Bridge (graph theory)

Template:More footnotes The Secure Remote Password protocol (SRP) is an augmented password-authenticated key agreement (PAKE) protocol, specifically designed to work around existing patents.

Like all augmented PAKE protocols, an eavesdropper or man in the middle cannot obtain enough information to be able to brute force guess a password without further interactions with the parties for each guess. This means that strong security can be obtained using weak passwords. Furthermore the server does not store password-equivalent data. This means that an attacker who steals the server data cannot masquerade as the client unless they first perform a brute force search for the password.

Overview

The SRP protocol has a number of desirable properties: it allows a user to authenticate themselves to a server, it is resistant to dictionary attacks mounted by an eavesdropper, and it does not require a trusted third party. It effectively conveys a zero-knowledge password proof from the user to the server. In revision 6 of the protocol only one password can be guessed per connection attempt. One of the interesting properties of the protocol is that even if one or two of the cryptographic primitives it uses are attacked, it is still secure. The SRP protocol has been revised several times, and is currently at revision 6a.

The SRP protocol creates a large private key shared between the two parties in a manner similar to Diffie–Hellman key exchange, then verifies to both parties that the two keys are identical and that both sides have the user's password. In cases where encrypted communications as well as authentication are required, the SRP protocol is more secure than the alternative SSH protocol and faster than using Diffie–Hellman key exchange with signed messages. It is also independent of third parties, unlike Kerberos. The SRP protocol, version 3 is described in RFC 2945. SRP version 6 is also used for strong password authentication in SSL/TLS (in TLS-SRP) and other standards such as EAP and SAML, and is being standardized in IEEE P1363 and ISO/IEC 11770-4.

Protocol

The following notation is used in this description of the protocol, version 6:

• q and N = 2q + 1 are chosen such that both are prime (which makes q a Sophie Germain prime and N a safe prime). N must be large enough so that computing discrete logarithms modulo N is infeasible.
• All arithmetic is performed in the ring of integers modulo N, ${\mathbb{\mathbb{Z}}}_N$.
• g is a generator of the multiplicative group.
• H() is a hash function; e.g., SHA-256.
• k is a parameter derived by both sides; for example, k = H(N, g). This creates an asymmetry between the client and server sides of the protocol, meaning a man-in-the-middle attacker only gets 1 verification attempt per impersonation, rather than 2.
• s is a small salt.
• I is an identifying username.
• p is the user's password.
• v is the host's password verifier, v = g^x, x = H(s, p). Usage of key derivation functions like PBKDF2 instead of simple hash functions for password hashing is highly recommended, taking into account that they only have to be computed on the client.
• a and b are random.
• | denotes concatenation.

All other variables are defined in terms of these.

First, to establish a password p with Steve, Carol picks a small random salt s, and computes x = H(s, p), v = g^x. Steve stores v and s, indexed by I, as Carol's password verifier and salt. x is discarded because it is equivalent to the plaintext password p. This step is completed before the system is used.

1. Carol → Steve: I and A = g^a
2. Steve → Carol: s and B = kv + g^b
3. Both: u = H(A, B)
4. Carol: S_Carol = (B − kg^x)^{(a + ux)} = (kv + g^b − kg^x)^{(a + ux)} = (kg^x − kg^x + g^b)^{(a + ux)} = (g^b)^{(a + ux)}
5. Carol: K_Carol = H(S_Carol)
6. Steve: S_Steve = (Av^u)^b = (g^av^u)^b = [g^a(g^x)^u]^b = (g^{a + ux})^b = (g^b)^{(a + ux)}
7. Steve: K_Steve = H(S_Steve) = K_Carol

Now the two parties have a shared, strong session key K. To complete authentication, they need to prove to each other that their keys match. One possible way is as follows:

1. Carol → Steve: M₁ = H[H(N) XOR H(g) | H(I) | s | A | B | K_Carol]. Steve verifies M₁.
2. Steve → Carol: M₂ = H(A | M₁ | K_Steve). Carol verifies M₂.

This method requires guessing more of the shared state to be successful in impersonation than just the key. While most of the additional state is public, private information could safely be added to the inputs to the hash function, like the server private key. The two parties also employ the following safeguards:

1. Carol will abort if she receives B == 0 (mod N) or u == 0.
2. Steve will abort if he receives A (mod N) == 0.
3. Carol must show her proof of K first. If Steve detects that Carol's proof is incorrect, he must abort without showing his own proof of K.

