# 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.