Montgomery Modular Exponentiation on Reconfigurable Hardware æ

 Colin Martin
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1 Montgomery Modlar Exponentiation on Reconfigrable Hardware æ Thomas Blm Worcester Polytechnic Institte ECE Department Worcester, MA , USA Christof Paar Abstract It is widely recognized that secrity isses will play a crcial role in the majority of ftre compter and commnication systems. Central tools for achieving system secrity are cryptographic algorithms. For performance as well as for physical secrity reasons, it is often advantageos to realize cryptographic algorithms in hardware. In order to overcome the wellknown drawback of redced flexibility that is associated with traditional ASIC soltions, this contribtion proposes arithmetic architectres which are optimized for modern field programmable gate arrays (FPGAs). The proposed architectres perform modlar exponentiation with very long integers. This operation is at the heart of many practical pblickey algorithms sch as RSA and discrete logarithm schemes. We combine the Montgomery modlar mltiplication algorithm with a new systolic array design, which is capable of processing a variable nmber of bits per array cell. The designs are flexible, allowing any choice of operand and modls. Unlike previos approaches, we systematically implement and compare several variants of or new architectre for different bit lengths. We provide absolte area and timing measres for each architectre. The reslts allow conclsions abot the feasibility and timespace tradeoffs of or architectre for implementation on Xilinx XC4000 series FPGAs. As a major practical reslt we show that it is possible to implement modlar exponentiation at secre bit lengths on a single commercially available FPGA. Introdction It is widely recognized that secrity isses will play a crcial role in many ftre compter and commnication systems. A central tool for achieving system secrity are æ The research was spported in part throgh an NSF CAREER award #CCR cryptographic algorithms. For performance as well as for physical secrity reasons it is often reqired to realize cryptographic algorithms in hardware. Traditional ASIC soltions, however, have the wellknown drawback of redced flexibility compared to software soltions. Since modern secrity protocols are increasingly defined to be algorithm independent, a high degree of flexibility with respect to the cryptographic algorithms is desirable. A promising soltion which combines high flexibility with the speed and physical secrity of traditional hardware is the implementation of cryptographic algorithms on reconfigrable devices sch as FPGAs and EPLDs. In the case of pblickey schemes, algorithm independence can mean not only a change of the actal crypto algorithm bt also change of parameters sch as bit length, modls, or exponents. This contribtion deals with arithmetic architectres for modlar exponentiation with very long integers which is at the heart of most modern pblickey schemes. Most notably, both RSA and discrete logarithmbased (e.g., DiffieHellman key exchange or the Digital Signatre Algorithm, DSA) schemes reqire modlar long nmber exponentiation. The challenge at hand is to design sch arithmetic architectres for operands with p to 024 bit on crrent FPGAs. The very long word lengths prohibit the application of many proposed architectres as they wold reslt in nrealistically large resorce reqirements. In this contribtion we derive a modlar exponentiation architectre which combines Montgomery s modlar redction scheme and a novel systolic array architectre. The systolic array architectre reqires considerably fewer logic resorces than many other systolic array architectres for modlar arithmetic. This is crcial, as one of or goals was to derive soltions that can fit into a single FPGA. Clearly a design which fits in a single FPGA has many cost and design advantages over mlti FPGA soltions. Another important objective was to systematically implement varios architectre options for different bit lengths. This contribtion is strctred as follows. In Section 2, we smmarize some of the previos work on modlar ex