Implementation example in Python

# An example SRP-6a authentication
# WARNING: Do not use for real cryptographic purposes beyond testing.
# based on http://srp.stanford.edu/design.html
import hashlib
import random

def global_print(*names):
    x = lambda s: ["%s", "0x%x"][isinstance(s, long)] % s
    print "".join("%s = %s\n" % (name, x(globals()[name])) for name in names)

# note: str converts as is, str( [1,2,3,4] ) will convert to "[1,2,3,4]" 
def H(*a):  # a one-way hash function
    return int(hashlib.sha256(str(a)).hexdigest(), 16)

def cryptrand(n=1024):  
    return random.SystemRandom().getrandbits(n) % N

# A large safe prime (N = 2q+1, where q is prime)
# All arithmetic is done modulo N
# (generated using "openssl dhparam -text 1024")
N = '''00:c0:37:c3:75:88:b4:32:98:87:e6:1c:2d:a3:32:
       4b:1b:a4:b8:1a:63:f9:74:8f:ed:2d:8a:41:0c:2f:
       c2:1b:12:32:f0:d3:bf:a0:24:27:6c:fd:88:44:81:
       97:aa:e4:86:a6:3b:fc:a7:b8:bf:77:54:df:b3:27:
       c7:20:1f:6f:d1:7f:d7:fd:74:15:8b:d3:1c:e7:72:
       c9:f5:f8:ab:58:45:48:a9:9a:75:9b:5a:2c:05:32:
       16:2b:7b:62:18:e8:f1:42:bc:e2:c3:0d:77:84:68:
       9a:48:3e:09:5e:70:16:18:43:79:13:a8:c3:9c:3d:
       d0:d4:ca:3c:50:0b:88:5f:e3'''
N = int(''.join(N.split()).replace(':', ''), 16)
g = 2        # A generator modulo N

k = H(N, g)  # Multiplier parameter (k=3 in legacy SRP-6)

print "#. H, N, g, and k are known beforehand to both client and server:"
global_print("H", "N", "g", "k")

print "0. server stores (I, s, v) in its password database"

# the server must first generate the password verifier
I = "person"         # Username
p = "password1234"   # Password
s = cryptrand(64)    # Salt for the user
x = H(s, p)          # Private key
v = pow(g, x, N)     # Password verifier
global_print("I", "p", "s", "x", "v")

print "1. client sends username I and public ephemeral value A to the server"
a = cryptrand()
A = pow(g, a, N)
global_print("I", "A")  # client->server (I, A)

print "2. server sends user's salt s and public ephemeral value B to client"
b = cryptrand()
B = (k * v + pow(g, b, N)) % N
global_print("s", "B")  # server->client (s, B)

print "3. client and server calculate the random scrambling parameter"
u = H(A, B)  # Random scrambling parameter
global_print("u")

print "4. client computes session key"
x = H(s, p)
S_c = pow(B - k * pow(g, x, N), a + u * x, N)
K_c = H(S_c)
global_print("S_c", "K_c")

print "5. server computes session key"
S_s = pow(A * pow(v, u, N), b, N)
K_s = H(S_s)
global_print("S_s", "K_s")

print "6. client sends proof of session key to server"
M_c = H(H(N) ^ H(g), H(I), s, A, B, K_c)
global_print("M_c")
# client->server (M_c) ; server verifies M_c

print "7. server sends proof of session key to client"
M_s = H(A, M_c, K_s)
global_print("M_s")
# server->client (M_s) ;  client verifies M_s

Implementations

• OpenSSL version 1.0.1 or later.
• TLS-SRP is a set of ciphersuites for transport layer security that uses SRP.
• srp-client SRP-6a implementation in Javascript (compatible with RFC 5054), open source, MPL licensed.
• The JavaScript Crypto Library includes a JavaScript implementation of the SRP protocol, open source, GPL licensed. Used in Clipperz online password manager.
• Gnu Crypto provide a Java implementation licensed under the GNU General Public License with the "library exception", which permits its use as a library in conjunction with non-Free software.
• The Legion of the Bouncy Castle provides Java and C# implementations under the MIT License.
• Nimbus SRP is a Java library providing a verifier generator, client and server-side sessions. Includes interfaces for custom password key, client and server evidence message routines. No external dependencies. Released under the GNU General Public License and a proprietary license.
• srplibcpp is a C++ implement base on MIRACL.
• DragonSRP is a C++ modular implementation currently works with OpenSSL
• Json2Ldap provides SRP-6a authentication to LDAP directory servers.
• csrp SRP-6a implementation in C.
• pysrp SRP-6a implementation in Python (compatible with csrp).
• srp-rb SRP-6a implementation in Ruby
• Meteor web framework's Accounts system implements SRP for password authentication.

External links

• Template:Official website
• SRP License -- BSD like open source.

Manual pages

• Template:Man
• Template:Man

RFCs

• RFC 2944 - Telnet Authentication: SRP
• RFC 2945 - The SRP Authentication and Key Exchange System
• RFC 3720 - Internet Small Computer Systems Interface (iSCSI)
• RFC 3723 - Securing Block Storage Protocols over IP
• RFC 3669 - Guidelines for Working Groups on Intellectual Property Issues
• RFC 5054 - Using the Secure Remote Password (SRP) Protocol for TLS Authentication

Other links

• IEEE 1363
• SRP Intellectual Property Slides (Dec 2001 - possible deprecated)
• Trusted HTTP -- website and wiki about implementing TLS-SRP in libraries (GnuTLS, OpenSSL, NSS, Python), servers (Apache), and clients (curl, Firefox, Chrome)