The Improved Montgomery Scalar Multiplication Algorithm with DPA Resistance Yanqi Xu, Lin Chen, Moran Li

Transcription

1 nd Internatonal Conference on Electrcal, Computer Engneerng and Electroncs (ICECEE 015) The Improved Montgomery Scalar Multplcaton Algorthm wth DPA Resstance Yanq Xu, Ln Chen, Moran L Informaton Scence and Technology Insttute, Zhengzhou, 45000, Chna e-mal: xyq_1@163com Keywords: ECC, DPA, Montgomery scalar multplcaton algorthm, securty system Abstract Wth Montgomery scalar multplcaton algorthm beng wdely used n ellptc curve cryptography systems, the researches on DPA attacks aganst Montgomery scalar multplcaton algorthm become more extensve, but few researches are about the DPA resstance algorthm Ths paper analyzed the DPA resstance capablty of the Montgomery scalar multplcaton algorthm frstly, and on ths bass, proposed an mproved algorthm wth random Z coordnate to resst the DPA attacks wth the characterstcs that the parameter Z only partcpates n the ntermedate operatons wthout affectng the fnal results Ths paper descrbed the orgnal algorthm and the mproved algorthm usng Verlog HDL targetng on the 65 nm standard cell lbrary Results show that the mproved Montgomery Scalar Multplcaton algorthm can dramatcally mprove the ant-dpa attack property wth only 036% performance and area 960% area penalty Introducton Ellptc Curve Cryptography (ECC)[1] s superor to other publc key cryptographes such as RSA n hgh bt strength, excellent calculaton speed, and small storage space Encrypton, decrypton, sgnng, verfcaton and other algorthms are all based on the scalar multplcaton of ponts on the ellptc curve[], so the calculaton speed and the ablty of attack resstance of the scalar multplcaton wll determne the calculaton capablty and safety of the whole curve cryptosystem The technology of Sde Channel Attacks posed a huge threat to the crypto chp because of ts smple attack equpment and measure Power Analyss Attacks[3] s a common technology of Sde Channel Attacks, ncludng Smple Power Analyss (SPA) and Dfferental Power Analyss (DPA) The desgn and mplementaton of scalar multplcaton algorthm should also consder to the threat of SPA and the DPA attacks to mprove the securty of the algorthm Reference[4] proved that Montgomery Ladder Scalar Multplcaton Algorthm can t resst the DPA attacks Reference [5] gave a way to resst the DPA by ntroducng random redundant operaton for decreasng S/N, but ths method largely ncreased the crcut area and power consumpton Ths paper proposed an mproved algorthm of random Z coordnate to resst the DPA attacks combnng the random thought wth the Montgomery Scalar Multplcaton Algorthm, and smulated the DPA resstance capablty of the mproved algorthm The Securty Analyss of the Montgomery Scalar Multplcaton Algorthm The Montgomery scalar multplcaton algorthm[6] s wdely used n engneerng feld because of ts hgh calculaton speed and small storage space Ths scalar multplcaton algorthm s descrbed based on projecton coordnate The algorthm s showed n Algorthm The authors - Publshed by Atlants Press 533

