Using Modular Extension to Provably Protect Edwards Curves Against Fault Attacks

International audienceFault injection attacks are a real-world threat to cryptosystems, in particular asymmetric cryptography. In this paper, we focus on countermeasures which guarantee the integrity of the computation result, hence covering most existing and future fault attacks. Namely, we study the modular extension protection scheme in previously existing and newly contributed variants of the countermeasure on elliptic curve scalar multiplication (ECSM) algorithms. We find that an existing countermeasure is incorrect and we propose new " test-free " variant of the modular extension scheme that fixes it. We then formally prove the correctness and security of modular extension: specifically, the fault non-detection probability is inversely proportional to the security parameter. Finally, we implement an ECSM protected with test-free modular extension during the elliptic curve operation to evaluate the efficient of this method on Edwards and twisted Edwards curves

Improvement of side-channel attack on asymmetric cryptography

Depuis les années 90, les attaques par canaux auxiliaires ont remis en cause le niveau de sécurité des algorithmes cryptographiques sur des composants embarqués. En effet, tout composant électronique produit des émanations physiques, telles que le rayonnement électromagnétique, la consommation de courant ou encore le temps d’exécution du calcul. Or il se trouve que ces émanations portent de l’information sur l’évolution de l’état interne. On parle donc de canal auxiliaire, car celui-ci permet à un attaquant avisé de retrouver des secrets cachés dans le composant par l’analyse de la « fuite » involontaire. Cette thèse présente d’une part deux nouvelles attaques ciblant la multiplication modulaire permettant d’attaquer des algorithmes cryptographiques protégés et d’autre part une démonstration formelle du niveau de sécurité d’une contre-mesure. La première attaque vise la multiplication scalaire sur les courbes elliptiques implémentée de façon régulière avec un masquage du scalaire. Cette attaque utilise une unique acquisition sur le composant visé et quelques acquisitions sur un composant similaire pour retrouver le scalaire entier. Une fuite horizontale durant la multiplication de grands nombres a été découverte et permet la détection et la correction d’erreurs afin de retrouver tous les bits du scalaire. La seconde attaque exploite une fuite due à la soustraction conditionnelle finale dans la multiplication modulaire de Montgomery. Une étude statistique de ces soustractions permet de remonter à l’enchaînement des multiplications ce qui met en échec un algorithme régulier dont les données d’entrée sont inconnues et masquées. Pour finir, nous avons prouvé formellement le niveau de sécurité de la contre-mesure contre les attaques par fautes du premier ordre nommée extension modulaire appliquée aux courbes elliptiques.: Since the 1990s, side channel attacks have challenged the security level of cryptographic algorithms on embedded devices. Indeed, each electronic component produces physical emanations, such as the electromagnetic radiation, the power consumption or the execution time. Besides, these emanations reveal some information on the internal state of the computation. A wise attacker can retrieve secret data in the embedded device using the analyzes of the involuntary “leakage”, that is side channel attacks. This thesis focuses on the security evaluation of asymmetric cryptographic algorithm such as RSA and ECC. In these algorithms, the main leakages are observed on the modular multiplication. This thesis presents two attacks targeting the modular multiplication in protected algorithms, and a formal demonstration of security level of a countermeasure named modular extension. A first attack is against scalar multiplication on elliptic curve implemented with a regular algorithm and scalar blinding. This attack uses a unique acquisition on the targeted device and few acquisitionson another similar device to retrieve the whole scalar. A horizontal leakage during the modular multiplication over large numbers allows to detect and correct easily an error bit in the scalar. A second attack exploits the final subtraction at the end of Montgomery modular multiplication. By studying the dependency of consecutive multiplications, we can exploit the information of presence or absence of final subtraction in order to defeat two protections : regular algorithm and blinding input values. Finally, we prove formally the security level of modular extension against first order fault attacks applied on elliptic curves cryptography

