A Practical DPA Countermeasure with BDD Architecture


 Georgiana Fisher
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1 A Prcticl DPA Countermesure with BDD Architecture Toru Akishit, Msnou Ktgi, Yoshikzu Miyto, Asmi Mizuno, nd Kyoji Shiutni System Technologies Lortories, Sony Corportion, 7 Konn, Mintoku, Tokyo 875, Jpn Astrct. We propose logiclevel DPA countermesure clled Dulril Prechrge circuit with Binry Decision Digrm rchitecture (DP BDD). The proposed countermesure hs dulril prechrge logic style nd cn e implemented using CMOS stndrd cell lirries, which is the similr property to Wve Dynmic Differentil Logic (WDDL). By using novel pproches, we cn successfully reduce the erly propgtion effect, which is one of the min fctors of DPA lekge of WDDL. DP BDD is suited to implementtion of Soxes. In our implementtions of the AES Sox, DPBDD cn reduce the mximum difference of trnsition timing t outputs of Sox to out /6.5 compred to tht of WDDL without dely djustment. Moreover, y pplying simple dely djustment to the inputs of the Sox, we cn reduce it to out /85 of tht without the djustment. We consider DPBDD is prcticl nd effective DPA countermesure for implementtion of Soxes. Keywords: DPA, countermesure, dulril prechrge logic, Binry Decision Digrm Introduction Differentil Power Anlysis (DPA) is serious thret to cryptogrphic devices such s smrt crds [8]. Recently, vrious countermesures hve een proposed to protect implementtions of cryptogrphic lgorithms ginst DPA t the logic level. Since the logiclevel countermesures cn e dpted to sic logicl gtes such s n AND gte, we cn pply them to implementtions of ny cryptogrphic lgorithms. These logiclevel countermesures re clssified into the following three groups: msking logics, dulril prechrge logics, nd hyridtype logics. Msking logics try to rndomize the ctivity t every node in circuit using rndom vlues in order to remove correltion etween keyrelted intermedite vlues nd power consumption of the circuit. MskedAND, type of msking logics, ws proposed y Trichin [2]. It hs een pointed out, however, tht MskedAND is not completely secure due to the effect of glitches [9, 4]. Recently, Rndom Switching Logic (RSL) ws proposed y Suzuki et l. [6]. RSL
2 is theoreticlly secure under the lekge models descried in [4], ut possesses two disdvntges: one is tht it cnnot e implemented using CMOS stndrd cell lirries nd the other is tht it reuires creful timing djustment of enle signls. A dulril prechrge logic ws first proposed y Tiri et l. s Sense Amplifier Bsed Logic (SABL) [7], where signl is represented y two complementry wires nd one of these two wires is chrged nd dischrged in every cycle. Considering tht SABL needs specil CMOS lirry, Tiri et l. lso proposed Wve Dynmic Differentil Logic (WDDL) [8] tht cn e implemented using CMOS stndrd cell lirries. WDDL is prcticl countermesure, ut it cnnot suppress two fctors of DPA lekge. The first one is due to unlnced lod cpcitnce of complementry logic gtes. In order to lnce it, WDDL reuires custom lyout for secure design [9, 7]. The other is due to the erly propgtion effect. This lekge is cused when input signls of WDDL gte hve difference of dely time [4]. The input signls generlly pss the different numer of logic gtes, nd then the difference of dely time inevitly occurs. Creful dely djustment cn reduce the difference, ut pplying it ll WDDL gtes in cryptogrphic circuits seems to e unrelistic. Hyridtype logics re comined with msking logics nd dulril prechrge logics. At CHES 25, Popp nd Mngrd proposed MDPL tht comines dulril prechrge circuits with rndom msking to improve the first disdvntge of WDDL []. They climed tht it cn chieve secure design using CMOS stndrd cell lirry without specil lyout constrint, ut in the next yer it ws pointed out tht MDPL cn e still insecure when there is reltively lrge difference in dely time etween the input signls of MDPL gtes [4, 5]. In ddition, the comintion of msking nd dulril ws shown to e unle to provide routinginsensitive logic style [6, 3]. At present, hyridtype logics seem to hve no dvntge over dulril prechrge logics. In this pper, we propose novel DPA countermesure clled Dulril Prechrge circuit with Binry Decision Digrm rchitecture (DPBDD). It is sed on Binry Decision Digrm (BDD) tht is direct cyclic grph used to represent Boolen function. DPBDD is composed of ANDOR gtes which re included in CMOS stndrd cell lirries. Due to the sed BDD rchitecture, the input signls of n ANDOR gte lwys pss the sme numer of ANDOR gtes, nd then the erly propgtion effect, which is one of the min fctors of DPA lekge of WDDL, is significntly reduced. This DPA countermesure is suited to implementtion of Soxes. In our implementtions of the AES [] Sox, DPBDD cn reduce the mximum difference of trnsition timing t the outputs of the Sox to out /6.5 compred to tht of WDDL without dely djustment. Moreover, y pplying simple dely djustment to the inputs of the Sox, we cn reduce it to out /85 of tht without the djustment. DPBDD reuires custom lyout to prevent the lekge cused y unlnced lod cpcitnce of complementry logic gtes the sme s WDDL, ut we consider tht DPBDD is prcticl nd effective DPA countermesure for implementtion of Soxes.
