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Guide to Elliptic Curve Cryptography.pdf
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List of Tables

 1.1 RSA, DL and EC key sizes for equivalent security levels . . . . . . . . . 19 2.1 OEF example parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.2 Computational details for inversion in OEFs . . . . . . . . . . . . . . . . 68 2.3 Computational details for inversion in OEFs . . . . . . . . . . . . . . . . 68 3.1 Admissible orders of elliptic curves over F37 . . . . . . . . . . . . . . . . 83 3.2 Isomorphism classes of elliptic curves over F5 . . . . . . . . . . . . . . . 85 3.3 2 = x 3 x + b . . . . . . . . . . . 92 Operation counts for arithmetic on y2 − 3 3 2 + b . . . . . . . . 3.4 Operation counts for arithmetic on y + x y = x + ax 96

3.5Point addition cost in sliding versus window NAF methods . . . . . . . . 102

3.6 Operation counts for computing k P + l Q . . . . . . . . . . . . . . . . . 113

3.7Operation counts in comb and interleaving methods . . . . . . . . . . . . 113

3.8Koblitz curves with almost-prime group order . . . . . . . . . . . . . . . 115

 3.9 Expressions for αu (for the Koblitz curve E0) . . . . . . . . . . . . . . . 121 3.1 Expressions for αu (for the Koblitz curve E1) . . . . . . . . . . . . . . . 122 3.11 Operation counts for point multiplication (random curve over F2163 ) . . . 140 3.12 Point multiplication costs for P-192 . . . . . . . . . . . . . . . . . . . . 143 3.13 Point multiplication costs for B-163 and K-163 . . . . . . . . . . . . . . 145 3.14 Point multiplication timings for P-192, B-163, and K-163 . . . . . . . . . 146 5.1 Partial history and features of the Intel IA-32 family of processors . . . . 207 5.2 Instruction latency/throughput for Pentium II/III vs Pentium 4 . . . . . . 208

5.3Timings for ﬁeld arithmetic (binary vs prime vs OEF) . . . . . . . . . . . 220

5.4Timings for binary ﬁeld arithmetic . . . . . . . . . . . . . . . . . . . . . 221

 5.5 Timings for prime ﬁeld arithmetic . . . . . . . . . . . . . . . . . . . . . 221 5.6 Multiplication and inversion times . . . . . . . . . . . . . . . . . . . . . 222 5.7 Multiplication times for the NIST prime p224 = 2224 − 296 + 1 . . . . . . 224 5.8 Priorities for hardware design criteria . . . . . . . . . . . . . . . . . . . 229 5.9 Operation counts for inversion via multiplication in binary ﬁelds . . . . . 238
 xiv List of Tables A.1 Irreducible binary polynomials of degree m, 2 ≤ m ≤ 300. . . . . . . . . 259 A.2 Irreducible binary polynomials of degree m, 301 ≤ m ≤ 600. . . . . . . . 260 A.3 NIST-recommended random elliptic curves over prime ﬁelds. . . . . . . . 262 A.4 NIST-recommended random elliptic curves over binary ﬁelds. . . . . . . 264 A.5 NIST-recommended Koblitz curves over binary ﬁelds. . . . . . . . . . . . 265 B.1 ECC standards and draft standards . . . . . . . . . . . . . . . . . . . . . 268 B.2 URLs for standards bodies and working groups. . . . . . . . . . . . . . . 268

List of Figures

 1.1 Basic communications model . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Symmetric-key versus public-key cryptography . . . . . . . . . . . . . . 4 2.1 Representing a prime-ﬁeld element as an array of words . . . . . . . . . . 29 2.2 Depth-2 splits for 224-bit integers (Karatsuba-Ofman multiplication) . . . 33 2.3 Depth-2 splits for 192-bit integers (Karatsuba-Ofman multiplication) . . . 34 2.4 Representing a binary-ﬁeld element as an array of words . . . . . . . . . 47 2.5 Right-to-left comb method for polynomial multiplication . . . . . . . . . 49 2.6 Left-to-right comb method for polynomial multiplication . . . . . . . . . 49 2.7 Left-to-right comb method with windows of width w . . . . . . . . . . . 51 2.8 Squaring a binary polynomial . . . . . . . . . . . . . . . . . . . . . . . . 52 2.9 Reduction of a word modulo f (z) = z163 + z7 + z6 + z3 + 1 . . . . . . . . 54 3.1 ECDSA support modules . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Elliptic curves over the real numbers . . . . . . . . . . . . . . . . . . . . 77 3.3 Geometric addition and doubling of elliptic curve points . . . . . . . . . 80

3.4Montgomery point multiplication . . . . . . . . . . . . . . . . . . . . . . 103

3.5 Fixed-base comb method for point multiplication . . . . . . . . . . . . . 107

3.6The exponent array in Lim-Lee combing methods . . . . . . . . . . . . . 108

3.7 Simultaneous point multiplication accumulation step . . . . . . . . . . . 109

3.8Interleaving with NAFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.1 Illustration of Pollard’s rho algorithm . . . . . . . . . . . . . . . . . . . 158

4.2Illustration of parallelized Pollard’s rho algorithm . . . . . . . . . . . . . 162

5.1Splitting of a 64-bit ﬂoating-point number . . . . . . . . . . . . . . . . . 211

5.2Hierarchy of operations in elliptic curve cryptographic schemes . . . . . . 226

 5.3 Elliptic curve processor architecture . . . . . . . . . . . . . . . . . . . . 227 5.4 Most signiﬁcant bit ﬁrst (MSB) multiplier for F25 . . . . . . . . . . . . . 231 5.5 Least signiﬁcant bit ﬁrst (LSB) multiplier for F25 . . . . . . . . . . . . . 232

xvi List of Figures

5.6MSB multiplier with ﬁxed reduction polynomial . . . . . . . . . . . . . . 232

 5.7 MSB multiplier for ﬁelds F2m with 1 ≤ m ≤ 10 . . . . . . . . . . . . . . 233 5.8 MSB multiplier for ﬁelds F25 , F27 , and F210 . . . . . . . . . . . . . . . . 234

5.9Multiplicand in a 2-digit multiplier for F25 . . . . . . . . . . . . . . . . . 235

 5.1 A 2-digit multiplier for F25 . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.11 Squaring circuit for F27 with ﬁxed reduction polynomial . . . . . . . . . 236 5.12 CMOS logic inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

5.13Power trace for a sequence of addition and double operations . . . . . . . 240

5.14Power trace for SPA-resistant elliptic curve operations . . . . . . . . . . . 241

5.15OAEP encoding function . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.16OAEP decoding function . . . . . . . . . . . . . . . . . . . . . . . . . . 247

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