Amélioration d'attaques par canaux auxiliaires sur la cryptographie asymétrique

: Since the 1990s, side channel attacks have challenged the security level of cryptographic algorithms on embedded devices. Indeed, each electronic component produces physical emanations, such as the electromagnetic radiation, the power consumption or the execution time. Besides, these emanations reveal some information on the internal state of the computation. A wise attacker can retrieve secret data in the embedded device using the analyzes of the involuntary “leakage”, that is side channel attacks. This thesis focuses on the security evaluation of asymmetric cryptographic algorithm such as RSA and ECC. In these algorithms, the main leakages are observed on the modular multiplication. This thesis presents two attacks targeting the modular multiplication in protected algorithms, and a formal demonstration of security level of a countermeasure named modular extension. A first attack is against scalar multiplication on elliptic curve implemented with a regular algorithm and scalar blinding. This attack uses a unique acquisition on the targeted device and few acquisitionson another similar device to retrieve the whole scalar. A horizontal leakage during the modular multiplication over large numbers allows to detect and correct easily an error bit in the scalar. A second attack exploits the final subtraction at the end of Montgomery modular multiplication. By studying the dependency of consecutive multiplications, we can exploit the information of presence or absence of final subtraction in order to defeat two protections : regular algorithm and blinding input values. Finally, we prove formally the security level of modular extension against first order fault attacks applied on elliptic curves cryptography.Depuis les années 90, les attaques par canaux auxiliaires ont remis en cause le niveau de sécurité des algorithmes cryptographiques sur des composants embarqués. En effet, tout composant électronique produit des émanations physiques, telles que le rayonnement électromagnétique, la consommation de courant ou encore le temps d’exécution du calcul. Or il se trouve que ces émanations portent de l’information sur l’évolution de l’état interne. On parle donc de canal auxiliaire, car celui-ci permet à un attaquant avisé de retrouver des secrets cachés dans le composant par l’analyse de la « fuite » involontaire. Cette thèse présente d’une part deux nouvelles attaques ciblant la multiplication modulaire permettant d’attaquer des algorithmes cryptographiques protégés et d’autre part une démonstration formelle du niveau de sécurité d’une contre-mesure. La première attaque vise la multiplication scalaire sur les courbes elliptiques implémentée de façon régulière avec un masquage du scalaire. Cette attaque utilise une unique acquisition sur le composant visé et quelques acquisitions sur un composant similaire pour retrouver le scalaire entier. Une fuite horizontale durant la multiplication de grands nombres a été découverte et permet la détection et la correction d’erreurs afin de retrouver tous les bits du scalaire. La seconde attaque exploite une fuite due à la soustraction conditionnelle finale dans la multiplication modulaire de Montgomery. Une étude statistique de ces soustractions permet de remonter à l’enchaînement des multiplications ce qui met en échec un algorithme régulier dont les données d’entrée sont inconnues et masquées. Pour finir, nous avons prouvé formellement le niveau de sécurité de la contre-mesure contre les attaques par fautes du premier ordre nommée extension modulaire appliquée aux courbes elliptiques

Amélioration d'attaques par canaux auxiliaires sur la cryptographie asymétrique

: Since the 1990s, side channel attacks have challenged the security level of cryptographic algorithms on embedded devices. Indeed, each electronic component produces physical emanations, such as the electromagnetic radiation, the power consumption or the execution time. Besides, these emanations reveal some information on the internal state of the computation. A wise attacker can retrieve secret data in the embedded device using the analyzes of the involuntary “leakage”, that is side channel attacks. This thesis focuses on the security evaluation of asymmetric cryptographic algorithm such as RSA and ECC. In these algorithms, the main leakages are observed on the modular multiplication. This thesis presents two attacks targeting the modular multiplication in protected algorithms, and a formal demonstration of security level of a countermeasure named modular extension. A first attack is against scalar multiplication on elliptic curve implemented with a regular algorithm and scalar blinding. This attack uses a unique acquisition on the targeted device and few acquisitionson another similar device to retrieve the whole scalar. A horizontal leakage during the modular multiplication over large numbers allows to detect and correct easily an error bit in the scalar. A second attack exploits the final subtraction at the end of Montgomery modular multiplication. By studying the dependency of consecutive multiplications, we can exploit the information of presence or absence of final subtraction in order to defeat two protections : regular algorithm and blinding input values. Finally, we prove formally the security level of modular extension against first order fault attacks applied on elliptic curves cryptography.Depuis les années 90, les attaques par canaux auxiliaires ont remis en cause le niveau de sécurité des algorithmes cryptographiques sur des composants embarqués. En effet, tout composant électronique produit des émanations physiques, telles que le rayonnement électromagnétique, la consommation de courant ou encore le temps d’exécution du calcul. Or il se trouve que ces émanations portent de l’information sur l’évolution de l’état interne. On parle donc de canal auxiliaire, car celui-ci permet à un attaquant avisé de retrouver des secrets cachés dans le composant par l’analyse de la « fuite » involontaire. Cette thèse présente d’une part deux nouvelles attaques ciblant la multiplication modulaire permettant d’attaquer des algorithmes cryptographiques protégés et d’autre part une démonstration formelle du niveau de sécurité d’une contre-mesure. La première attaque vise la multiplication scalaire sur les courbes elliptiques implémentée de façon régulière avec un masquage du scalaire. Cette attaque utilise une unique acquisition sur le composant visé et quelques acquisitions sur un composant similaire pour retrouver le scalaire entier. Une fuite horizontale durant la multiplication de grands nombres a été découverte et permet la détection et la correction d’erreurs afin de retrouver tous les bits du scalaire. La seconde attaque exploite une fuite due à la soustraction conditionnelle finale dans la multiplication modulaire de Montgomery. Une étude statistique de ces soustractions permet de remonter à l’enchaînement des multiplications ce qui met en échec un algorithme régulier dont les données d’entrée sont inconnues et masquées. Pour finir, nous avons prouvé formellement le niveau de sécurité de la contre-mesure contre les attaques par fautes du premier ordre nommée extension modulaire appliquée aux courbes elliptiques