2 Algorthm 1: Montgomery Scalar Multplcaton Input: k = ( kt 1,, k1, k0), k t-1 =1, P= ( xy, ) E( m ) Output: kp 4 1 X1 xz, 1 1, X x + bz, x { ( P, P )} For from t to 0, do 1 If k = 1,then T Z, Z ( X Z + X Z ), X xz + X X TZ ,, T X X X + bz Z T Z else T Z, Z ( X Z + X Z ), X xz + X X TZ T X1, X1 X1 + bz1, Z1 T Z 1 3 x3 X1 / Z1 4 y = ( x + X / Z )[( X + xz )( X + xz ) ( x + y)( ZZ )]( xzz ) + y 5 Return ( x3, y 3) The SPA and the tme attack requre that the key has an extraordnary nfluence on power consumpton The attacker needs to grasp the exhaustve mplementaton detals and then surmse the relatve key nformaton related to operaton[7], accordng to the metrc power consumpton track and the analyss of one moment s arthmetc operatons of encrypton devces along the tme axs Through the Montgomery scalar multplcaton algorthm s second step we can fnd, ether k = 1or k = 0, the algorthm wll execute the same calculaton so we can t surmse the key by drectly analyzng the power consumpton track So ths algorthm can resst the SPA and the tme attack The DPA can utlze the data dependence by large amount of energy track to analyze the devce s power consumpton of the fxed tme Then the attacker obtans the nvarants of the calculaton process(such as key s nformaton)[7] So as to analyss convenently, we use ths form whch s equal to the second step of Algorthm 1 as follows: If k = 1, then P1 P1+ P, P P If k = 0, then P P1+ P, P1 P1 Because kt-1 = 1, P1 = P, P = P, the next, when kt- = 0, P1 = P, P = 3P; when kt- = 1, P1 = 3P, P = 4P Thus we can see that the dfferent value of kt- can generate dfferent mddle consequence, and dfferent mddle consequence wll generate dfferent power consumpton whch can be utlzed by DPA Document [4] smulated the DPA attacks aganst Montgomery scalar multplcaton algorthm n the envronment of EDA and proved that ths algorthm can t resst the DPA attacks The Improved Algorthm of Montgomery Scalar Multplcaton Wth DPA Resstance Montgomery scalar multplcaton algorthm can be realzed by projecton coordnates--turn nverson to multplcaton to mprove the effcency of the algorthm The result of operaton s unrelated to the value of Z Combnng wth ths character, ths artcle added a random number generator n the algorthm s hardware mplementaton, and the generator can generate a random Z before every dot product, thus the ntermedate result ntroduces a random ngredent because of Z Power consumpton s related to data and n ths algorthm even we perform twce dentcal nput operatons the power consumpton won t be the same, so t can resst the DPA attacks The mproved algorthm of random Z coordnate s as follows: 534

3 Algorthm :The Improved Montgomery Scalar Multplcaton Input: k = ( kt 1,, k1, k0), k t-1 =1, P= ( xy, ) E( m ) k<n Output: kp X1 xz1, Z1 random, X X1 + bz1, Z X1Z1 For from t to 0, do 1 If k = 1,then T Z, Z ( X Z + X Z ), X xz + X X TZ ,, T X X X + bz Z T Z else T Z, Z ( X Z + X Z ), X xz + X X TZ T X1, X1 X1 + bz1, Z1 T Z 1 3 x3 X1 / Z1 4 y = ( x + X / Z )[( X + xz )( X + xz ) ( x + y)( ZZ 1 )]( xzz 1 ) + y 5 Return ( x3, y 3) We used hardware to realze Algorthm n the doman of GF(19), and adapt the mproved algorthm of Montgomery FIOS whch s sutable for hardware to realze the core multplcaton unt We adapt sngle port wrtng and dual port readng regster fle to nject parameter and dspatch the data convenently In order not to affect the operatonal effcency, the random generator and the arthmetc unt perform n parallel and store the random number for the next round operaton The hardware archtecture of random Z coordnate mproved algorthm s as follows: Key Reg Man Control Unt Random Number Generator F S M Wrtng Control Unt Regster Fle T y x Z X Z1 X1 Readng Readng Control Control Unt 1 Unt Modular Multplcaton Unt Modular Inverse Unt Modular Add Unt Modular Arthmetc Control Unt Fg 1 The hardware of random Z coordnate mproved algorthm Ths paper compled the hardware module n the 018um technology by DC and compared wth Algorthm One The result s as follows: Table I The Area and Effcency s Comparson Between Algorthm 1 and Algorthm Algorthm 1 Algorthm Comparson Gates Increased by 960% Clock Cycle Increased by 036% Algorthm adds the random number generator, so the area s ncreased by 960% Because the random number generatng and the mathematcal unt don t affect the effcency, the algorthm only adds twce modulus square operatons and three tmes modulus multplcaton, the effcency of the algorthm s only decreased by 036% Test the DPA Resstance Capablty of the Improved Algorthm To test the DPA resstance ablty of the mproved algorthm, ths paper attacked the mproved algorthm by DPA attacks n the envronment of EDA Processes are as follows: 1 Buld power analyss smulaton system Frst, we descrbed the 19-bt length algorthm wth Verlog HDL, and wrote the testbench, then ntegrated the hardware descrpton fle n the envronment of DC tool to generate gate-level netlst fle, and then changed the gate-level netlst fle 535

