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Enhanced Doublng Attacks on SignedEnhanced Doublng Attacks on Signed-All-Bits Set Recoding-All-Bits Set Recoding
1 Graduate School of Information Management and Security,
Korea University, Korea
http://cist.korea.ac.kr
Hee-seok KimHee-seok Kim11, Tae Hyun Kim, Tae Hyun Kim11, , Jeong Choon RyooJeong Choon Ryoo11, , Dong-Guk HanDong-Guk Han22, , Ho Won KimHo Won Kim22, , and Jongin Limand Jongin Lim11
2 Electronics and Telecommunications Research Institute(ETRI), Korea
http://www.etri.re.kr/
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Side Channel attacks-Power analysisSide Channel attacks-Power analysis Scalar multiplication & Simple power analysis to ECCScalar multiplication & Simple power analysis to ECC Countermeasures & Original Doubling Attack ( DA )Countermeasures & Original Doubling Attack ( DA )
Countermeasure1 - Coron’s dummy method Countermeasure2 - sABS recoding method DA & Weakness of Coron’s dummy method Security of sABS recoding against DA
Proposed AttacksProposed Attacks Recursive attack Initializing attack
Experiments & Statistical approach of noise reductionExperiments & Statistical approach of noise reduction Countermeasures & ConclusionCountermeasures & Conclusion
ContentsContents
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Which are Side Channel Attacks Which are Side Channel Attacks
1. Timing Attacks
- Kocher (1996)
2. Differential Fault Analysis (DFA)
- Biham-Shamir (1997)
3. Simple Power Analysis (SPA)
- Kocher, Jaffe, Jun (1998)
4. Differential Power Analysis (DPA)
- Kocher, Jaffe, Jun (1998)
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Power attacksPower attacks
Kocher et al., June 1998: Measure instantaneous power Kocher et al., June 1998: Measure instantaneous power consumption of a device while it runs a cryptographic algorithmconsumption of a device while it runs a cryptographic algorithm
Different power consumption when operating on logical oDifferent power consumption when operating on logical onnes vs. es vs. logical zeroes.logical zeroes.
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In general, Addition has different power consumption from Doubling. – C. Clavier et al. [3]
Simple Power analysis to ECCSimple Power analysis to ECC
Point Doubling ( D ) : Execution in all bit values of secret key
d : secret exponent
Point Addition ( A ) : Execution when bit value is only ‘1’
D D DAA AD
(2)11101d
General scalar multiplication algorithm
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Countermeasure against SPA-Coron’s methodCountermeasure against SPA-Coron’s method
d 1 1 1 0 1
P 2P 6P 14P
28P
3P 7P 29P
Point Doubling ( D ) , Point Addition ( A ) : Execution in all bit values of secret key
Coron’s dummy method
d 1 1 1 0 1
P 2P 6P 14P
28P
3P 7P 29P
D A D A D D A15P
D A D A D D AA
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Countermeasure against SPA-sABS recodingCountermeasure against SPA-sABS recoding
1 11 111 1111 ..... 1 1where
sABS recoding 1 1 0 0 1 0 1
1 1 1 1 1 1 1
d 1 1 1 -1 -1 1 -1
P 2PD
6PD
14PD
26PD
50PD
102PD
3PA
7PA
13PS
25PS
51PA
101PS
D : Doubling, A : Addition, S : Subtraction
The power consumption of
Addition is similar to that of
Subtraction !!
It’s secure against original
SPA .
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Doubling Attack ( DA ) – Fouque et al.Doubling Attack ( DA ) – Fouque et al.
Characteristics
Assumption Attacker has an ability to decide whether A=B or not when a smartcard computes ECDBL(A) and ECDBL(B).
When input values are P and 2P, Coron’s dummy method carries out the same doubling in the vicinity of the bit value ‘0’.
Attack method d 1 0 1 0 0 1
PPP
2P3P
4P5P
10P11P
2OP21P
40P41P
2P2P2P
4P6P
8P10P
20P22P
40P42P
80P82P
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Doubling Attack ( DA ) – Fouque et al.Doubling Attack ( DA ) – Fouque et al.
P
2P
D A D A D A D A
D A D A D A D A
Key : 1 Key : 1 00 11 00 . . . . . . ..