2 ponentiation. Section 3 describes algorithms for modlar exponentiation and mltiplication and some simplifications and speedps for their hardware implementation. In this section we also describe some of the relevant featres of the Xilinx XC4000 FPGA series. Section 4 otlines or architectre for modlar exponentiation. Section 5 briefly describes or methodology and tools that were sed for this research. Section 6 of this contribtion posts the timing and area reslts obtained. A comparison to other architectres and an otlook conclde this contribtion. 2 Previos Work In the following, we will smmarize relevant previos work in the field of modlar mltiplication. Most proposed approaches are based on Montgomery s algorithm [0], either in conjnction with a redndant nmber representation or in an systolic array architectre. Soltions sing other algorithms have also been presented. To avoid the carry propagation in mltiplication/addition architectres several soltions have been proposed in the literatre. They either se Montgomery s algorithm, in combination with a redndant radix nmber system [5, 3, 7, 4, 8, 6] or the Reside Nmber System []. The Research Laboratory of Digital Eqipment Corp. in Paris implemented modlar exponentiation in architectres on FPGAs [7, 3]. They tilized an array of 6 XILINX 3090 FPGAs. Compared to XILINX 4000 series in terms of flip flops, this is eqivalent to a chip with 500 configrable logic blocks (CLBs). In terms of logic resorces this is eqivalent to a chip of 4000 CLBs. In their work they sed several speedp methods [3] inclding the Chinese remainder theorem, an asynchronos carry completion adder, and a windowing method. The implementation comptes a 970bit RSA decryption at a rate of 85kb/s (5.2ms per 970 bit decryption) and a 52 bit RSA decryption in excess of 300 kb/s (.7ms per 52 bit decryption). A drawback of this soltion is that the binary representation of the modls is hardwired into the logic representation so that the architectre has to be reconfigred with every new modls. There has been a nmber of proposals for systolic array architectres for modlar arithmetic. However, no implementations have been reported to or knowledge. In [5] a VLSI soltion is presented where a modlar mltiplication is calclated in è4n +èæ 3n=2 clock cycles (n is the nmber of bits of the modls). That is approximately for times more cycles than in a conventional soltion. In terms of resorces this design wold be sitable for FPGA. Similar twodimensional systolic arrays are presented in [7, 9, 20, 6]. For a radix of two they all propose an n æ n matrix of one bit processing elements. With this configration 2n modlar mltiplications are calclated at the same time and the theoretical throghpt is one modlar mltiplication per clock cycle. In terms of resorces, sch a soltion is not feasible in either VLSI or FPGA for the bit length reqired in pblickey algorithms. Even implementing only onerowofprocessingelements, (resltinginn times slower throghpt) into presently available FPGAs is difficlt in terms of resorces. We tried to overcome the shortage of resorces per chip by sing larger processing elements and ths saving overhead. Reference [2] provides a good overview of previosly presented architectres for VLSI implementations of modlar integer arithmetic. Reference [3] smmarizes the chips available in 990 for performing RSA encryption. More recently an approach [23] has been presented that tilizes precompted complements of the modls and is based on the iterative Horner s rle. Compared to Montgomery s algorithms these approaches se the most significant bits of an intermediate reslt to decide which mltiples of the modls to sbtract. The drawback of these soltions is that they either need a large amont of storage space or many clock cycles to complete a modlar mltiplication. The athors attempted to overcome the later problem by a higher clock freqency which is possible de to a simplified modlo redction operation. 