From 8797c84f899c56c8d6073cca7e85cf0c527263c3 Mon Sep 17 00:00:00 2001 From: gogo24 Date: Sat, 19 Oct 2024 01:32:40 +0200 Subject: [PATCH] finish article --- .../episode-2-proof-demonstration.md | 26 +++-- .../latex-reverse-type7.pdf | Bin 104940 -> 101796 bytes .../latex-reverse-type7.tex | 94 +++++------------- .../reversing-exclusive-or.png | Bin 0 -> 130849 bytes 4 files changed, 41 insertions(+), 79 deletions(-) create mode 100644 static/from-0-to-crypto-by-projects/episode-2-proof-demonstration/reversing-exclusive-or.png diff --git a/content/from-0-to-crypto-by-projects/episode-2-proof-demonstration.md b/content/from-0-to-crypto-by-projects/episode-2-proof-demonstration.md index d5cabe6..db94044 100644 --- a/content/from-0-to-crypto-by-projects/episode-2-proof-demonstration.md +++ b/content/from-0-to-crypto-by-projects/episode-2-proof-demonstration.md @@ -1,5 +1,5 @@ --- -title: "Episode 2: Reversing cryptography algorithm made to be unreversables (checksum). Method by mathematical proof: disproof" +title: "Episode 2: Reversing cryptography algorithm made to be unreversable (checksum). Method by mathematical proof: disproof" date: 2024-10-13T1:25:30+02:00 weight: 2 draft: false @@ -11,8 +11,11 @@ According to the documentation and as it is mentionned that type 7 is an hashing A secure hash algorithm is an hash algorithm so that for any function hash that transform the original (plaintext) value $ hased = H(plain) $ there does not exist a function $ rev(hashed) $ so that $ rev(hashed) = plain $. -## 1 - Analysis under mathematical thinking +## II - Notes: +I really definitely insist on this point: `It is crucial for a cryptologist to PROOVE his statement. Not just calculating.` If you only calculate, you could reach some proprietary algorithms such as this one but you will never ever be able to code CVE exploits on modern algorithms. I insist in the point you have to read [book fo proof](https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf) if you did not do it yet. It is to do theorem proving. + +## III - Analysis under mathematical thinking The reverse engineering of the hash of vigenere cisco has permitted to deduct the method taken by this algorithm. @@ -20,18 +23,23 @@ We could then guess that the researchers thanks then that: ![image](/gogo-s-blog-cpe/from-0-to-crypto-by-projects/episode-2-proof-demonstration/theory-behind-type7-hash.png) -The question is to proove that there exists a function $ rev(hashed) $ so that $ \forall plain [rev(H(plain)) = plain] $ then $ \forall x [x = H(plain)]$ - +The question is to proove that there exists a function $ rev(hashed) $ so that $ \forall plain [rev(H(plain)) = plain] $ then $ \forall x [x = H(plain)] $ -We intuitevely see points to split the issue into easier pieces: -The algorith treat bigrams (blocks of two opcodes) as following: -- the two opcodes are both xored to the hardcoded password. -- +## IV/ 1- solving the theorem finding a way to proove the case. +There are a lot of different method to proove a theorem. You could pick the one you prefer or the one you find easier. +The big picture is to split the proof into several cases. +There a serveral various operations including: +- splitting number between 0 and 256 to two differnt more little number (the shift: $ \ggg $ and the logical and: $ \land 0xf0 $ ). Reversable by mergingtwo numbers in a single one with same algortihm. +- adding. You could simply substract to reverse. +- doing an boolean exclusive logical or to a known password. + - as each number exclusively logically set to logical or (xored) with itself has the final value of 0 and as 0 set to logical or with another number will return this number, ![image](/gogo-s-blog-cpe/from-0-to-crypto-by-projects/episode-2-proof-demonstration/reversing-exclusive-or.png) it follows that logically set to logical or to the hardcoded password one time will change the values but logically set to logical or a second time to the same hardcoded value will change it to the original value. See [boolean