Mathematics of Public Key Cryptography (Version 1.1) by Steven Galbraith 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  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.