3 Preliminaries 3. Modlar Exponentiation and RSA We start this section with a short description of the RSA algorithm, proposed by Rivest, Shamir and Adleman [2] in 978. The algorithm is based on modlar exponentiation of integers. The private key of a ser consists of two large primes p and q and an exponent D. The pblic key consists of the modls N = p æ q and an exponent E sch that E = D, mod èp, èèq, è. In the remainder of the article we always assme that N can be represented by n bits. To encrypt a message X the ser comptes: Y = X E mod N Decryption is done by calclating: X = Y D mod N The identical operations are tilized for the RSA digital signatre scheme. In order to thwart crrently known attacks, the modls N and ths X and Y shold have a length of bits. Both encryption and decryption reqire algorithms for compting a modlar exponentiation. This can be realized by sing the sqare and mltiply algorithm [4]. To compte sqaring and mltiplication in parallel we can se the following version [20]: 2
3 Algorithm : comptes P = X E mod N, where E = P n, i=0 e i2 i, e i 2f0; g. P 0 =, Z 0 = X 2. for i = 0 to n, do 3. Z i+ = Z 2 i mod N 4. if e i =then P i+ = P i æ Z i mod N Algorithm takes 2n operations in the worst case and :5n on average. For speeding p encryption the se of a short exponent E has been proposed [8]. Recommended by ITU is the the Fermat prime F 4 = Using F 4, the encryption is exected in only 7 operations. Other short exponents proposed inclde E =3and E =7. Obviosly the same trick can not be sed for decryption, as the decryption exponent D mst be kept secret. Bt sing the knowledge of the factors of N = q æ p, thechinese Remainder Theorem [] can be applied by the decrypting party. Two n=2 size modlar exponentiations and an additional recombination instead of one n size modlar exponentiations are compted in this case. Each modlar exponentiation of length n=2 takes =4 of the time reqired for an n bit exponentiation. If both exponentiations are performed serially, an over all speed p factor of two is achieved. If they are performed in parallel, a speed p factor of for is achieved. 3.2 Montgomery Modlar Mltiplication As shown in the previos section, modlar exponentiation is redced to a series of modlar mltiplications and sqarings. The algorithm for modlar mltiplication described below has been proposed by P. L. Montgomery in 985 [0]. Several optimizations were taken from reference [9]: Algorithm 2: Montgomery Modlar Mltiplication (radix 2) for compting A æ B mod N, where B = P P n+ b i=0 i2 i n+2, b i 2f0; g, b 0 = 0, A = a i=0 i2 i, a i 2 f0; g, a n+ =0, a n+2 =0. R 0 =0 2. for i =0to n +2do 3. q i = R i è0è 4. R i+ =èr i + a i æ B + q i æ N è=2 B is shifted p one bit with b 0 =0. This measre simplifies the comptation of q i, compared to the original algorithm. The loop of Algorithm 2 is exected three more times than originally proposed. With this step we make sre the ineqalities R i é 3N and R n+3 é 2N always hold. The reslt of a modlar mltiplication R n+3 can ths be resed as inpt A and B for the next mltiplication. We avoid the originally proposed final comparison and sbtraction and make a pipelined exection of the algorithm possible. A precondition for the algorithm to work is that the modls N has to be relatively prime to the radix. In RSA this is always satisfied as N is a mltiple of two primes and therefore odd. The algorithm above calclates R n = è2,n,3 ABè modn. To get the right reslt we need an extra Montgomery modlar mltiplication by 2 2n+6 mod N. However if frther mltiplications are reqired as for exponentiation it is better to pre mltiply all inpts by the factor 2 2n+6 mod N. Ths every intermediate reslt carries a factor 2 n+3. We jst need to Montgomery mltiply the reslt bytoeliminatethatfactor. The final Montgomery mltiplication with makes sre or final reslt is smaller than N. Consider Algorithm 2 with Bé 4N (B shifted p) and A =è0;:::;0; è. We will get R = B=2 é 2N. As all remaining a i =0,we getatmostr i+ =èr i + N è=2! N. If only one q i =0 èi =; 2 :::n+2è,thenr i+ = R i =2 én(probability:, 2,èn+2è ). The whole comptational complexity of Algorithm 2 lies in the three additions of n bit operands for compting R i+. As the propagation of n carries is too slow and an eqivalent carry look ahead logic reqires too many resorces, two different strategies have been prsed in the literatre:. Redndant representation: The intermediate reslts are kept in redndant form. Resoltion into binary representation is only done at the very end and for feeding the intermediate reslt back as a i in Algorithm Systolic Arrays: n processing nits calclate bit per clock cycle. The compted carries, q i and a i are pmped throgh the processing nits. As these signals have to be distribted only between adjacent processing nits, a faster clock speed and a reslting higher throghpt shold be possible. The cost is a higher latency and possibly more resorces. 