Golden Sequence for the PPSS Broadcast Encryption Scheme with an Asymmetric Pairing

Broadcast encryption is conventionally formalized as broadcast encapsulation in which, instead of a cipher- text, a session key is produced, which is required to be indistinguishable from random. Such a scheme can provide public encryption functionality in combination with a symmetric encryption through the hybrid en- cryption paradigm. The Boneh-Gentry-Waters scheme of 2005 proposed a broadcast scheme with constant-size ciphertext. It is one of the most efficient broadcast encryption schemes regarding overhead size. In this work we consider the improved scheme of Phan-Pointcheval-Shahandashi-Ste er [PPSS12] which provides an adaptive CCA broadcast encryption scheme. These two schemes may be tweaked to use bilinear pairings[DGS]. This document details our choices for the implementation of the PPSS scheme. We provide a complete golden sequence of the protocol with efficient pairings (Tate, Ate and Optimal Ate). We target a 128-bit security level, hence we use a BN-curve [BN06]. The aim of this work is to contribute to the use and the standardization of PPSS scheme and pairings in concrete systems

Correlated Extra-Reductions Defeat Blinded Regular Exponentiation

International audienceWalter & Thomson (CT-RSA '01) and Schindler (PKC '02) have shown that extra-reductions allow to break RSA-CRT even with message blinding. Indeed, the extra-reduction probability depends on the type of operation (square, multiply, or multiply with a constant). Regular exponentiation schemes can be regarded as protections since the operation sequence does not depend on the secret. In this article, we show that there exists a strong negative correlation between extra-reductions of two consecutive operations, provided that the first feeds the second. This allows to mount successful attacks even against blinded asymmetrical computations with a regular exponentiation algorithm, such as Square-and-Multiply Always or Montgomery Ladder. We investigate various attack strategies depending on the context - known or unknown modulus, known or unknown extra-reduction detection probability, etc.-and implement them on two devices: a single core ARM Cortex-M4 and a dual core ARM Cortex M0-M4

Correlated Extra-Reductions Defeat Blinded Regular Exponentiation

International audienceWalter & Thomson (CT-RSA '01) and Schindler (PKC '02) have shown that extra-reductions allow to break RSA-CRT even with message blinding. Indeed, the extra-reduction probability depends on the type of operation (square, multiply, or multiply with a constant). Regular exponentiation schemes can be regarded as protections since the operation sequence does not depend on the secret. In this article, we show that there exists a strong negative correlation between extra-reductions of two consecutive operations, provided that the first feeds the second. This allows to mount successful attacks even against blinded asymmetrical computations with a regular exponentiation algorithm, such as Square-and-Multiply Always or Montgomery Ladder. We investigate various attack strategies depending on the context - known or unknown modulus, known or unknown extra-reduction detection probability, etc.-and implement them on two devices: a single core ARM Cortex-M4 and a dual core ARM Cortex M0-M4

Side-Channel Analysis of SM2: A Late-Stage Featurization Case Study

SM2 is a public key cryptography suite originating from Chinese standards, including digital signatures and public key encryption. Ahead of schedule, code for this functionality was recently mainlined in OpenSSL, marked for the upcoming 1.1.1 release. We perform a security review of this implementation, uncovering various deficiencies ranging from traditional software quality issues to side-channel risks. To assess the latter, we carry out a side-channel security evaluation and discover that the implementation hits every pitfall seen for OpenSSL's ECDSA code in the past decade. We carry out remote timings, cache timings, and EM analysis, with accompanying empirical data to demonstrate secret information leakage during execution of both digital signature generation and public key decryption. Finally, we propose, implement, and empirically evaluate countermeasures.publishedVersionPeer reviewe