4 and testbench nto vcd fle through NC smulaton At last, we analyzed the power consumpton of the vcd fle wth PrmePower and recorded the result Collect the data Based on the Power Consumpton Data Acquston System n Step 1, we used fxed key to encrypt the 1000 random ponts P, P,, P separately and recorded the 1000 groups power consumpton data S0[ j], S1[ j],, S999[ j] ( j s the sample ponts) 3 Generate dscernblty functon Known k = 1 191, we could speculate k = 1 190, and other bts are any k g = (1,1,x,,x,x) kg value that the speculated key s Then we used Modelsm to calculate the and the 1000 random ponts P to get a group of dscernblty functon values D, D,, D Deal wth the data We used D to dvde the power consumpton curve n Step nto two groups accordng to these standards: S0 = { S[ j] D=0} S1 = { S[ j] D=1} Then we calculated every sample pont s average value of two group s power consumpton curve and subtract [ j] = E( S ) E( S ) = S [ j] S[ j] S 1 S S S 1 0 S S0 We can repeat these steps and get the power consumpton s dfference curves of the orgnal Montgomery scalar multplcaton algorthm and the mproved algorthm As shown n Fgure and Fgure 3 (a) The dfference curve of power consumpton when guessng the key rght (b) The dfference curve of power consumpton when guessng the key wrong Fgure The result of DPA s attack to Montgomery scalar multplcaton algorthm 536

5 (a) The dfference curve of power consumpton when guessng the key rght (b) The dfference curve of power consumpton when guessng the key wrong Fgure 3 The DPA attack to the mproved algorthm basng on random Z coordnate Comparng the fgure (a) and fgure (b) we can see that there are apparent peaks when the speculated key s rght and there sn t any peak when the speculated key s wrong, so we can conclude that the Montgomery scalar multplcaton algorthm can t resst DPA attacks and we can attack the algorthm one bt by one bt for rght key accordng to the above steps From the Fgure 3 we can see there won t be any peak when the speculated key s rght or wrong and we can conclude that the random Z coordnate can change the related features of the key and the power consumpton so as to dsable statstcal means and resst the DPA attacks n the end Concluson Ths paper analyzed the safety of the Montgomery scalar multplcaton algorthm, proposed an mproved algorthm basng on random Z coordnate Its ablty of DPA resstance s verfed by EDA smulaton tools Results show that the mproved Montgomery Scalar Multplcaton algorthm can dramatcally mprove the ant-dpa attack property Otherwse, many scalar multplcaton algorthms adopt standard projecton coordnate, Jacob projecton coordnates and Lopez-Dahab projecton coordnates to mprove the effcency Because all these projecton coordnates use the parameter Z only partcpates n the ntermedate operatons wthout affectng the fnal results, we can transplant the random Z coordnate nto other algorthms basng on projecton coordnate Consequently, the mproved algorthm has mportant practcal value 537

6 References [1] Kobltz N Ellptc curve crytosystems[j] Mathematcs of Computaton, vol48, pp 03-09,1987 [] Chung S C, Lee J W, Chang H C, et al A hgh-performance ellptc curve cryptographc processor over GF (p) wth SPA resstance [C]//Crcuts and Systems (ISCAS), 01 IEEE Internatonal Symposum on IEEE,pp ,01 [3] Kocherp, Jaffe J, Jun B Dfferental Power Analyss[C] Proceedngs of Advances n CRYPTO 99, LNCS 1666 Sprnger-Verlag, Berln Hedelberg, pp , 1999 [4] Deng Qu Cheng, Ba Xue Fe, Guo L Research on Dfferental Power Analyss Attack on ECC Algorthm[J] Mcroelectoncs & Computer, vol,8(),pp 3-5, 011 [5] Dan Yong Png Chp Implementaton and t s Securty Defence for Ellptc Curve Cryptosystems over GF(m)[D] Wuhan: Huazhong Unversty of Scence and Technology, 008 [6] Darrel Hankerson, Alfred Menezes, Scott Vanstone Gude To Ellptc Curve Cryptography[M] Zhang Huan Guo, transl Bejng: Publshng House of Electroncs Industry, 005 [7] Stefan Mangard, Elsabeth Oswald, Thomas Popp Power Analyss Attacks Feng Deng Guo, Zhou Yong Bn, Lu J Ye, transl Bejng:Scence Press,