3.3 Xilinx XC4000 Series FPGAs In this section we present some of the relevant featres of the Xilinx XC4000 Series FPGAs and introdce a metric for FPGA cost and performance evalation. An FPGA device consists of three types of reconfigrable elements, the Configrable Logic Blocks (CLBs), I/O blocks (IOBs) and roting resorces [22]. An XC4000 CLB is made p of 3 look p tables, two flipflops and programmable mltiplexers. Any boolean fnction of 5 inpts, 3
4 any 2 fnctions of 4 inpts and some fnctions of p to 9 inpts can be compted in one CLB. The mltiplexers can rote these signals directly to the otpts or to the flipflops. In the first case the flipflops can be tilized to store direct inpts. Programmable roting resorces connect the CLBs and IOBs into a network. For signal distribtion all over the device there are 8 global nets available. Another featre of the CLB is its dedicated hardware to accelerate the carry path of adders and conters [22]. An n bit ripple carry adder is implemented in n=2 +2CLBs. As the carry signal ses dedicated interconnects, there is no roting delay in the path and the total delay is fixed: t pd =4:5+n æ 0:35 ënsë. On chip RAM redces the cost of data storage. A single CLB can be sed for a 6 æ 2 bit or 32 æ bit ROM/RAM or for a 6 æ bit Dal Port RAM. In previos work [20, 9, 4] the gate cont model has been sed for cost evalation and the gate delay model for speed evalation. This is not appropriate for FPGAs. As the fnctional nit of an FPGA is the CLB, we evalate the cost (C) in nmber of CLBs. The operation time (T) consists of logic delay in the CLBs and roting delay and is obtained from Xilinx s Timing Analyzer software. As a third parameter we se the time area prodct (TA). It is defined by time mltiplied by cost. 4 A New Architectre 4. Design Overview As described in Section 3.2, there have been two principle approaches proposed to compte Montgomery modlar mltiplication. A soltion following approach has already been implemented in FPGA [7]. The second approach sing systolic arrays has drawn considerable attention in the research commnity. However, no architectres that specifically target FPGAs have been reported, nor are there reports of implementations of sch systolic architectres. Or contribtion targets these two goals. Or system can be divided hierarchically into three levels.. Processing Element: Compte bits of a modlar mltiplication. 2. Modlar Mltiplication: An array of processing elements comptes a modlar mltiplication. 3. Modlar Exponentiation: Combine modlar mltiplications to modlar exponentiation according to Algorithm. In the following we describe the system with a bottom p approach. 4.2 Processing Elements A general radix 2 systolic array as proposed in [7, 9, 6, 5] tilizes n times n processing elements. As this approach wold reslt in nrealistically large CLB conts for the bit length reqired in modern pblic key schemes, we implemented only one row of processing elements. To frther redce the reqired nmber of CLBs we implemented processing elements (nits) of =4,8,6 bits. Withthisapproach we need onlyn= instead of n processing elements, and a considerable amont of overhead can be saved. Similar to the approach in [9] we compte sqarings and mltiplications of Algorithm in parallel. As explained in Section 4.3, this measre flly tilizes every cycle. Mx_B B_Reg "0" B+N_Reg N_Reg B_In N_In Res_0_In Mx_ bit Adder + Add_Reg  Add_Reg_2 2 "0" Mx_2 Control  Figre. Processing Element (nit) Decode Mx_Res Control_Reg q_i, a_ireg Reslt_Reg Control_Ot q_i, a_iin 2 q_i, a_iot Carry_In Reslt_Ot Reslt_In Carry_Ot Res_0_Ot In the processing elements we need the following registers: æ NReg ( bits): storage of the modls æ BReg ( bits): storage of the B mltiplier æ B+NReg ( bits): storage of the intermediate reslt B + N æ AddReg ( +bits): storage of the intermediate reslt æ AddReg2 (, bits): storage of the intermediate reslt æ ControlReg (3 bits): control of the mltiplexers and clock enables æ a i,q i (2 bits): mltiplier A, qotient Q, according to Algorithm 2 æ ResltReg ( bits): storage of the reslt at the end of a mltiplication 4
