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ECDSA is unsafe to use if an entropy source produces predictable output for the ephemeral nonces. E.g., [Nguyen]. A simple countermeasure is to hash the secret key, the message, and entropy together to seed a CSPRNG, from which the ephemeral key is derived. -- This is a minimalist (in terms of patch size) solution, though not the most parsimonious in its use of primitives: - csprng_key = ChopMD-256(SHA2-512(priv.D||entropy||hash)) - reader = AES-256-CTR(k=csprng_key) This, however, provides at most 128-bit collision-resistance, so that Adv will have a term related to the number of messages signed that is significantly worse than plain ECDSA. This does not seem to be of any practical importance. ChopMD-256(SHA2-512(x)) is used, rather than SHA2-256(x), for two sets of reasons: *Practical:* SHA2-512 has a larger state and 16 more rounds; it is likely non-generically stronger than SHA2-256. And, AFAIK, cryptanalysis backs this up. (E.g., [Biryukov] gives a distinguisher on 47-round SHA2-256 with cost < 2^85.) This is well below a reasonable security-strength target. *Theoretical:* [Coron] and [Chang] show that Chop-MD(F(x)) is indifferentiable from a random oracle for slightly beyond the birthday barrier. It seems likely that this makes a generic security proof that this construction remains UF-CMA is possible in the indifferentiability framework. -- Many thanks to Payman Mohassel for reviewing this construction; any mistakes are mine, however. And, as he notes, reusing the private key in this way means that the generic-group (non-RO) proof of ECDSA's security given in [Brown] no longer directly applies. -- [Brown]: http://www.cacr.math.uwaterloo.ca/techreports/2000/corr2000-54.ps "Brown. The exact security of ECDSA. 2000" [Coron]: https://www.cs.nyu.edu/~puniya/papers/merkle.pdf "Coron et al. Merkle-Damgard revisited. 2005" [Chang]: https://www.iacr.org/archive/fse2008/50860436/50860436.pdf "Chang and Nandi. Improved indifferentiability security analysis of chopMD hash function. 2008" [Biryukov]: http://www.iacr.org/archive/asiacrypt2011/70730269/70730269.pdf "Biryukov et al. Second-order differential collisions for reduced SHA-256. 2011" [Nguyen]: ftp://ftp.di.ens.fr/pub/users/pnguyen/PubECDSA.ps "Nguyen and Shparlinski. The insecurity of the elliptic curve digital signature algorithm with partially known nonces. 2003" Fixes #9452 Tests: TestNonceSafety: Check that signatures are safe even with a broken entropy source. TestINDCCA: Check that signatures remain non-deterministic with a functional entropy source. Change-Id: Ie7e04057a3a26e6becb80e845ecb5004bb482745 Reviewed-on: https://go-review.googlesource.com/2422 Reviewed-by: Adam Langley <agl@golang.org> |
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