8 + Introduction Higher Order Masking Side Channel Cryptanalysis of a Higher Order Schemes Generic Masking Scheme Scheme Improved Scheme Experimental Results J.-S. Coron 1 E. Prouff 2 M. Rivain 1 , 2 Conclusion 1 University of Luxembourg 2 Oberthur Card Systems CHES 2007 Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 1 / 23
Contents 8 + Introduction Introduction Higher Order 1 Masking Schemes Generic Higher Order Masking Schemes 2 Scheme Improved Scheme Generic Scheme 3 Experimental Results Conclusion Improved Scheme 4 Experimental Results 5 Conclusion 6 Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 2 / 23
Differential Power Analysis (DPA) 8 + Introduction Higher Order Masking Schemes The physical leakage of the execution of any algorithm Generic Scheme depends on the intermediate variables Improved Scheme DPA exploits leakage on sensitive variables that depends Experimental Results on the secret key Conclusion Common countermeasure: masking ◮ A random value is added to every sensitive variable ◮ ⇒ Instantaneous leakage independent of sensitive variables Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 3 / 23
Higher Order DPA (HO-DPA) Against First Order Masking 8 + Y : sensitive variable, M : mask Introduction ◮ Y ⊕ M processed at t 0 Higher Order Masking ◮ M processed at t 1 Schemes Generic Scheme First order DPA attack not feasible Improved Second order DPA attack feasible Scheme Experimental Results Conclusion Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 4 / 23
Higher Order DPA (HO-DPA) Against d -th Order Masking 8 + Y : sensitive variable, M i ’s: masks Introduction ◮ Y ⊕ M 1 ⊕ · · · ⊕ M d processed at t 0 Higher Order Masking ◮ M i ’s processed at t i Schemes Generic d -th order DPA attack not feasible Scheme Improved ( d + 1) -th order DPA attack feasible Scheme Experimental Results Conclusion Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 5 / 23
Higher Order DPA (HO-DPA) 8 + Introduction Higher Order Masking Schemes Generic The complexity of an HO-DPA is exponential with its Scheme order (Chari et al. in CRYPTO’99) Improved Scheme The order d is a good security parameter Experimental Results Conclusion A generic masking scheme must ◮ involve d random masks per sensitive variable ◮ thwart d -th order DPA Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 6 / 23
Higher Order Masking Schemes 8 + Introduction Higher Order Masking Schemes Formalizing the security: Generic Scheme sensitive variable : depends on both the plaintext and the Improved Scheme secret key Experimental Results Conclusion Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 7 / 23
Higher Order Masking Schemes 8 + Introduction Higher Order Masking Schemes Formalizing the security: Generic Scheme sensitive variable : depends on both the plaintext and the Improved Scheme secret key Experimental Results d -th order flaw : a d -tuple of intermediate variables Conclusion statistically dependent on a sensitive variable Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 7 / 23
Higher Order Masking Schemes 8 + Introduction Higher Order Masking Schemes Formalizing the security: Generic Scheme sensitive variable : depends on both the plaintext and the Improved Scheme secret key Experimental Results d -th order flaw : a d -tuple of intermediate variables Conclusion statistically dependent on a sensitive variable security against d -th order DPA : no d -th order flaw Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 7 / 23
Higher Order Masking Schemes 8 + Introduction Higher Order Masking Each sensitive variable Y is masked with d masks M i ’s Schemes Generic Scheme completeness : the masked variable M V and the masks Improved M i ’s must always satisfy: Scheme Experimental Results M V ⊕ M 1 ⊕ · · · ⊕ M d = Y Conclusion security : M V and all the M i ’s must be processed separately Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 8 / 23