5 The registers need a total of è6 +5è=2 CLBs. Instead of compting èr + a i æ B + q i æ N è=2 in each iteration, we compte N + B once and store the reslt in the B+NReg. Mltiplexer Mx selects one of its inpts 0, N, B, B + N to be added to R according to the vale of the binary variables a i and q i. The additional cost is a bitregister,a slightly more complicated mltiplexer Mx, and two more clock cycles per mltiplication. The advantage is that only a two operand adder is needed that can be implemented with the ripple carry adder optimized for the Xilinx XC4000 series (see Section 3.3). Also we need only one carry instead of two between nits. The carry propagation delay of a 6 bit adder is eqivalent to only one additional CLB delay. The adder can be combined into the CLBs of the AddReg; we need therefore no additional CLBs. An additional register AddReg2 allows storage of a mltiplication while a sqaring is compted and vice versa. The decoded control register signals and the a i, q i signals control the mltiplexers Mx B, Mx, Mx 2, Mx Res and the clock enables of the registers. NReg is loaded only when the modls is changed, BReg and B+NReg after each completion of Algorithm 2. Mx feeds 0, B, N or B + N into the adder according to the a i and q i bits. Mx 2 feeds N (for calclation of N +B)orthe, most significant bits of the reslt pls the least significant reslt bit of the next nit (division by two / shift right) back into the adder. Mx Res selects either the reslt of this nit or the one to the left to be stored into ResltReg. Theoretically the implementation of the mltiplexers and decoders wold cost additional 4 +4CLBs. The possibility of re sing registers for combinatorial logic allows some savings of CLBs. Mx B and Mx Res are implemented in the CLBs of BReg and ResltReg, Mx and Mx 2 partially in NReg and B+NReg. The resltingcostsare approximately 3+4 CLBs per bit processing nit. We compare this expense to the resorces needed for a one bit nit implementation. The B + N register wold not be needed, as a ripple carry adder for sch a small adder makes no sense. We wold need a total of seven bit register space (N, B, a i, q i, control(2) and reslt) and a 4bit inpt 3 bit otpt (2 carries, reslt) adder. Together with one or two CLBs for decoding the control word and mltiplexing, we wold have a total of 6 or 7 CLBs per nit. A device that spports sch a 024 bit implementation wold need 6:5æ 0 3 to 7:5æ 0 3 CLBs, inclding overhead. 4.3 Modlar Mltiplication Figre 2 shows how the processing elements are connected to an array for compting an n bit modlar mltiplication. Starting at the rightmost nit 0, the control word, a i,andq i are fed into their registers. The adder comptes AddReg2 pls B/N/B +N in one clock cycle according to N_In B_In q_i, a_iin Carry_In Res_0_Ot Reslt_Ot Unit_(n/) Units_(n/)..2 N_In Control_ot q_i, a_iot Carry_Ot Res_0_In Reslt_In B_In Unit_ N_Bs B_Bs q_i, a_iin Carry_In Res_0_Ot Reslt_Ot N_In B_In Control_ot q_i, a_iot Carry_Ot a_iin q_iin Res_0_In Res_0_Ot Reslt_In Reslt_Ot Unit_0 Figre 2. Systolic Array for modlar mltiplication N_In B_In a_in Reslt_Ot a i and q i. The least significant bit of the reslt is read back as q i+ for the next comptation. The reslting carry bit, the control word, a i and q i are pmped into the nit to the left, where the same comptation takes place in the next clock cycle. In sch a systolic fashion the control word, a i, q i, and the carry bits are pmped from right to left throgh the whole nit array. The division by two in Algorithm 2 leads also to a shift right operation. The least significant bit of a nit s addition (Res 0 ) is always fed back into the nit to the right. After a modlar mltiplication is completed, the reslts are pmped from left to right throgh the nits and consectively stored in RAM for frther processing. A single processing element comptes bits of R i+ = èr i + a i æ B + q i æ N è=2 of Algorithm 2. In clock cycle i, nit 0 comptes bits 0 :::, of R i.incyclei +, nit ses the reslting carry and comptes bits :::2, of R i. Unit 0 ses the right shifted (division by 2) bit of R i (Res 0 ) to compte bits 0 :::, of R i+ in clock cycle i +2. Clock cycle i +is nprodctive in nit 0 while waiting for the reslt of nit. This inefficiency is avoided by compting sqares and mltiplications in parallel according to Algorithm 2. Both p i+ and z i+ depend on z i.wetherefore store the intermediate reslt z i in the B Registers and feed z i and p i into the a i inpt of the nits for sqaring and mltiplication. 