Advances in Cryptology - CRYPTO 2009: 29th Annual by Shai Halevi

By Shai Halevi

This e-book constitutes the refereed lawsuits of the twenty ninth Annual overseas Cryptology convention, CRYPTO 2009, held in Santa Barbara, CA, united states in August 2009.
The 38 revised complete papers provided have been rigorously reviewed and chosen from 213 submissions. Addressing all present foundational, theoretical and learn points of cryptology, cryptography, and cryptanalysis in addition to complex purposes, the papers are equipped in topical sections on key leakage, hash-function cryptanalysis, privateness and anonymity, interactive proofs and zero-knowledge, block-cipher cryptanalysis, modes of operation, elliptic curves, cryptographic hardness, merkle puzzles, cryptography within the actual global, assaults on signature schemes, mystery sharing and safe computation, cryptography and game-theory, cryptography and lattices, identity-based encryption and cryptographers’ toolbox.

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Extra resources for Advances in Cryptology - CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2009, Proceedings (Lecture ... Computer Science / Security and Cryptology)

Example text

If E is a supersingular curve, then the group structure of E(Fq ) is determined by the next result. 13 ([137]) Let #E(Fq ) =q +1 - t. Ift 2 = q, 2q, or 3q, then E(Fq ) is cyclic. :::! ;q+1l depending on whether t = 2vq or t = -2vq respectively. (i) (iii) 1ft = 0 and q ¢ 3 (mod 4), then E(Fq ) is cyclic. l(q+1)/2 E9 0 ~. If I is a prime, then let vl(n) be the largest integer with Itll(n)ln. 14) 26 CHAPTER 2. INTRODUCTION TO ELLIPTIC CURVES with al 2: b/, al + bl = vl(N), and bl ~ v/(q - 1). For example, if gcd(N, q - 1) = 1 then E(Fq) is cyclic.

The slope of I is A= { :: =::. 3x~ + 2a2xl + a4 2Yl = + alxl + a3 if P al Yl :f: Q, , if P = = Q. Yl - AXl, then the equation defining 1 is Y AX +f3. 3. 1) to get a cubic polynomial equation x3 +a2x2 +a4x +a6 - (AX +{3)2 - alx(Ax +{3) - a3(Ax +{3) = O. 7) are X}, X2 and X3. 7) factors as (x - xt}(x - X2)(X - X3) = O. 8), we obtain -(Xl + X2 + X3) = a2 - A2 - alA. Hence and Y3 = -(A + at}x3 - {3 - a3· If P, Q E E(J(), then computing P+Q involves just a few arithmetic operations in the field J(. Hence if J( is a finite field, then computing P + Q takes (deterministic) polynomial time.

Yd E E, then -P = (Xl, -yd. If Q ¥= -P, then P + Q = (X3, Y3), where E E, A2 - :1:1 - X2 A(Xl-X3)-Yl, X3 Y3 and A= = (X2, Y2) { Y2 - Yl , ifP¥=Q, X2 - Xl 3x~ + a if P 2Yl ' = Q. 7 The equation E : y2 = x 3 +x+6 over the finite field 7111 (the integers modulo 11) defines an elliptic curve since its discriminant is ~ = 4 ¥= 0. The 7l11 -rational points on E are E(7111) = {O, (2,4), (2,7), (3,5), (3,6), (5,2), (5,9), (7,2), (7,9), (8,3), (8,8), (10,2), (10,9)}. Some applications of the addition law are (2,4) (3,5) = (7,2), and (2,4) + (2,4) = (5,9).

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