Higher Order Masking Schemes 8 + In network of linear layers and non-linear SBoxes Introduction Higher Order ◮ Propagation through a linear layer Masking Schemes Generic Scheme M V M 1 · · · M d Improved Scheme Experimental Results L L L Conclusion L ( M V ) L ( M 1 ) L ( M d ) · · · Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 9 / 23
Higher Order Masking Schemes 8 + In network of linear layers and non-linear SBoxes Introduction Higher Order ◮ Propagation through a linear layer Masking Schemes Generic Scheme = Y M V M 1 · · · M d Improved Scheme Experimental Results L L L Conclusion L ( M V ) L ( M 1 ) L ( M d ) · · · Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 9 / 23
Higher Order Masking Schemes 8 + In network of linear layers and non-linear SBoxes Introduction Higher Order ◮ Propagation through a linear layer Masking Schemes Generic Scheme = Y M V M 1 · · · M d Improved Scheme Experimental Results L L L Conclusion L ( M V ) L ( M 1 ) L ( M d ) = L ( Y ) · · · Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 9 / 23
Higher Order Masking Schemes 8 + In network of linear layers and non-linear SBoxes Introduction Higher Order ◮ Propagation through a non-linear SBox Masking Schemes Generic Scheme = Y M V M 1 · · · M d Improved Scheme Experimental Results S S S Conclusion S ( M V ) S ( M 1 ) S ( M d ) � = S ( Y ) · · · Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 9 / 23
Higher Order Masking Schemes 8 + In network of linear layers and non-linear SBoxes Introduction Higher Order ◮ Propagation through a non-linear SBox Masking Schemes Generic Scheme = Y M V M 1 · · · M d Improved Scheme Experimental Results ?? ?? ?? Conclusion = S ( Y ) N V N 1 · · · N d Problem How to securely compute ( N V , N ′ i s ) from ( M V , M ′ i s ) ? Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 9 / 23
Higher Order Masking Schemes 8 + Introduction Higher Order Problem widely investigated for 1 -st order masking Masking Schemes ◮ Efficient and widely used method: the table Generic re-computation Scheme Improved Scheme For d -th order masking: one single proposal in the Experimental Literature Results ◮ [SP06] - K. Schramm and C. Paar, “Higher Order Masking Conclusion of the AES” in CT-RSA 2006. ◮ Principle: adapt the table re-computation method to d -th order masking Our paper: [SP06] is broken by 3-rd order DPA for any value of the masking order d Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 10 / 23
Table re-computation method For 1 -st order masking 8 + Introduction = Y M V M 1 Higher Order Masking Schemes Generic Scheme S ∗ S re-computation Improved Scheme Experimental Results N 1 ← rand () Conclusion For all x : S ∗ ( x ) ← S ( x ⊕ M 1 ) ⊕ N 1 Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 11 / 23
Table re-computation method For 1 -st order masking 8 + Introduction = Y M V M 1 Higher Order Masking Schemes Generic Scheme S ∗ Improved Scheme Experimental Results Conclusion N V N 1 For all x : S ∗ ( x ) ← S ( x ⊕ M 1 ) ⊕ N 1 N V ← S ∗ ( M V ) Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 11 / 23
Table re-computation method For 1 -st order masking 8 + Introduction = Y M V M 1 Higher Order Masking Schemes Generic Scheme S ∗ Improved Scheme Experimental Results = S ( Y ) Conclusion N V N 1 For all x : S ∗ ( x ) ← S ( x ⊕ M 1 ) ⊕ N 1 N V ← S ∗ ( M V ) = S ( M V ⊕ M 1 ) ⊕ N 1 = S ( Y ) ⊕ N 1 Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 11 / 23
Table re-computation method For d -th order masking [SP06] 8 + Introduction M V M 1 · · · M d = Y Higher Order Masking Schemes Generic Scheme S ∗ S d-th order re-computation Improved Scheme Experimental Results N 1 · · · N d Conclusion � � x ⊕ � d ⊕ � d For every x : S ∗ ( x ) = S i =1 M i i =1 N i Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 12 / 23
Table re-computation method For d -th order masking [SP06] 8 + Introduction M V M 1 · · · M d = Y Higher Order Masking Schemes Generic Scheme S ∗ Improved Scheme Experimental Results N V N 1 · · · N d = S ( Y ) Conclusion � � x ⊕ � d ⊕ � d For every x : S ∗ ( x ) = S i =1 M i i =1 N i Coron - Prouff - Rivain (UoL, OCS) S.C. Cryptanalysis of a H.O. Masking CHES 2007 12 / 23
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