By Steven Galbraith

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**Extra info for Mathematics of Public Key Cryptography (Version 1.1)**

**Example text**

K, each step taking some shifts and one multiplication in the field. In other words, the complexity in this case is O(km2 log(q)2 ) = O(log(m)m2 log(q)2 ) field operations. For details of the general case, and further discussion we refer to [305, 269]. See, for example, Fong, Hankerson, Lopez and Menezes [206] for more discussion about inversion for the fields relevant for elliptic curve cryptography. Finally we remark that, for some computational devices, it is convenient to use finite fields Fpm where p ≈ 232 or p ≈ 264 .

Then one runs a variant of Algorithm 2 where w (or more) squarings are performed followed by one multiplication corresponding to a w-bit sub-string of the binary expansion of m that corresponds to an odd integer. , right to left). 5. Let w = 2 so that one precomputes u1 = g and u3 = g 3 . Suppose m has binary expansion (10011011)2. By parsing the binary expansion starting with the least significant bits one obtains the representation 10003003 (we stress that this is still a representation in base 2).

Bm )t = Mi,j ai bj i=1 j=1 where (a1 , . . , am ) and (b1 , . . , bm ) are the coefficient vectors for the representation of a and b with respect to the basis. m−1 } then multipliIn particular, if Fqm is represented by a normal basis {θ, θq , . . , θq cation of elements in normal basis representation is given by ai θ q i=0 m−1 m−1 m−1 m−1 j i j=0 bj θ q = ai b j θ q i=0 j=0 i +qj 62 CHAPTER 2. BASIC ALGORITHMIC NUMBER THEORY i j so it is necessary to precompute the representation of each term Mi,j = θq +q over the normal basis.