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DSA key recovery from nonce

Step 1: Relocate so that you are out of easy travel distance of us.

Step 2: Implement DSA, up to signing and verifying, including parameter generation.

Hah-hah you're too far away to come punch us.

Just kidding you can skip the parameter generation part if you want; if you do, use these params:

p = 800000000000000089e1855218a0e7dac38136ffafa72eda7
    859f2171e25e65eac698c1702578b07dc2a1076da241c76c6
    2d374d8389ea5aeffd3226a0530cc565f3bf6b50929139ebe
    ac04f48c3c84afb796d61e5a4f9a8fda812ab59494232c7d2
    b4deb50aa18ee9e132bfa85ac4374d7f9091abc3d015efc87
    1a584471bb1

q = f4f47f05794b256174bba6e9b396a7707e563c5b

g = 5958c9d3898b224b12672c0b98e06c60df923cb8bc999d119
    458fef538b8fa4046c8db53039db620c094c9fa077ef389b5
    322a559946a71903f990f1f7e0e025e2d7f7cf494aff1a047
    0f5b64c36b625a097f1651fe775323556fe00b3608c887892
    878480e99041be601a62166ca6894bdd41a7054ec89f756ba
    9fc95302291

("But I want smaller params!" Then generate them yourself.)

The DSA signing operation generates a random subkey "k". You know this because you implemented the DSA sign operation.

This is the first and easier of two challenges regarding the DSA "k" subkey.

Given a known "k", it's trivial to recover the DSA private key "x":

    (s * k) - H(msg)
x = ----------------  mod q
            r

Do this a couple times to prove to yourself that you grok it. Capture it in a function of some sort.

Now then. I used the parameters above. I generated a keypair. My pubkey is:

y = 84ad4719d044495496a3201c8ff484feb45b962e7302e56a392aee4
    abab3e4bdebf2955b4736012f21a08084056b19bcd7fee56048e004
    e44984e2f411788efdc837a0d2e5abb7b555039fd243ac01f0fb2ed
    1dec568280ce678e931868d23eb095fde9d3779191b8c0299d6e07b
    bb283e6633451e535c45513b2d33c99ea17

I signed

For those that envy a MC it can be hazardous to your health
So be friendly, a matter of life and death, just like a etch-a-sketch

(My SHA1 for this string was d2d0714f014a9784047eaeccf956520045c45265; I don't know what NIST wants you to do, but when I convert that hash to an integer I get: 0xd2d0714f014a9784047eaeccf956520045c45265).

I get:

r = 548099063082341131477253921760299949438196259240
s = 857042759984254168557880549501802188789837994940

I signed this string with a broken implemention of DSA that generated "k" values between 0 and 2^16. What's my private key?

Its SHA-1 fingerprint (after being converted to hex) is:

0954edd5e0afe5542a4adf012611a91912a3ec16

Obviously, it also generates the same signature for that string.