1 Analysis of Fractional Window Recoding Methods and Their Application to Elliptic Curve Cryptosystems ECC IEEE Transactions on Computers, VOL. 55, NO. 1, JAN 2006 Author Katja Schmidt-Samoa, Olivier Semay, and Tsuyoshi Takagi Adviser , Reporter 2 Outline Introduction Elliptic curve cryptography (ECC) Nonadjacent Form (NAF) window NAF (wNAF) mutual opposite form (MOF) window MOF (wMOF) Fractional wNAF Fractional wMOF Conclusions 3 Introduction(1/13) Elliptic curve cryptography (ECC) shorter key-size and faster computation suitable for small-memory device Time of crack (ns)RSA bit-lengthECC bit-lengthRSA/ECC 1 6 1 7 1 10 1 35 1 4 Introduction(4/13) Elliptic curve on prime field ECADD ECDBL 5 Introduction(2/13) EC Doubling (ECDBL) EC Addition (ECADD) 6 Introduction(3/13) Scalar Multiplication Binary Method 1. 2.For down to ECDBL if, ECADD 3.Return binary representation Ex. D D DADADA 7 Introduction(5/13) Example: {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)} 8 Introduction(6/13) Nonadjacent Form (NAF) Input: A positive integer Output: A signed digit representation 9 Introduction(7/13) Example: = 10 Introduction(8/13) window NAF (wNAF) The most significant non-zero bit is positive. Among any consecutive digits, at most one is non- zero. Each non-zero digit is odd and less than in absolute value. 11 Introduction(9/13) 12 Introduction(10/13) Example: w=5 13 Introduction(11/13) mutual opposite form (MOF) recoding stage can be done Left-to-Right The signs of adjacent non-zero bits (without considering 0 bits) are opposite. The most non-zero bit and the least non-zero bit are 1 and -1, respectively. 14 Introduction(12/13) Example: 15 Introduction(13/13) window MOF(wMOF) - The most significant non-zero bit is positive. - Each non-zero digit is odd and less than in absolute value. EX : 16 Fractional wNAF w=2 | 0 3| 0| 1 0| 0 3| 0 0| 0 3| 0 0| 0 3| 1 0| 1 0| 0| 0 3| 0 1 w=3 | 0 3 0| 0 0 5| 1 0 0| 0 3 0| 0| 0 0 7| 0 |1 0 0| 0 3 0| 1 17 Fractional wMOF First Phase: the table entries are precompute Second Phase: merges recoding and evaluation 18 conclusions proved that the proposed Frac-wMOF has the same nonzero density as Frac-wNAF using identical table sizes Frac-wMOF recoding requiring less working memory than the Frac-wNAF approach