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If we consider a group G with modulus p, order q with $p=2*q+1$, and generator $g=2$ ($ p$, $q$ huge prime numbers), is there a way to solve the discrete log problem $ g^x = y $ for a y given, using the baby steps giant step algorithm AND the fact that $x$ is of the form: $ x = \sum_{i=0}^{10} \alpha_i * 2^i $ $:\alpha_i \in \mathbb{N} $ without requiring to store all small steps in a hashtable, which is impossible with the magnitude of the numbers $p$ and $q$.

edit: The $ \alpha_i $ values are unknown to the attacker.

edit 2: $\alpha_i \in [0,\infty] $

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  • $\begingroup$ Does the attacker know the $\alpha_i$ values? $\endgroup$
    – poncho
    Oct 25, 2021 at 15:10
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    $\begingroup$ Does actually $\alpha_i \in \{0,1\}$? Does $\mathbb{N} = \mathbb{Z}^+$ or $\mathbb{N} = \mathbb{Z}^+ \cup \{0\}$, there is no common agreement on this. $\endgroup$
    – kelalaka
    Oct 25, 2021 at 21:50

1 Answer 1

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is there a way to solve the discrete log problem $g^x=y$ for a $y$ given, using the baby steps giant step algorithm

No practical way if $p, q$ are large.

Even if $x$ is known to be in the form $ x = \sum_{i=0}^{10} \alpha_i * 2^i $ $:\alpha_i \in \mathbb{N} $ ?

Every potential $x$ can be expressed in that form; consider $\alpha_1 = \alpha_2 = ... = \alpha_{10} = 0$ and $\alpha_{0} = x$. Hence, knowing that $x$ is expressable in that form does not provide any additional information; BsGs is usable only if the modulus is small enough (and the question assumes it is not)

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  • $\begingroup$ No $x< 2048$. The problem is much easier, you can do Bruteforce attack (but without babystep/giant step). $\endgroup$
    – Ievgeni
    Oct 25, 2021 at 16:04
  • $\begingroup$ @levgeni: if we're talking about values that can be expressed in that form, we have $x \ge 2047$ (and that's with the convention that $0 \not\in \mathbb{N}$; otherwise, all $x$ values can be expressed in that form). $\endgroup$
    – poncho
    Oct 25, 2021 at 16:36
  • $\begingroup$ @levgeni: and, if $x < 2048$ and $p$ is cryptographically sized (i.e. at least 2048 bits), the discrete log can be extracted directly from the value $2^x$; that has bit $x$ as 1, and all the other bits as 0... $\endgroup$
    – poncho
    Oct 25, 2021 at 16:38
  • $\begingroup$ @poncho I think levgeni argues that even if $x>2046$, it doesn't imply that $x$ is not trivially small. The large is clear, however the small need a metric. $\endgroup$
    – kelalaka
    Oct 25, 2021 at 20:40
  • $\begingroup$ @kelalaka: I don't understand your argument; if we interpret the question of "is DLog solvable if we assume the exponent is random, conditioned by being expressable in the given form?" With that understanding, my answer is correct - how do you interpret the question? $\endgroup$
    – poncho
    Oct 25, 2021 at 21:33

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