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Diffie-Hellman

Subgroup confinement attacks

The papers by van Oorshot and Wiener [OW96] rsp. Lim and Lee [LL98] show that Diffie-Hellman keys can be found much faster if the short exponents are used and if the multiplicative group modulo p contains small subgroups. In particular an attacker can try to send a public key that is an element of a small subgroup. If the receiver does not check for such elements then may be possible to find the private key modulo the order of the small subgroup. Several countermeasures against such attacks have been proposed: For example IKE uses fields of order p where p is a safe prime (i.e. $$q=(p-1)/2),$$ hence the only elements of small order are 1 and p-1.

[NIST SP 800-56A] rev. 2, Section 5.5.1.1 only requires that the size of the subgroup generated by the generator g is big enough to prevent the baby-step giant-step algorithm. I.e. for 80-bit security p must be at least 1024 bits long and the prime q must be at least 160 bits long. A 2048 bit prime p and a 224 bit prime q are sufficient for 112 bit security. To avoid subgroup confinment attacks NIST requires that public keys are validated, i.e. by checking that a public key y satisfies the conditions $$2 \leq y \leq p-2$$ and $$y^q \mod p = 1$$ (Section 5.6.2.3.1). Further, after generating the shared secret $$z = y_a^{x_b} \mod p$$ each party should check that $$z \neq 1.$$ RFC 2785 contains similar recommendations. The public key validation described by NIST requires that the order q of the generator g is known to the verifier. Unfortunately, the order q is missing in [PKCS #3]. [PKCS #3] describes the Diffie-Hellman parameters only by the values p, g and optionally the key size in bits.

The class DHParameterSpec that defines the Diffie-Hellman parameters in JCE contains the same values as [PKCS #3]. In particular, it does not contain the order of the subgroup q. Moreover, the SUN provider uses the minimal sizes specified by NIST for q. Essentially the provider reuses the parameters for DSA.

Therefore, there is no guarantee that an implementation of Diffie-Hellman is secure against subgroup confinement attacks. Without a key validation it is insecure to use the key-pair generation from [NIST SP 800-56A] Section 5.6.1.1 (The key-pair generation there only requires that static and ephemeral private keys are randomly chosen in the range \(1..q-1)\).

To avoid big disasters the tests below require that key sizes are not minimal. I.e., currently the tests require at least 512 bit keys for 1024 bit fields. We use this lower limit because that is what the SUN provider is currently doing.

TODO(bleichen): Find a reference supporting or disproving that decision.

Weak parameters

The DH parameters must be carefully chosen to avoid security issues. A panel at Eurocrypt'92 discussed the possiblity of trapdoors in DL based primitives [Eurocrypt92 panel]. A. Lenstra pointed out that the primes chould be chosen such that the special number field sieve can be used to compute discrete logarithms. Gordon has analyzed methods to generate and detect weak parameters [G92]. Section 4 of Gordons paper describes a method that can detect some special cases, but no general method was given. Recently Fried et al. showed that 1024 bit discrete logarithms with the special number field sieve are feasible [FGHT16]. Moreover some libraries use primes that are susceptible to this attack [FGHT16].

TODO(bleichen): So far not test for weak DH parameters has been implemented. Possibly we should at least implement a test that detects special cases, so that weak primes (such as the one used in libtomcrypt) are detected.

DH implementations are sometimes misconfigured. Adrian et al. [WeakDh] analyzed various implementations and found for example the following problems in the parameters: p is sometimes composite, p-1 contains no large prime factor, q is used instead of the generator g.

References

[Eurocrypt92 panel]: "The Eurocrypt'92 Controversial Issue Trapdoor Primes and Moduli", EUROCRYPT '92, LNCS 658, pp. 194-199.

[G92]: D. M. Gordon. "Designing and detecting trapdoors for discrete log cryptosystems." CRYPTO’92, pp. 66–75.

[FGHT16]: J. Fried, P. Gaudry, N. Heininger, E. Thome. "A kilobit hidden SNFS discrete logarithm computation". http://eprint.iacr.org/2016/961.pdf

[OW96]: P. C. van Oorschot, M. J. Wiener, "On Diffie-Hellman key agreement with short exponents", Eurocrypt 96, pp 332–343.

[LL98]: C.H. Lim and P.J. Lee, "A key recovery attack on discrete log-based schemes using a prime order subgroup", CRYPTO' 98, pp 249–263.

[WeakDh]: D. Adrian, K. Bhargavan, Z. Durumeric, P. Gaudry, M. Green, J. A. Halderman, N. Heninger, D. Springall, E. Thomé, Luke Valenta, B. VanderSloot, E. Wustrow, S. Zanella-Béguelink, P. Zimmermann, "Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice" https://weakdh.org/imperfect-forward-secrecy-ccs15.pdf

[NIST SP 800-56A], revision 2, May 2013 http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Ar2.pdf

[PKCS #3]: "Diffie–Hellman Key Agreement", http://uk.emc.com/emc-plus/rsa-labs/standards-initiatives/pkcs-3-diffie-hellman-key-agreement-standar.htm

[RFC 2785]: R. Zuccherato, "Methods for Avoiding 'Small-Subgroup' Attacks on the Diffie-Hellman Key Agreement Method for S/MIME", March 2000 https://www.ietf.org/rfc/rfc2785.txt