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Attacks on Diffie-Hellman Protocols

This paper contains high-level instructions to perform practical labs within the workshop. Some tasks were taken from the Cryptopals challenges, some tasks are original.

Labs

Basic Diffie-Hellman Protocol

Implement the GenerateKey and DH methods of dhgroup interface on GroupParams type.

Run TestDH to be sure your implementation is correct:

go test ./dhgroup -run TestDH

Small-subgroup Attack

Realize newDHOracle function in oracle.go and its API. Then read the description of the Small-subgroup attack.

To perform the attack implement runDHSmallSubgroupAttack function and run the TestSmalSubgroupAttack test:

go test -run TestSmallSubgroupAttack

Small-subgroup and Pollard's Kangaroo Attacks

Implement the catchKangaroo function following the instructions. To verify the implementation of the Pollard's algorithm run the TestKangarooAlgorithm test:

go test -run TestKangarooAlgorithm

Then develop runDHKangarooAttack and verify the results:

go test -run TestKangarooAttack

P.S. It may take several minutes to complete the attack.

Additional task (optional):

Add protection against the small-subgroup attack into DH method. Run the tests and make sure that the attacks above do not work anymore.

Elliptic Curve Cryptography

Implement the Curve interface defined in elliptic/elliptic.go.

You may use instructions from the docs/elliptic_curves or any other mathematical papers.

elliptic_test contains multiple tests for different real and custom curves. The real test vectors of P-256 and P-224 curves are used to verify the correctness of multiplication operation.

It should be noted, that the elliptic curve design follows the Golang's approach. It is highly recommended to employ that as a reference code.

Elliptic-curve Diffie Hellman Protocol

Now implement GenerateKey function and use it to implement elliptic-curve Diffie-Hellman protocol.

Run TestECDH to verify your function:

go test ./elliptic -run TestECDH

Invalid Curve Attack

Review the ECDH oracle located in the oracle.go and its API.

Read the description of the Invalid curve attack.

Implement runECDHInvalidCurveAttack function and run TestECDHInvalidCurveAttack:

go test -run TestECDHInvalidCurveAttack

Insecure Twist Attack

Implement the single-coordinate Montgomery's ladder using the instructions from the docs/twist_attack.

Verify the implementation running the following tests:

go test ./x128
go test -run TestCurvesP128AndX128

Review the x128 oracle located in the oracle.go and its API.

Implement Pollard's Kangaroo algorithm for elliptic curves and make sure it works properly:

go test -run TestECKangarooAlgorithm

Now, you have got all necessary primitives to implement the attack against the twist.

Implement runECDHTwistAttack function and verify the solution:

go test -run TestTwistAttack

Caveats

The attack may take 10-15 minutes.

The original challenge has the following HINT:

You may come to notice that ku = -ku, resulting in a combinatorial explosion of potential CRT
outputs. Try sending extra queries to narrow the range of possibilities.

To simplify your life, let's imagine that the ECDH implementation has another vulnerability which allows you to learn some bits of information. This vulnerable feature accessible via privateKeyOracle function. So, it allows you to filter incorrect points. If you think it was cheating you may don't use this method and check all combinations using more sophisticated methods.

Key-compromise Impersonation

Read Tox Handshake Vulnerable to KCI and try to understand the security model of the KCI.

The task suggests that you have compromised Bob's server and got the private key. The aim of the task is to impersonate Alice. You need to send a message to Bob. Bob will think the he will received the message from Alice, but he actually will received the message from Mallory.

Implement runKCIAttack function and verify the solution:

go test -run TestKCIAttack

References

  1. J.M. Pollard. Monte Carlo Methods for Index Computation
  2. Nigel Smart. Introduction to ECC