Guide to Elliptic Curve Cryptography (Springer Professional by Darrel Hankerson, Alfred J. Menezes, Scott Vanstone

By Darrel Hankerson, Alfred J. Menezes, Scott Vanstone

After 20 years of analysis and improvement, elliptic curve cryptography now has common publicity and recognition. undefined, banking, and govt criteria are in position to facilitate wide deployment of this effective public-key mechanism.

Anchored by way of a entire remedy of the sensible features of elliptic curve cryptography (ECC), this advisor explains the elemental arithmetic, describes cutting-edge implementation equipment, and provides standardized protocols for public-key encryption, electronic signatures, and key institution. additionally, the booklet addresses a few matters that come up in software program and implementation, in addition to side-channel assaults and countermeasures. Readers obtain the theoretical basics as an underpinning for a wealth of functional and obtainable wisdom approximately effective application.

Features & Benefits:

  • Breadth of assurance and unified, built-in method of elliptic curve cryptosystems
  • Describes very important and executive protocols, akin to the FIPS 186-2 normal from the U.S. nationwide Institute for criteria and Technology
  • Provides complete exposition on concepts for successfully imposing finite-field and elliptic curve arithmetic
  • Distills advanced arithmetic and algorithms for simple understanding
  • Includes invaluable literature references, an inventory of algorithms, and appendices on pattern parameters, ECC criteria, and software program tools

This entire, hugely targeted reference is an invaluable and integral source for practitioners, pros, or researchers in laptop technological know-how, laptop engineering, community layout, and community info protection.

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Additional resources for Guide to Elliptic Curve Cryptography (Springer Professional Computing)

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Since b ≥ 3 and p < bk , we have 3 p < bk+1 . Thus 0 ≤ z −q p < bk+1 , and so r = z −q p after step 3. Hence, at most two subtractions at step 4 are required to obtain 0 ≤ r < p, and then r = z mod p. 14) (i) A natural choice for the base is b = 2 L where L is near the word size of the processor. (ii) Other than the calculation of µ (which is done once per modulus), the divisions required are simple shifts of the base-b representation. (iii) Let z = z/bk−1 . Note that z and µ have at most k + 1 base-b digits.

3. Depth-2 splits for 192-bit integers. The product x y using (a) has three 96×96 multiplications. Each is performed with a 32×32 and two 64×64 (each requiring three 32×32) multiplications, for a total of 21 multiplications of size 32 ×32. Using (b) or (c), only 18 multiplications of size 32×32 are required. As a second illustration, consider Karatsuba-Ofman applied to 192-bit integers, again with W = 32. 3. 3(a) will require 21, while (b) and (c) use 18. The basic idea is that multiplication of 3l-bit integers x = x2 22l + x1 2l + x0 and y = y2 22l + y1 2l + y0 can be done as x y = (x2 22l + x1 2l + x0 ) · (y2 22l + y1 2l + y0 ) = x2 y2 24l + (x2 y1 + x1 y2 )23l + (x2 y0 + x0 y2 + x1 y1 )22l + (x1 y0 + x0 y1 )2l + x0 y0 = x2 · y2 24l + [(x2 + x1 ) · (y2 + y1 ) − x2 y2 − x1 · y1 ]23l + [(x2 + x0 ) · (y2 + y0 ) − x2 y2 − x0 · y0 + x1 y1 ]22l + [(x1 + x0 ) · (y1 + y0 ) − x1 y1 − x0 y0 ]2l + x0 y0 for a total of six multiplications of l-bit integers.

The reduction step can be accelerated considerably when the modulus p has a special form. 6. The algorithms presented here are well suited for software implementation. We assume that the implementation platform has a W -bit architecture where W is a multiple of 8. Workstations are commonly 64- or 32-bit architectures. Low-power or inexpensive components may have smaller W , for example, some embedded systems are 16-bit and smartcards may have W = 8. The bits of a W -bit word U are numbered from 0 to W − 1, with the rightmost bit of U designated as bit 0.

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