3 The rest of the pper is orgnized s follows: Section 2 presents WDDL nd its security prolem. Section 3 gives rief introduction of BDD tht is the sic rchitecture of our method. In Section 4 we present the proposed DPA countermesure clled DPBDD. In Section 5, we pply WDDL nd DPBDD to implementtions of AES Sox nd compre their effectiveness. We introduce simple dely djustment of DPBDD to reduce the difference of trnsition timing further in Section 6. Finlly we drw our conclusion nd discuss further work in Section 7. 2 Wve Dynmic Differentil Logic (WDDL) Tiri et l. proposed Wve Dynmic Differentil Logic (WDDL) s logiclevel countermesure of DPA [8]. WDDL hs the following fetures: WDDL gtes hve complementry inputs nd outputs. WDDL hs the prechrge phse to trnsmit (, ) nd the evlution phse to trnsmit (, ) or (, ), nd performs these phses mutully. WDDL cn construct comintionl logics y using only AND gtes, OR gtes, nd NOT opertions (signl swpping). A vlue is represented (, ) in WDDL, where is the negtion of. An ctivity fctor within WDDL circuits is constnt independent of the input signls due to the ove fetures. Since power consumption t CMOS gtes generlly depends on the trnsition proility of the gtes, WDDL is considered to e effective s DPA countermesure. However, the power consumption t CMOS gtes lso depends on lod cpcitnce of the gtes. If there is difference of lod cpcitnce etween complementry logic gtes of WDDL, the difference of power consumption occurs. The numer of gtes connected to complementry logic gtes of WDDL is siclly eul, nd then the difference of lod cpcitnce is cused y the difference of plcendroute. The lekge due to the plcendroute is clled s incidentl lekge [5]. It cn e improved y the plcendroute in the mnul or semiutomtic opertion using specil constrints such s Ft Wire [9] nd Bckend Dupliction [7]. Another lekge is due to the erly propgtion effect [4, 5]. This lekge is cused when there is the difference of dely time etween the input signls of WDDL gte. In Fig., we illustrte WDDL AND gte nd its signl trnsitions ccording to the inputs (, ). Here, we ssume tht the trnsition of or reches the gte erlier thn the trnsition of or oth on the evlution phse nd on the prechrge phse. The trnsition timing of the complementry output or on the evlution phse depends on the input. On the other hnd, the trnsition timing of or on the prechrge phse depends on the input. Therefore, the difference of dely time etween the inputs nd my lek the vlues nd. Since sic cryptogrphic components including Soxes of lockciphers reuire complicted logic circuits, the input signls of WDDL gte generlly pss different numer of logic gtes. Therefore, the difference of
4 WDDL AND gte (, )=(, ) evlution (, )=(, ) evlution prechrge prechrge (, )=(, ) evlution (, )=(, ) evlution prechrge prechrge Fig.. The erly propgtion effect of WDDL AND gte dely time etween these signls inevitly occurs. This type of lekge is clled s inevitle lekge [5]. The lekge cn e improved y djusting dely time etween the input signls, ut very high effort nd mny constrints in the circuit design re reuired to djust dely time of ll WDDL gtes in complicted logic circuits including Soxes. 3 Binry Decision Digrm A Binry Decision Digrm (BDD) is direct cyclic grph tht is used to represent Boolen function [], nd one of most commonly used synthesis tools for logic optimiztion of digitl systems [22]. We riefly explin BDD ccording to Fig. 2. The left figure is truth tle representing the function f(a, B, C) nd the right figure shows lock digrm of inry decision tree corresponding to the truth tle. In the right figure, n isosceles trpezoid represents 2to multiplexer, nd we cll signl A, B, C s nonterminl node, signl,,, t the lowest prt s terminl node, nd signl connecting two multiplexers s n internl node. The outputs f in the truth tle re locted in regulr order from the left to the right of terminl nodes. Generlly the term BDD refers to Reduced Ordered Binry Decision Digrm (ROBDD) [2]. A inry decision tree is uniuely trnsformed into ROBDD y merging ny isomorphic sugrphs nd eliminting ny redundnt nodes. In this pper, however, we cll s BDD the lock digrm in which we only merge ny isomorphic sugrphs on inry decision tree. In this BDD rchitecture, since the sme numer of multiplexers must e pssed from ny terminl node to the output, the difference of propgtion dely dependent of inputs is reltively smll.
