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For textbook DSA, as defined below. If $r, s, k, g, p, q, m$ are all known, is it possible to efficiently compute $x$ from all these given values? How does the math work out?

  • Secret key: $x\stackrel R\gets \mathbb{Z}_{q}$
  • Private key: $X=g^{x}\in \mathbb{Z}^{*}_{p}$
  • Signature:
    $\text{Algorithm }S_x(M)\\\quad{ m\gets H(M)\\ k\stackrel\\\$\gets\mathbb Z_q^*\\ r\gets(g^k\bmod p)\bmod q\\ s\gets(m+x\,r)\,k^{-1}\bmod q\\ \text{return }(r,s)}$
  • Verification:
    $\text{Algorithm }V_X(M,(r,s))\\\quad{ m\gets H(M)\\ w\gets s^{-1}\bmod q\\ u_1\gets m\,w\bmod q\\ u_2\gets r\,w\bmod q\\ v\gets(g^{u_1}\,X^{u_2}\bmod p)\bmod q\\ \text{if }(v=r)\text{ return }1\\ \text{else return }0}$

where $H$ is just SHA-1.

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  • $\begingroup$ Better use \stackrel r\gets as here $\endgroup$
    – kelalaka
    Commented Nov 14, 2020 at 10:04
  • $\begingroup$ Is this a homework question? $\endgroup$
    – kelalaka
    Commented Nov 14, 2020 at 10:28
  • $\begingroup$ I corrected the algorithms previously specified as images. $r$ is computed $\bmod p$, not $\bmod q$. And Algorithms $V$ is parametrized by $X$, not $x$. Also improved typo. $\endgroup$
    – fgrieu
    Commented Nov 14, 2020 at 15:36
  • $\begingroup$ Hint: the critical difference with normal DSA is that in the question, $k$ is known. $\endgroup$
    – fgrieu
    Commented Nov 14, 2020 at 19:34

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