relationship USHORT × USHORT ≤ ULONG holds. h (cf. 1). h for the GNU C/C++ compiler (cf. [Stlm]) the following appears: #define UCHAR_MAX 0xffU #define USHRT_MAX 0xffffU #define UINT_MAX 0xffffffffU #define ULONG_MAX 0xffffffffUL One should note that with respect to the number of binary places there are actually only three sizes that are distinguished.