5 input output A B C f nonterminl node A B C f output terminl node Fig. 2. A truth tle nd inry decision tree 2to multiplexer c ANDOR c Fig. 3. A 2to multiplexer nd n ANDOR gte 4 DulRil PreChrge Circuit with Binry Decision Digrm Architecture In this section, we propose novel DPA countermesure to reduce the inevitle lekge t logic level, clled Dulril Prechrge circuit with Binry Decision Digrm rchitecture (DPBDD). It is sed on BDD nd constructed in the following steps. Prechrged ANDOR gtes. We void the existence of glitches to control the trnsition proility of ll signls in BDD circuit. In order to prevent glitches, we firstly replce 2to multiplexers in BDD to 2wy 2nd 4input ANDOR (shortly, ANDOR) gtes. As shown in Fig. 3, n ANDOR gte is euivlent to 2to multiplexer except the negtion of ect signl eing input. Fig. 4() shows modified BDD circuit. In the figure n isosceles trpezoid represents n ANDOR gte. Nonterminl nodes (A, A), (B, B), or (C, C) re connected to ech ANDOR gte s (, ) in Fig. 3. Next, we pply soclled prechrge mechnism to the terminl nodes (, ) nd the nonterminl nodes (A, A), (B, B), (C, C); these signls re set to on the prechrge phse nd evlute to the corresponding vlue on the evlution
6 f f f f A, A A, A A, A B, B B, B B, B C, C C, C C, C () BDD circuit () complementry BDD circuit (c) DPBDD Fig. 4. Constructing DPBDD phse. We consider the output of n ANDOR gte t the lowest stge. On the evlution phse, ll four inputs of n ANDOR gte perform either ( ) or ( ), then the output lso performs either ( ) or ( ). On the prechrge phse, ll four inputs perform either ( ) or ( ), then the output lso performs either ( ) or ( ). By dpting these trnsitions to the inputs of ANDOR gtes t the next stge, we cn confirm tht ll internl nodes nd outputs of BDD hve t most one trnsition oth on the evlution phse nd on the prechrge phse. Therefore, we cn prevent glitches in the BDD circuit. Appending complementry circuit. Preventing glitches doesn t give ny gurntee to DPA resistnce ecuse the distriution of the trnsition ctivity depends on the inputs A, B, C. In order to mke it independent of the inputs, we construct the complementry BDD circuit to the originl BDD circuit. It cn e simply creted y exchnging nd which re input to the terminl nodes s shown in Fig. 4(). By ppending the complementry circuit to the originl circuit nd merging them s shown in Fig. 4(c), one of the complementry AND OR gtes perform trnsition oth on the evlution phse nd on the prechrge phse. Therefore, the ctivity fctor within the merged circuit is constnt independent of the input signls. We cll such merged circuit s Dulril Prechrge circuit with Binry Decision Digrm rchitecture (DPBDD). We consider the inevitle lekge, which is lekge cused y the difference of dely time etween the input signls of complementry ANDOR gtes shown in Fig. 5. We ssume tht ll inputs of DPBDD, nonterminl nodes nd terminl nodes, re directly connected to registers nd hve no propgtion dely except their setup time. By inputting rndom it m nd its negtion m to the terminl nodes insted of nd, ll internl nodes nd output of DPBDD re esily msked y m. The ddition of rndom msking, however, does not chieve secure design without specil lyout constrint ccording to the oservtion in [6, 3].