algebra](https://en.wikipedia.org/wiki/Exclusive_or#Definition), and see this schems provided with the tool name `cryptool-2`. +All of these are reversables. +Then I decide to choose a proof in the form: as $ A \implies B \implies C $, then $ A \implies C $. -We now have the mathematical proof (demonstration) that the algorithm is vulnerable [in this paper]( /gogo-s-blog-cpe/from-0-to-crypto-by-projects/episode-2-proof-demonstration/latex-reverse-type7.pdf ). \ No newline at end of file +Let's check it out that [in this paper]( /gogo-s-blog-cpe/from-0-to-crypto-by-projects/episode-2-proof-demonstration/latex-reverse-type7.pdf )! \ No newline at end of file diff --git a/static/from-0-to-crypto-by-projects/episode-2-proof-demonstration/latex-reverse-type7.pdf b/static/from-0-to-crypto-by-projects/episode-2-proof-demonstration/latex-reverse-type7.pdf index a7a8ca954c4170a35cb1279395e2b552fd1dedcf..4623743d7fb56acba9bada6f8128fcf114661c77 100644 GIT binary patch delta 51155 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z+jC*PFMD8>-^3~Ef!~;no(94C-2i3wcCVU-Z9KzKjV=4cbEllohJ%OV<|T@Bf12>z zf1k$j%YNG6em+Brza|YhI!hw`gf+`~(!k}shxU!-$9Ihk@6MG1UNlG9j z;ZV3F3@XR*|Cii!(YfX5>Fnpi0TmNNKlK3d0o;SKx`g_SQNMllY)+XJ^v7sORKWJT zglk)TNtw4l+!!=+G1J=V6nx2w9;gBFH!74lQ9%v`33kq0@>=GHWeI8$KyqNR<+l|H z(PZ8AWpNcvG>wHySIMZSJjp7F8gkP}3l`d(PIR#bb$8!L#$N2gKy~?!)NRCFdPRFOP99p7|Y2^fZyRENeLWMG}aO0uGmuB 0xa } \implies y = x + 0x37 $ $\\$ It follows: -$ \forall y = H(x), y \in { y | 0 < y \leq 0xa + 0x30 } \implies x = y - 0x30 $ then $ 0 < x < 0xa $ $\\$ -$ \forall y = H(x), y \in { y | 0 < y \leq 0xa + 0x37 } \implies x = y - 0x30 $ then $ 0xa < x < 0x13 $ $\\$ +$ \forall y = H(x), y \in { y | 0 < y \leq 0x0a + 0x30 } \implies x = y - 0x30 $ then $ 0 < x < 0x0a $ $\\$ +$ \forall y = H(x), y \in { y | 0 < y \leq 0x0a + 0x37 } \implies x = y - 0x30 $ then $ 0x0a \leq x $ $\\$ # V /communtativity: -Addition, substraction and $ \oplus $ are commutative. \\ \\ +Addition, substraction and $ \oplus $ are commutative. \\ # VI / proof Then we have already proven each piece of the theorem so that: - -$hp = (d, s, f, d, ;, k, f, o, A, ,, ., i, y, e, w, r, k, l, d, J, K, D, H, S, U, B, s, g, v, c, a, 6, 9, 8, 3, 4, n, c, x , v) \implies (\forall x \in hp[0 \geq x 0 \geq 256 \implies x \in hp]) $ +$hp = (d, s, f, d, ;, k, f, o, A, ,, ., i, y, e, w, r, k, l, d, J, K, D, H, S, U, B, s, g, v, c, a, 6, 9, 8, 3, 4, n, c, x , v) \implies (\forall x \in hp[0 \geq x 0 \geq 256 \implies x \in hp]) $ \\ then: @@ -198,15 +152,15 @@ h_{1} = 8, \\ h = \Sigma_{i=2}^{lp} \begin{cases} - (((p_{i} \oplus hp_{i+8}) \lll 4) - 0x30), & \text{if } p_i < 0xa0 \text{ and if } i \equiv 0 \pmod 2 \\ - (((p_{i} \oplus hp_{i+8}) \lll 4) - 0x37), & \text{if } p_i \geq 0x0a0 \text{ and if } i \equiv 0 \pmod 2 \\ - (((p_{i} \oplus hp_{i+8}) \land 0xffffffff0) - 0x30), & \text{if } p_i < 0x0a \text{ and if } i \equiv 1 \pmod 2 \\ - (((p_{i} \oplus hp_{i+8}) \land 0xffffffff0) - 0x37), & \text{if } p_i \geq 0x0a \text{ and if } i \equiv 1 \pmod 2 + (((p_{i} \oplus hp_{i+8}) \lll 4) - 0x30), & \text{if } h_i < 0xa0 \text{ and if } i \equiv 0 \pmod 2 \\ + (((p_{i} \oplus hp_{i+8}) \lll 4) - 0x37), & \text{if } h_i \geq 0xa0 \text{ and if } i \equiv 0 \pmod 2 \\ + (((p_{i} \oplus hp_{i+8}) \land 0xffffffff0) - 0x30), & \text{if } h_i < 0x0a \text{ and if } i \equiv 1 \pmod 2 \\ + (((p_{i} \oplus hp_{i+8}) \land 0xffffffff0) - 0x37), & \text{if } h_i \geq 0x0a \text{ and if } i \equiv 1 \pmod 2 \end{cases} \\ ) \implies \forall p[p = \mathbf{rev}(h)] \\ \] -\end{flushleft} -\end{multline}\\ +\end{multline} +\end{flushleft}\\ $$ \end{document} \ No newline at end of file diff --git a/static/from-0-to-crypto-by-projects/episode-2-proof-demonstration/reversing-exclusive-or.png b/static/from-0-to-crypto-by-projects/episode-2-proof-demonstration/reversing-exclusive-or.png new file mode 100644 index 0000000000000000000000000000000000000000..b8aeb05598e399f3747562cf8af070ed764914a7 GIT binary patch literal 130849 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