= =≠
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Security of sABS recoding against DASecurity of sABS recoding against DA
Characteristics
Because sABS recoded value has not ‘0’ bit, it is secure against original DA
Example
d 1 1 -1 1 -1 -1
PPP
2P3P
6P5P
10P11P
22P21P
42P41P
2P2P2P
4P6P
12P10P
20P22P
44P42P
84P82P
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Characteristics
Feasible attack – Supporting a concrete method for experiment
Object New power attacks on scalar multiplication using recoding countermeasures (sABS recoding)
Proposed ‘initializing attack’ - Combination of ‘doubling attack’ and ‘Goubin’s attack’
SPA-based attacks on one-bit of key
Proposed attacksProposed attacks
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Proposed attack 1 - Recursive AttackProposed attack 1 - Recursive Attack
Object New power attack on scalar multiplication using recoding countermeasures (sABS recoding)
If an attacker knows upper n bits of secret key, he can find the upper (n+1)-th bit by this attack. By this method, attacker can find all bits of secret key in sequence.
Characteristic
An attacker that knows upper n bits of secret key ( = d’ ) selects two inputs A, B for originating same ECDBL in the vicinity of upper (n+1)-th bit ( = t ) .
A = d’P, B = (2d’+1) P if t = 1, (2d’+1)A = d’B if t = -1, (2d’+1)A ≠d’B
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Proposed attack 1 - Recursive AttackProposed attack 1 - Recursive Attack
d 1 1 -1 1 1 1 -1
A = d’P, B = (2d’+1) P if t = 1, (2d’+1)A = d’B if t = -1, (2d’+1)A ≠d’B
d’=11
11P 11P 22P33P
66P55P
110P121P
242P253P
506P517P
1034P1023P
23P 23P 46P69P
138P115P
230P253P
506P529P
1058P1081P
2162P2139P
1
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Proposed attack 2 - Initializing AttackProposed attack 2 - Initializing Attack
An attacker that knows upper n bits of secret key ( = d’ ), he selects one input A for originating ECDBL(P) in the upper (n+1)-th bit ( = t ) .
A = (2d’+1)-1P if t = 1, (2d’+1)A = P if t = -1, (2d’+1)A ≠P An attacker acquires the first doubling signal-
ECDBL(P) in the signal according to input point ‘P’. the first doubling signal-ECDBL(P) in the signal according to input point ‘P’ compares with the (n+1)-th doubling signal-ECDBL(P) in the power signal according to input point ‘(2d’+1)-1P’
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Proposed attack 2 - Initializing AttackProposed attack 2 - Initializing Attack
d 1 1 -1 1 1 1 -1
d’=11
54P 54P 35P16P
32P51P
29P10P
20PP
2P56P
39P20P
1
A = (2d’+1)-1P if t = 1, (2d’+1)A = P if t = -1, (2d’+1)A ≠PThe order
of curve : 73 (2*11+1)-1 mod 73
= 54
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Experiments & Statistical approach of noise Experiments & Statistical approach of noise reductionreduction
SettingSetting
PIC Microcontroller Power supply – 5VFunction generator – 1MHz
Oscilloscope
…………
Splitting 1 trace into n- 1 pieces
1 ECDBL+1 ECADD
…………
INPUT : P
INPUT : P
…………
INPUT : Q
Disc. Disc. Disc. Disc.Disc. Disc. Disc. Disc.
Disc. Disc. Disc. Disc.Disc. Disc. Disc. Disc.
X1
X2
Ambiguous area
k
1m 2m1a1b
=m
X1 X2
k points
Experiments & Statistical approach of noise Experiments & Statistical approach of noise reductionreduction
21 2 1 2
1
1.( , , ) ( ( ) ( ))
k
j
Disc S S t S t j S t jk
21 1
2 1
( 1)a b
m km m
Ambiguous area
k
1m 2m1a1b 2a 2b
Eliminateambiguous area
1m 2m
=m
X1 X2 X1 X2
Experiments & Statistical approach of noise Experiments & Statistical approach of noise reductionreduction
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………………
………………
………………
INPUT : 3P
Key : 1 1 -Key : 1 1 -1 . . . .1 . . . .
1 2 1 1
1 1 2 1
a m bmD
a b m m
KeKeyy
1 ??
Disc < D
1
INPUT : P
1 ??
INPUT : 7P
Disc > D
-1 k pointsuk points
Experiments & Statistical approach of noise Experiments & Statistical approach of noise reductionreduction
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Countermeasures & ConclusionCountermeasures & Conclusion
Characteristics of proposed attacksCharacteristics of proposed attacks These new attacks is applicable to sABS recoding countermeasure. These new attacks is applicable to sABS recoding countermeasure. SPA-based attacks on one-bit of key.SPA-based attacks on one-bit of key. Initializing attack is more powerful than Goubin’s attack.Initializing attack is more powerful than Goubin’s attack.
CountermeasuresCountermeasures Using a Projective coordinates – affine coordinates is not secure.Using a Projective coordinates – affine coordinates is not secure. BRIP can be applied to our attacks [BRIP can be applied to our attacks [13]13] .