7 Fig. 5. Complementry ANDOR gtes The difference of dely time etween input signls of ANDOR gtes my led the difference of trnsition timing t the output which depends on some secret informtion. Since signls nd re directly connected to inputs of DPBDD, the trnsition of nd occurs soon fter the trnsition from the prechrge phse to the evlution phse, nd the reverse trnsition. On the prechrge phse, the trnsition of or occurs t the time when the trnsition of or whether = or. On the evlution phse, if =, the trnsition of the output signl or occurs t the time when the trnsition of the input or occurs; if =, the trnsition of or occurs t the time when the trnsition of the input or occurs. Therefore, the difference of dely time etween nd (or nd ) my lek the vlue on the evlution phse. However, since the signls nd (or nd ) pss the sme numer of ANDOR gtes, the difference of dely time etween these signls is reltively smll, nd then detecting the inevitle lekge y DPA is more difficult. 5 Appliction to AES Sox In order to protect hrdwre implementtions of the Advnced Encryption Stndrd (AES) [], the Sox is the most criticl opertion ecuse it is the only nonliner opertion in AES. In this section, we pply oth WDDL nd DPBDD to implementtions of AES Sox, nd compre their effectiveness. 5. AES Sox sed on WDDL (WDDL Sox) There re vrious wys to implement the AES Sox. The most compct implementtion of AES Sox is tht using composite fields [2, 2, 3]. We pply WDDL to the AES Sox descried in [2], whose overll mount of gtes is 3 XORs + 57 ANDs, ecuse of its reltively short criticl pth. Fig. 6 shows the schemtic circuit of AES Sox using composite fields. There re severl opertions including n isomorphic mpping, multiplictions nd dditions over Glois field. We notice pth nd pth 2 which oth re the pths to the multipliction circuit over GF(2 4 ). Pth hs reltively short propgtion dely ecuse it psses only the isomorphic mpping circuit. On the other hnd, pth 2 hs long propgtion dely ecuse it psses lso the suring, constnt multipliction, ddition, nd inversion circuits over GF(2 4 ) except the isomorphic mpping circuit. Thus, since the difference of dely time etween pth nd 2 re lrge, we guess the inevitle lekge cused y this difference cn e detected y DPA.
8 in [7] isomorphism pth 2 x 2 X x λ pth x  X X  isomorphism x ffine out [7] Fig. 6. AES Sox using composite fields out[] out[] out[7] out[] out[] out[7] in[7], in[7] ORNAND x 6 in[6], in[6] 2 ANDNOR x 6 in[5], in[5] 4 ORNAND x 6 in[4], in[4] 8 ANDNOR x 6 in[3], in[3] 6 ORNAND x 6 in[2], in[2] 2 ANDNOR in[], in[] 6 ORNAND in[], in[] 4 ANDNOR Fig. 7. AES Sox sed on DPBDD (DPBDD Sox) 5.2 AES Sox sed on DPBDD (DPBDD Sox) Since the AES Sox hs n 8it input nd n 8it output, we firstly rrnge eight inry decision trees of eight stges ccording to the truth tles of AES S ox. Then, AES Sox sed on DPBDD (DPBDD Sox) cn e constructed in the wy descried in Section 4. Fig. 7 shows the constructed DPBDD Sox, where in[i] denotes ith it of the input of the Sox nd out[i] denotes ith it of the output. In CMOS positive gte is usully constructed out of negtive gte nd n inverter, nd then the use of positive gtes is disdvntge in terms of gte size. In order to reduce the gte size of DPBDD Sox, we replce ANDOR gtes to ANDNOR gtes t the odd stges nd to ORNAND gtes t the even stges, nd then the input of ORNAND gtes re prechrged to on the prechrge phse. Its overll mount of gtes is 374 ANDNORs ORNANDs. Since ny pth from the terminl node nd to two input signls of n ANDNOR/ORNAND
9 2 WDDL out[6] DPBDD out[3] 5 Propgtion Dely [nsec] Input of Sox Fig. 8. Propgtion dely of n output it of WDDL Sox nd DPBDD Sox gte psses the sme numer of ANDNOR/ORNAND gtes, the difference of dely time etween the input signls of the gte is reltively smll. 5.3 Experimentl Results We implemented oth WDDL Sox nd DPBDD Sox, nd performed netlist timing simultions to evlute their effectiveness. The environment of our evlution is s follows: Lnguge VerilogHDL Design Lirry.8 µm CMOS stndrd cell lirry Simultor VCS version 26.6 Logic Synthesis Design Compiler version 26.6 One gte is euivlent to 2wy NAND nd the speed is evluted under the worstcse conditions. In the lirry, n AND/OR gte, n ANDOR/ORAND gte, nd n ANDNOR/ORNAND gte re euivlent to 5/4 gtes, 9/4 gtes, nd 7/4 gtes, respectively. These simultions re sed on prerouting dely, nd then free from the incidentl lekge cused y the utomtiztion of the plcendroute. We firstly evlute the gte counts of WDDL Sox nd DPBDD Sox. An AND gte in the AES Sox is implemented using n AND gte nd n OR gte in WDDL Sox s shown in Fig., while n XOR gte in the AES Sox cn e implemented using n ANDOR gte nd n ORAND gte. Thus the gte count of WDDL Sox is euivlent to 3 9/ /2 = 66 excluding uffers. On the other hnd, the gte count of DPBDD Sox is euivlent to 374 7/ /4 = 27 excluding uffers.
