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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

<!--
## Sources that might be used for additional tests:

CVE-2015-3193: The Montgomery squaring implementation in crypto/bn/asm/x86_64-mont5.pl
in OpenSSL 1.0.2 before 1.0.2e on the x86_64 platform, as used by the BN_mod_exp function,
mishandles carry propagation
https://blog.fuzzing-project.org/31-Fuzzing-Math-miscalculations-in-OpenSSLs-BN_mod_exp-CVE-2015-3193.html

CVE-2016-0739: libssh before 0.7.3 improperly truncates ephemeral secrets generated for the
(1) diffie-hellman-group1 and (2) diffie-hellman-group14 key exchange methods to 128 bits ...

CVE-2015-1787 The ssl3_get_client_key_exchange function in s3_srvr.c in OpenSSL 1.0.2 before
1.0.2a, when client authentication and an ephemeral Diffie-Hellman ciphersuite are enabled,
allows remote attackers to cause a denial of service (daemon crash) via a ClientKeyExchange
message with a length of zero.

CVE-2015-0205 The ssl3_get_cert_verify function in s3_srvr.c in OpenSSL 1.0.0 before 1.0.0p
and 1.0.1 before 1.0.1k accepts client authentication with a Diffie-Hellman (DH) certificate
without requiring a CertificateVerify message, which allows remote attackers to obtain access
without knowledge of a private key via crafted TLS Handshake Protocol traffic to a server that
recognizes a Certification Authority with DH support.

CVE-2016-0701 The DH_check_pub_key function in crypto/dh/dh_check.c in OpenSSL 1.0.2 before
1.0.2f does not ensure that prime numbers are appropriate for Diffie-Hellman (DH) key exchange,
which makes it easier for remote attackers to discover a private DH exponent by making multiple
handshakes with a peer that chose an inappropriate number, as demonstrated by a number in an
X9.42 file.

CVE-2006-1115 nCipher HSM before 2.22.6, when generating a Diffie-Hellman public/private key
pair without any specified DiscreteLogGroup parameters, chooses random parameters that could
allow an attacker to crack the private key in significantly less time than a brute force attack.

CVE-2015-1716 Schannel in Microsoft Windows Server 2003 SP2, Windows Vista SP2, Windows Server
2008 SP2 and R2 SP1, Windows 7 SP1, Windows 8, Windows 8.1, Windows Server 2012 Gold and R2, and
Windows RT Gold and 8.1 does not properly restrict Diffie-Hellman Ephemeral (DHE) key lengths,
which makes it easier for remote attackers to defeat cryptographic protection mechanisms via
unspecified vectors, aka "Schannel Information Disclosure Vulnerability.

CVE-2015-2419: Random generation of the prime p allows Pohlig-Hellman and probably other
stuff.
-->