Paper 2012/318

Non-uniform cracks in the concrete: the power of free precomputation

Daniel J. Bernstein and Tanja Lange


AES-128, the NIST P-256 elliptic curve, DSA-3072, RSA-3072, and various higher-level protocols are frequently conjectured to provide a security level of 2^128. Extensive cryptanalysis of these primitives appears to have stabilized sufficiently to support such conjectures. In the literature on provable concrete security it is standard to define 2^b security as the nonexistence of high-probability attack algorithms taking time \le 2^b. However, this paper provides overwhelming evidence for the existence of high-probability attack algorithms against AES-128, NIST P-256, DSA-3072, and RSA-3072 taking time considerably below 2^128, contradicting the standard security conjectures. These attack algorithms are not realistic; do not indicate any actual security problem; do not indicate any risk to cryptographic users; and do not indicate any failure in previous cryptanalysis. Any actual use of these attack algorithms would be much more expensive than the conventional 2^128 attack algorithms. However, this expense is not visible to the standard definitions of security. Consequently the standard definitions of security fail to accurately model actual security. The underlying problem is that the standard set of algorithms, namely the set of algorithms taking time \le 2^b, fails to accurately model the set of algorithms that an attacker can carry out. This paper analyzes this failure in detail, and analyzes several ideas for fixing the security definitions.

Note: Major revision. Abbreviated version will appear at Asiacrypt 2013; this is the full version.

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Publication info
A major revision of an IACR publication in ASIACRYPT 2013
provable securityconcrete securityalgorithm cost metricsnon-uniform algorithmsnon-constructive algorithms
Contact author(s)
tanja @ hyperelliptic org
2013-09-14: last of 4 revisions
2012-06-05: received
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      author = {Daniel J.  Bernstein and Tanja Lange},
      title = {Non-uniform cracks in the concrete:  the power of free precomputation},
      howpublished = {Cryptology ePrint Archive, Paper 2012/318},
      year = {2012},
      note = {\url{}},
      url = {}
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