10 Next, we evlute the difference of trnsition timing t the output of logic gtes in oth WDDL Sox nd DPBDD Sox. Since we guessed the lrgest difference will occur t the output of the Sox, we serched the output it of Sox tht hs the lrgest difference of trnsition timing for ll possile 256 Sox inputs; out[6] (or out[6]) nd out[3] (or out[3]) re the corresponding its of WDDL Sox nd DPBDD Sox respectively. Fig. 8 shows the propgtion dely of these its for ll 256 inputs; the ove line shows tht of WDDL S ox nd the elow line shows tht of DPBDD Sox. We confirmed tht the mximum difference of trnsition timing t the output of DPBDD Sox (.526 ns) is out /6.5 of tht of WDDL Sox (9.855 ns). 6 Towrds Less Difference of Trnsition Timing DPBDD reduces the difference of trnsition timing t the output of ANDOR gtes. It is, however, desirle to reduce this difference ll the more since it could e detected y DPA. We consider tht the difference occurs y the ccumultion of the following fctors: difference of propgtion dely etween input ports of ech ANDOR gte, difference of lod cpcitnce etween input ports of ech ANDOR gte, difference of the numer of fnout etween output signls of ANDOR gtes. In order to reduce the influence of these fctors, we pply dely djustment to inputs of DPBDD shown in Fig. 9. On the prechrge phse, we don t reuire ny dely djustment cell ecuse the difference of trnsition timing t the output of ech ANDOR gte is euivlent to the difference of propgtion dely etween input port of the ANDOR gte. On the evlution phse, we insert dely cells of dely(), dely(), nd dely(c) to (A, A), (B, B), nd (C, C) respectively. By inserting the dely cell of dely(c) to (C, C), trnsition of the output of ANDOR gtes t stge occurs t the time when trnsition of C or C reches their input ports. Next, we set dely() tht stisfies dely() dely(c) is lrger thn the propgtion dely from ny input ports of ANDOR gtes t stge to ny input ports of AND OR gtes t stge 2. Tht indictes tht trnsition of the output of ANDOR gtes t stge 2 occurs t the time when trnsition of B or B reches their input ports. Similrly, we set dely() tht stisfies dely() dely() is lrger thn the propgtion dely from ny input ports of ANDOR gtes t stge 2 to ny input ports of ANDOR gtes t stge 3. Therefore, we cn reduce the difference of trnsition timing t the outputs of ll ANDOR gtes to the difference of propgtion dely etween input port of the ANDOR gte lso on the evlution stge. It is very esy to stisfy these dely conditions ecuse we hve only to mke the difference of dely etween ny two djcent its of the input sufficiently lrge. By switching the input signls without dely nd those with dely using AND gtes, we cn successfully reduce the difference of trnsition timing t ll signls
11 A, A dely() f f stge 3 B, B dely() stge 2 C, C dely(c) stge Fig. 9. Dely djustment for DPBDD in DPBDD in oth the prechrge stge nd the evlution stge. We confirmed tht this dely djustment reduced the mximum difference of trnsition timing in DPBDD Sox to.8 ns (out /85 of tht without dely djustment), which is just the difference of propgtion dely etween the input ports nd of n ORNAND gte. 7 Conclusion In this pper we presented the logiclevel DPA countermesure clled DPBDD. DPBDD hs dulril logic style nd cn e implemented using CMOS stndrd cell lirries. Our experimentl results showed tht DPBDD cn significntly reduce the difference of trnsition timing t the outputs of AES Sox compred to WDDL. We consider tht DPBDD is prcticl nd effective DPA countermesure for implementtions of Soxes. At CHES 26, Homm et l. presented highresolution wveform mtching sed on PhseOnly Correltion (POC) techniues nd its ppliction to DPA [5]. They climed tht the POCsed techniues cn evlute the displcement etween signl wveforms with higher resolution thn the smpling resolution. One of further works we need to crry out is how lrge difference of the dely time etween the input signls leds to DPA lekge in rel devices using such techniues. References. S.B. Akers, Binry Decision Digrm, IEEE Trns. on Computers, Vol.C27, No.6, pp.5956, R.E. Brynt, GrphBsed Algorithm for Boolen Function Mnipultion, IEEE Trns. on Computers, Vol.C35, No.8, pp , D. Cnright, A Very Compct SBox for AES, CHES 25, LNCS 3659, pp , SpringerVerlg, 25.