4.4 Modlar Exponentiation Figre 3 shows how the array of nits is tilized for modlar exponentiation. First, the exponent E and the pre comptation factor 2 2n+6 mod N are read from I/O and stored into RAM (Exp and Prec). Then the modls N is read from I/O and fed on the bit wide N bs to the N registers of the nits. These steps have to be exected only if the system parameters need to be changed. Next we read the X vale from I/O, bits per clock cycle, and store it into the dal port (DP) RAM Z. Atthesame time the precomptation factor 2 2n+6 mod N is read from Prec RAM and fed bits per clock cycle via the B bs to the B registers of the nits. 5
6 X_In N_In Prec_In E_In N_In Units_(n/)...0 B_In a_iin DP RAM X Reslt_Ot Shift X TDM Prec RAM DP RAM Z State machine Exp RAM Shift Z Figre 3. Design for a modlar exponentiation Exection of Algorithm begins in parallel to the reading of X. Initially we have P 0 = and Z 0 = X. First we mltiply both vales by the pre comptation factor 2 2n+6 mod N. This is done by time mltiplexing X and ; 0 :::0 in the time division mltiplexing nit (TDM), pmping the reslt as a i into the nits and mltiplying it by 2 2n+6 mod N that is already stored in the B registers. The reslts of the two pre comptations are stored into DP RAM Z and DP RAM P. Sqaring is now straightforward: The intermediate reslt Z i is always stored into the B registers and into DP RAM Z and fed via a i back into the nits. Mltiplication is done almost the same way. P i+ is always compted by feeding P i into the nits, bt the reslt is stored into DP RAM P only if the exponent e i is eqal to. In this way always the last stored P i is pmped back into the nits. To eliminate the factor 2 n+3 (see Section 4.3) from the reslt P n, we compte a final Montgomery mltiplication with inpts P n and. 0; 0;:::0; is stored via the B bs into the B registers, P n is fed from DP RAM P as a i into the nits. A fll modlar exponentiation is compted in 2èn + 2èèn+4èclock cycles. That is the delay it takes from inserting the first bits of X into the device ntil the first reslt bits appear at the otpt. At that point, another X vale can enter the device. With a latency of n= clock cycles the last bits appear on the otpt bs. 5 Methodology In or implementation we adopted the following design flow approach that reslted in fast verification of gate level netlists as well as back annotated designs:. Design entry 2. Logic verification 3. Synthesis 4. Place and Rote 5. Timing Verification The entire design, with the exception of vendor specific soft macros, was entered in VHDL format. Once the design was developed in VHDL, boolean logic and major timing errors were verified by simlating the gate level description with Synopsys VHDL analyzer (vhdlan) and VHDL debgger (vhdldbx) version The next step involved the synthesis of the VHDL code with Synopsys Design Compiler (fpga analyzer) version The otpt of this step was an optimized netlist describing the gate level design in XILINX format. The most time consming step was the compilation of the synthesized design with the place and rote tools available from Xilinx. This process was accomplished with the XILINX Design Manager tools version M.5.9. The final step of the design flow was to verify the design once again bt this time with the physical net, CLB, and pad delays introdced when the design was placed into a specific device. This was accomplished with the same test benches and simlation models that were sed dring the logic verification stage. Synopsys (vhdldbx) was sed once again to verify backannotated designs. The timing reslts from Section 6 were all compted by the Xilinx timing analyzer and verified by the Synopsis vhdl debgger. They were not verified with an actal chip. 6 Reslts 6. Modlar Exponentiation We implemented or design for varios bit lengths and nit widths. Table shows or reslts in terms of sed CLBs (C), clock cycle time (T) and the time area prodct (TA). 256 bit 52 bit C T TA C T TA [CLBs] [ns] [CLB æ ns] [CLBs] [ns] [CLB æ ns] bit 024 bit C T TA C T TA [CLBs] [ns] [CLB æ ns] [CLBs] [ns] [CLB æ ns] Table. CLB sage, minimal clock cycle time, and time area prodct of modlar exponentiation architectres on Xilinx FPGAs 6
