Recovering OpenSSL ECDSA Nonces Using the FLUSH+RELOAD Cache Side-channel Attack

We illustrate a vulnerability introduced to elliptic curve cryptographic protocols when implemented using a function of the OpenSSL cryptographic library. For the given implementation using an elliptic curve E over a binary field with a point G ∈ E, our attack recovers the majority of the bits of a scalar k when kG is computed using the OpenSSL implementation of the Montgomery ladder. For the Elliptic Curve Digital Signature Algorithm (ECDSA) the scalar k is intended to remain secret. Our attack recovers the scalar k and thus the secret key of the signer and would therefore allow unlimited forgeries. This is possible from snooping on only one signing process and requires computation of less than one second on a quad core desktop when the scalar k (and secret key) is around 571 bits.

[1]  Nicolas Thériault,et al.  Solving Discrete Logarithms from Partial Knowledge of the Key , 2007, INDOCRYPT.

[2]  Anne Canteaut,et al.  Understanding cache attacks , 2006 .

[3]  Michael J. Wiener,et al.  Faster Attacks on Elliptic Curve Cryptosystems , 1998, Selected Areas in Cryptography.

[4]  Igor E. Shparlinski,et al.  The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces , 2003, Des. Codes Cryptogr..

[5]  Daniel M. Gordon,et al.  A Survey of Fast Exponentiation Methods , 1998, J. Algorithms.

[6]  Serge Vaudenay,et al.  Security Flaws Induced by CBC Padding - Applications to SSL, IPSEC, WTLS , 2002, EUROCRYPT.

[7]  Onur Aciiçmez,et al.  Yet another MicroArchitectural Attack:: exploiting I-Cache , 2007, CSAW '07.

[8]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[9]  No License,et al.  Intel ® 64 and IA-32 Architectures Software Developer ’ s Manual Volume 3 A : System Programming Guide , Part 1 , 2006 .

[10]  Yuval Yarom,et al.  FLUSH+RELOAD: A High Resolution, Low Noise, L3 Cache Side-Channel Attack , 2014, USENIX Security Symposium.

[11]  Billy Bob Brumley,et al.  Remote Timing Attacks Are Still Practical , 2011, ESORICS.

[12]  Risto M. Hakala,et al.  Cache-Timing Template Attacks , 2009, ASIACRYPT.

[13]  Augustus K. Uht,et al.  Disjoint eager execution: an optimal form of speculative execution , 1995, Proceedings of the 28th Annual International Symposium on Microarchitecture.

[14]  Nigel P. Smart,et al.  Lattice Attacks on Digital Signature Schemes , 2001, Des. Codes Cryptogr..

[15]  Dawn Xiaodong Song,et al.  Timing Analysis of Keystrokes and Timing Attacks on SSH , 2001, USENIX Security Symposium.

[16]  Xiong Li,et al.  Improvement of trace-driven I-Cache timing attack on the RSA algorithm , 2013, J. Syst. Softw..

[17]  Tibor Juhas.  The use of elliptic curves in cryptography , 2007 .

[18]  Kouichi Sakurai,et al.  Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications , 2000, Public Key Cryptography.

[19]  Hovav Shacham,et al.  Hey, you, get off of my cloud: exploring information leakage in third-party compute clouds , 2009, CCS.

[20]  Tanja Lange,et al.  The Security Impact of a New Cryptographic Library , 2012, LATINCRYPT.

[21]  Taher El Gamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, IEEE Trans. Inf. Theory.

[22]  尚弘 島影.  National Institute of Standards and Technologyにおける超伝導研究及び生活 , 2001 .

[23]  Michael K. Reiter,et al.  Cross-VM side channels and their use to extract private keys , 2012, CCS.

[24]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[25]  Jacco Krijnen,et al.  Software Reverse Engineering , 2022 .

[26]  Leonard M. Adleman,et al.  A Subexponential Algorithm for Discrete Logarithms over All Finite Fields , 1993, CRYPTO.

[27]  D. Shanks.  Class number, a theory of factorization, and genera , 1971 .

[28]  Onur Aciiçmez,et al.  New Results on Instruction Cache Attacks , 2010, CHES.

[29]  Scott A. Vanstone,et al.  Improving the parallelized Pollard lambda search on anomalous binary curves , 2000, Math. Comput..

[30]  R. Balasubramanian,et al.  The Improbability That an Elliptic Curve Has Subexponential Discrete Log Problem under the Menezes—Okamoto—Vanstone Algorithm , 1998, Journal of Cryptology.

[31]  Marc Joye,et al.  The Montgomery Powering Ladder , 2002, CHES.

[32]  Onur Aciiçmez,et al.  A Vulnerability in RSA Implementations Due to Instruction Cache Analysis and Its Demonstration on OpenSSL , 2008, CT-RSA.

[33]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[34]  Kenneth G. Paterson,et al.  Plaintext-Recovery Attacks Against Datagram TLS , 2012, NDSS.

[35]  Alfred Menezes,et al.  The Elliptic Curve Digital Signature Algorithm (ECDSA) , 2001, International Journal of Information Security.

[36]  Helmut Hasse.  The Class Number , 1980 .

[37]  Adi Shamir,et al.  Efficient Cache Attacks on AES, and Countermeasures , 2010, Journal of Cryptology.

[38]  C. D. Walter,et al.  Longer Keys May Facilitate Side Channel Attacks , 2003, Selected Areas in Cryptography.

[39]  P. L. Montgomery.  Speeding the Pollard and elliptic curve methods of factorization , 1987 .

[40]  Stephan Krenn,et al.  Cache Games -- Bringing Access-Based Cache Attacks on AES to Practice , 2011, 2011 IEEE Symposium on Security and Privacy.