12 4. Z. Chen nd Y. Zhou, DulRil Rndom Switching Logic: A Countermesure to Reduce Side Chnnel Lekge, CHES 26, LNCS 4249, pp , Springer Verlg, N. Homm, S. Ngshim, Y. Imi, T. Aoki, nd A. Stoh, HighResolution SideChnnel Attck Using PhseBsed Wveform Mtching, CHES 26, LNCS 4249, pp.872, SpringerVerlg, B. Gierlichs, DPAResistnce Without Routing Constrints?, CHES 27, LNCS 4727, pp.72, SpringerVerlg S. Guilley, P. Hoogvorst, Y. Mthieu, nd R. Pclet, The Bckend Dupliction Method, CHES 25, LNCS 3659, pp , SpringerVerlg, P. Kocher, J. Jffe, nd B. Jun, Differentil Power Anlysis, Crypto 99, LNCS 666, pp , SpringerVerlg, S. Mngrd, T. Popp, nd B.M. Gmmel, SideChnnel Lekge of Msked CMOS Gtes, CTRSA 25, LNCS 3376, pp , SpringerVerlg, 25.. Ntionl Institute of Stndrd nd Technology (NIST), Advnced Encryption Stndrd (AES), FIPS Puliction 97, 2.. T. Popp nd S. Mngrd, Msked DulRil PreChrge Logic: DPAResistnt without Routing Constrints, CHES 25, LNCS 3659, pp.7286, Springer Verlg, A. Stoh, S. Moriok, K. Tkno, nd S. Munetoh, A Compct Rijndel Hrdwre Architecture with Sox Optimiztion, ASIACRYPT 2, LNCS 2248, pp , SpringerVerlg, P. Schumont nd K. Tiri, Msking nd DulRil Logic Don t Add Up, CHES 27, LNCS 4727, pp.956, SpringerVerlg, D. Suzuki, M. Seki, nd T. Ichikw, DPA Lekge Models for CMOS Logic Circuits, CHES 25, LNCS 3659, pp , SpringerVerlg, D. Suzuki nd M. Seki, Security Evlutions of DPA Countermesures Using DulRil PreChrge Logic Style, CHES 26, LNCS 4249, pp , Springer Verlg, D. Suzuki, M. Seki, nd T. Ichikw, Rndom Switching Logic: A New Countermesure ginst DPA nd SecondOrder DPA t the Logic Level, IEICE Trnsctions 9A(), pp.668, K. Tiri, M. Akml, nd I. Veruwhede, A Dynmic nd Differentil CMOS Logic with Signl Independent Power Consumption to Withstnd Differentil Power Anlysis on Smrt Crds, ESSCIRC 22, pp.4346, K. Tiri nd I. Veruwhede, A Logic Level Design Methodology for A Secure DPA Resistnt ASIC or FPGA Implementtion, DATE 24, pp , K. Tiri nd I. Veruwhede, Plce nd Route for Secure Stndrd Cell Design, CARDIS 24, pp.4358, E. Trichin, Comintionl Logic Design for AES SuByte Trnsformtion on Msked Dt, IACR Cryptology eprint Archive 23/236, J. Wolkerstorfer, E. Oswld, nd M. Lmerger, An ASIC Implementtion of the AES Soxes, CTRSA 22, LNCS 227, pp.6778, SpringerVerlg, C. Yng, M. Ciesielski, V. Singhel, BDS: A BDD Bsed Logic Optimiztion System, Proc. of the 37th ACM/IEEE DAC 2, pp , 2.
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