7 The majority of CLBs is expended in the nits. In Section 4.2 we derived an approximation of 3 +4CLBs per nit. The overhead consists mainly of RAM, dal port RAM, shift registers, conters and the state machine. An n bit RAM is implemented in n=32 CLBs, a dal port RAM in n=6 CLBs. Conters and their decoding for addressing RAM and dal port RAM are more costly for larger designs. On the other hand, we sed the same state machine for all designs in Table. The clock cycle time T in Table is the propagation delay from BReg throgh Mx and the carries of the adder to the registered carry, pls the setp time of the flipflop. We compare this delay to the optimal cycle time calclated by the Xilinx timing analyzer; for a 4 bit nit the delay with optimal roting is 0.5ns (256 and 52 bit designs) and 2.7ns (768 and 024 bit designs); for an 8 bit nit.2ns and 3.7ns and for a 6 bit nit 2.8ns and 5.5ns. The larger designs were implemented in larger FPGA devices featring different delay specifications. Otherwise we expect the same cycle times for designs with the same nit size. The additional roting delay is between 50% and 80% above the optimal propagation delay. For designs p to 768 and 024 ( =4) bits it remains approximately constant; it deteriorates for 024 bit designs with nit sizes =8and = 6. The same can be said abot the place and rote time: we experienced rn times of a cople of hors on a AMD K6 2/300 MHz PC for designs p to 768 and 024 ( =4) bits, p to a week for the 024 ( =8and =6) bit designs. Different design methods, sch as hard macros for a single nit, wold probably improve roting delay and place and rote time. The time area prodct shows that designs with 8 bit nits are generally most efficient. 52 bit 768 bit 024 bit C T C T C T CLBs [ms] CLBs [ms] CLBs [ms] Table 2. CLB sage and exection time for a fll modlar exponentiation Table 2 shows the application of or reslts to pblic key schemes where the Chinese remainder theorem cannot be applied. A fll modlar exponentiation with an n bit exponent is compted in 2èn + 2èèn +4èclock cycles. 6.2 Application to RSA Table 3 shows or reslts from the tables above, applied to RSA. The encryption time is calclated for the F 4 exponent, reqiring 2 æ 9èn +4èclock cycles. Using the F 4 exponent, only one mltiplication can be calclated in parallel to a sqaring. 52 bit 024 bit C T C T CLBs [ms] CLBs [ms] Table 3. Application to RSA: Encryption For decryption we apply the Chinese remainder theorem. We either decrypt n bits with an n=2 bit architectre serially, or with two n=2 bit architectres in parallel. The first approach ses only half as many resorces, the later is twice as fast. 52 bit 52 bit 024 bit 024 bit 2 æ 256 serial 2 æ 256 parallel 2 æ 52 serial 2 æ 52 parallel C T C T C T C T CLBs [ms] CLBs [ms] CLBs [ms] CLBs [ms] Table 4. Application to RSA: Decryption 6.3 Comparison and Otlook We compare or fastest RSA 52/024 bit designs of Table 4 to the fastest soft and hardware soltions we fond in the literatre [7, 3, 2]. Or 2.37ms decryption time is abot for times faster than the 52 bit software implementation (9.ms) on a 50MHz Alpha [3]. The fastest 024 bit software implementation [2] of 43.3ms rnning on a PPro 200 based PC is abot 4 times slower than or best reslt (0.2ms). The fastest reported hardware design [7] (.7ms for a 52 bit modls and 5.2ms for a 970 bit modls) is a factor.4/.7 faster than ors (9.ms for a 970 bit modls). A drawback of the soltion in [7] is, however, that the binary representation of the modls is hardwired into the logic representation so that the architectre has to be reconfigred with every new modls. The ser of sch an implementation needs to own the fll development tools for synthesis, placing and roting of FPGAs, if RSA with different modli shold be exected. Or design stores the modls, the exponent and the pre comptation factor in registers and RAM. A second advantage of or design is that it is implemented into one device instead of a matrix of 6 devices. Using crrently available FPGA technology, however, the design [7] wold probably also fit in a single device. 7
