FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

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2.3. SEQUENTIAL MULTIPLICATION SCHEMES 13 Due to the complexity of a reduction step, in the following this work diverts the arithmetic operations in into two successive parts. The arithmetical operation:hÁ5.Â corresponding to that in- and with a degreeÕ ( and the subsequent reduction step. 2.3 Sequential Multiplication Schemes In order to achieve a reasonable level of security for an EC cryptosystem, the extension degree ( of the underlying finite field has to be in the hundreds. Due to chip area limitations an application of combinational multipliers of full bit width ( is usually not feasible. Instead, a sequential multiplication scheme, which is based on a reasonable sized purely combinational multiplier unit, has to be utilized. In Sec. 3.4.4 the architecture of such a combinationalhÁ5.Â multiplier is presented, which is scalable and highly efficient in terms of required logic resources. In the remainder of this section it is assumed that a reasonable sized combinational multiplier, which computes the unreduced product of two degree +êé ( polynomials, is part of the design. Some wellknown methods for sequential multiplication are introduced first. Then, in Sec. 2.3.3, the Multi-Segment Karatsuba multiplication scheme is detailed and compared to classical approaches. 2.3.1 Schoolbook Multiplication Given two polynomialsv @ |ëE -hÁ5ÃÂ of degree( and a combinational multiplier of size+ 6ëì(b†¦kí , the product} 6 v | can be computed as follows: Firstv and| are split each into two segments of equal size. Then the product can be computed as 6 v YC5 î 3 ¾ v ® v 6 | YC5 î 3 ¾ | ® | 6 v | } v Yg5 î 3 ¾ v ®ZaP| Yg5 î 3 ¾ | ® 6 v YÚ | YC5 ¾Æv YÚ | ®Ú¾ v ®j | Y[º5 î 3 ¾ v ®j | ®k\ (2.10) 6 Please note that in the context of hardware implementations the5 ± factors correspond to position offsets, which can simply be implemented by appropriate wiring. Generally, the polynomials can be split into an arbitrary number of segments E”© ª . It is selected such that the resulting segments are small enough to be multiplied on the combinational multiplier (+ ³ (b†¦ ). The number of necessary multiplications is given by 3 . Since the additions can be computed combinationally in the same cycle, the cycle count for a complete multiplication is also given by 3 . A variation of the schoolbook method splits each of the two polynomialsv and| into different numbers of segments (=ïÚ@A=ð .) Of course, in this case an appropriate asymmetric combinational multiplier is necessary 2 . The number of required multiplications is given here by.ïˆBPð . In the extreme case of4ï 6 ( and =ð 6 c this scheme is called bit serial multiplication. 2 This topic is extensively treated in [16].
2.3. SEQUENTIAL MULTIPLICATION SCHEMES 14 x^ 8 x^ 7 x^ 6 x^ 5 x^ 4 x^ 3 x^ 2 x^ 1 x^ a) 0 x^ b) 8 x^ 7 x^ 6 x^ 5 x^ 4 x^ 3 x^ 2 x^ 1 x^ 0 3 23 A 3*B3 A 3*B2 A 2*B3 A 3*B1 A 2*B2 A 1*B3 A 3*B0 A 2*B1 A 1*B2 A 0*B3 A 2*B0 A 1*B1 A 0*B2 A 1*B0 A 0*B1 A 0*B0 13 0123 2 02 1 01 0 A*B A*B c) x^ 8 x^ 7 x^ 6 x^ 5 x^ 4 x^ 3 x^ 2 x^ 1 x^ 0 x^ 8 x^ 7 x^ 6 x^ 5 x^ 4 x^ 3 x^ 2 x^ 1 x^ d) 3 0 23 3 123 23 12 23 2 3 0123 123 012 1 2 012 01 12 01 1 0 0 123 0123 2 12 012 1 01 0 A*B A*B Figure 2.4: Sequential Multiplication Schemes: a) Schoolbook method; b) unrolled Karatsuba for 2 recursion steps; c)! #"%$ before reordering of the subterms; d)& œ"H$ after reordering of the subterms
Page 1 and 2: Fachbereich Informatik Integrierte
Page 3 and 4: Contents List of Figures iv 1 Intro
Page 5 and 6: List of Figures c)! #"%$ d)& #"'$ 2
Page 7 and 8: 1.2. PREVIOUS WORK 2 of dividing a
Page 9 and 10: Chapter 2 Mathematical Background T
Page 11 and 12: Ymonpn _ #qkr9;=stPuiY[ v l _ wr9xK
Page 13 and 14: 2.2. FINITE FIELD ARITHMETIC 8 k P
Page 15 and 16: 2.2.4 Addition in-·‰¸º¹ » 2
Page 17: Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì
Page 21 and 22: 6 v |!6 v 3 5 3 î 8 ¾ v Yg5 î 8
Page 23 and 24: 2.3. SEQUENTIAL MULTIPLICATION SCHE
Page 25 and 26: 3.2. REGISTER FILE 20 3.2 Register
Page 27 and 28: 3.4. FINITE FIELD ARITHMETIC 22 3.4
Page 29 and 30: Ì Ì Ì Ì Ì 3.4. FINITE FIELD AR
Page 31 and 32: 3.6. EVALUATION SOFTWARE 26 used fo
Page 33 and 34: 4.2. XILINX XCV405E 28 implementati
Page 35 and 36: Chapter 5 Conclusions and Outlook 5
Page 37 and 38: Bibliography [1] National Institute
Page 39 and 40: Ô Ô bY©®v @ | ò 6 6 Y©® v
Page 41: 3 m°e'Ỹ %jijY l Y Ym°5 l l $ƒm

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