8 To improve or design in terms of speed, three approaches can be taken:. Comptation of one bit per processing nit (25% improvement estimated). 2. Montgomery mltiplication with a radix r =2 ; ç 2. Comptation of a fll modlar exponentiation in Oèn 2 =è cycles instead of Oèn 2 è. Both approaches have the major disadvantage that considerably more resorces will be sed. We will concentrate or ftre research on trying to implement a higher radix design according to approach 3). The challenge at hand is to accommodate simplifications as proposed in [6] to systolic array and FPGA technology. References [] J. Bajard, L. Didier, and P. Kornerp. An RNS Montgomery modlar mltiplication algorithm. IEEE Transactions on Compters, 47(7):766 76, Jly 998. [2] T. Beth and D. Gollmann. Algorithm engineering for pblic key algorithms. IEEE Jornal on Selected Areas in Commnications, 7(4):458 65, May 989. [3] E. Brickell. A srvey of hardware implementations of RSA. In Advances in Cryptology CRYPTO 89, pages SpringerVerlag, 990. [4] S. E. Eldridge and C. D. Walter. Hardware implementation of Montgomery s modlar mltiplication algorithm. IEEE Transactions on Compters, 42(6): , Jly 993. [5] W. Gai and H. Chen. A systolic linear array for modlar mltiplication. In 2nd International Conference on ASIC, pages 7 4, 996. [6] H.Orp. Simplifying qotient determination in highradix modlar mltiplication. In Proceedings 2th Symposim on Compter Arithmetic, pages 93 9, 995. [7] K. Iwamra, T. Matsmoto, and H. Imai. Montgomery modlarmltiplication method and systolic arrays sitable for modlar exponentiation. Electronics and Commnications in Japan, Part 3, 77(3):40 5, March 994. [8] D. Knth. The Art of Compter Programming. Volme 2: Seminmerical Algorithms. AddisonWesley, Reading, Massachsetts, 2nd edition, 98. [9] P. Kornerp. A systolic, lineararray mltiplier for a class of rightshift algorithms. IEEE Transactions on Compters, 43(8):892 8, Agst 994. [0] P. Montgomery. Modlar mltiplication withot trial division. Mathematics of Comptation, 44(70):59 2, April 985. [] J. Qisqater and C. Covrer. Fast decipherment algorithm for RSA pblic key cryptosystem. Electronics Letters, 8:905 7, October 982. [2] R. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatres and pblic key cryptosystems. Commnications of the ACM, 2(2):20 6, Feb [3] M. Shand and J. Villemin. Fast implementations of RSA cryptography. In Proceedings th IEEE Symposim on Compter Arithmetic, pages , 993. [4] D. R. Stinson. Cryptography, Theory and Practice. CRC Press, 995. [5] N. Takagi. A radix4 modlar mltiplication hardware algorithm efficient for iterative modlar mltiplications. In Proceedings 0th IEEE Symposim on Compter Arithmetic, pages 35 42, 99. [6] A. Tiontchik. Systolic modlar exponentiation via Montgomery algorithm. Electronic Letters, 34(9):874 5, April 998. [7] J. Villemin, P. Bertin, D. Roncin, M. Shand, H. Toati, and P. Bocard. Programmable active memories: Reconfigrable systems come of age. IEEE Transactions on VLSI Systems, 4():56 69, Mar 996. [8] C. Walter. Fast modlar mltiplication sing 2power radix. International Jornal of Compter Mathematics, 39( 2):2 8, 99. [9] C. Walter. Systolic modlar mltiplication. IEEE Transactions on Compters, 42(3):376 8, March 993. [20] P. Wang. New VLSI architectres of RSA pblic key cryptosystems. In Proceedings of 997 IEEE International Symposim on Circits and Systems, volme 3, pages , 997. [2] E. D. Win, S. Mister, B. Preneel, and M. Wiener. On the performance of signatre schemes based on elliptic crves. In Algorithmic Nmber Theory Symposim III, pages SpringerVerlag, 998. [22] Xilinx Inc., San Jose, CA. The Programmable Logic Data Book [23] J. YongYin and W. Brleson. VLSI array algorithms and architectres for RSA modlar mltiplication. IEEE Transactions on VLSI Systems, 5(2):2 7, Jne
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