# DSA/DSS
**Public** key `(p, q, g, y)` where `y = g·x mod p` \
**Private** key `(p, q, g, x)`
```
SIGN:
k = random in [1, q-1]
r = g^k mod p mod q
s = k-1 (H(M) + x·r) mod q
--> (r, s)
VERIFY:
a = g^( H(M) · s-1 mod q ) mod p
b = y^( r · s-1 mod q ) mod p
r == (a·b mod p) mod q
```
## Repeat K Attack
```
s1 = k-1 (H(M1) + x·r) mod q
s2 = k-1 (H(M2) + x·r) mod q
x = (H(M1)s2 - H(M2)s1) · (r·(s1 - s2))-1 mod q
```
Demonstration
```
s1 = k-1 (H(M1) + x·r) mod q
s2 = k-1 (H(M2) + x·r) mod q
s1·k = H(M1) + x·r mod q
s2·k = H(M2) + x·r mod q
s1·k - x·r = H(M1) mod q
s2·k - x·r = H(M2) mod q
H(M1) - H(M2) = (s1·k - x·r)-(s2·k - x·r) mod q
H(M1) - H(M2) = s1·k - x·r -s2·k + x·r mod q
H(M1) - H(M2) = s1·k - s2·k mod q
H(M1) - H(M2) = (s1 - s2)·k mod q
k = (H(M1) - H(M2)) · (s1 - s2)-1 mod q
x·r = s1·k - H(M1) mod q
x = s1·k - H(M1) · r-1 mod q
x = s1·( (H(M1) - H(M2)) · (s1 - s2)-1 ) - H(M1) · r-1. mod q
x = (H(M1)s2 - H(M2)s1) · (r·(s1 - s2))-1 mod q
```
## K with Linear Increment Attack
### K, K + 1, K + 2, K + 3, ...
```
s1·k = H(M1) + x·r1 mod q
s2·k + s2 = H(M2) + x·r2 mod q
x = s1·( (H(M2) - s2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 )·r1-1 - H(M1)·r1-1 mod q
```
Demonstration
```
s1·k = H(M1) + x·r1 mod q
s2·k + s2 = H(M2) + x·r2 mod q
Gaussian elimination (x)
s1·k·r1-1 = H(M1)·r1-1 + x mod q # Poniamo sola la x, divido per r1
s2·k + s2 - s1·k·r1-1 = H(M2) + x·r2 - (H(M1)·r1-1 + x) mod q # secondo passaggio meno terzo
s2·k + s2 - s1·k·r1-1·r2 = H(M2) + x·r2 - (H(M1)·r1-1 + x)·r2 mod q # moltiplico per r2 la seconda componente
s2·k + s2 - s1·k·r1-1·r2 = H(M2) + x·r2 - H(M1)·r1-1·r2 - x·r2 mod q
s2·k + s2 - s1·k·r1-1·r2 = H(M2) - H(M1)·r1-1·r2 mod q
x = s1·k·r1-1 - H(M1)·r1-1 mod q # Guardando la terza
# Divido la terza con s2 - s1·r1-1·r2
k = (H(M2) - s2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q # Divido la terza con s2 - s1·r1-1·r2
k = (s2·k + s2 - s2 - s1·k·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = (s2·k - s1·k·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = k·(s2 - s1·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = k
x = s1·( (H(M2) - s2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 )·r1-1 - H(M1)·r1-1 mod q
```
### K, K + N, K + 2N, K + 3N, ...
```
s1·k = H(M1) + x·r1 mod q
s2·k + Ns2 = H(M2) + x·r2 mod q
x = s1·( (H(M2) - Ns2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 )·r1-1 - H(M1)·r1-1 mod q
```
Demonstration
```
s1·k = H(M1) + x·r1 mod q
s2·k + Ns2 = H(M2) + x·r2 mod q
Gaussian elimination (x)
s1·k·r1-1 = H(M1)·r1-1 + x mod q # Poniamo sola la x, divido per r1
s2·k + Ns2 - s1·k·r1-1 = H(M2) + x·r2 - (H(M1)·r1-1 + x) mod q # secondo passaggio meno terzo
s2·k + Ns2 - s1·k·r1-1·r2 = H(M2) + x·r2 - (H(M1)·r1-1 + x)·r2 mod q # moltiplico per r2 la seconda componente
s2·k + Ns2 - s1·k·r1-1·r2 = H(M2) + x·r2 - H(M1)·r1-1·r2 - x·r2 mod q
s2·k + Ns2 - s1·k·r1-1·r2 = H(M2) - H(M1)·r1-1·r2 mod q
x = s1·k·r1-1 - H(M1)·r1-1 mod q # Guardando la terza
# Divido la terza con s2 - s1·r1-1·r2
k = (H(M2) - Ns2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q # Divido la terza con s2 - s1·r1-1·r2
k = (s2·k + Ns2 - Ns2 - s1·k·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = (s2·k - s1·k·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = k·(s2 - s1·r1-1·r2) · (s2 - s1·r1-1·r2)-1 mod q
k = k
x = s1·( (H(M2) - Ns2 - H(M1)·r1-1·r2) · (s2 - s1·r1-1·r2)-1 )·r1-1 - H(M1)·r1-1 mod q
```
---
# Agent Instructions: Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.
Perform an HTTP GET request on the current page URL with the `ask` query parameter:
```
GET https://ivalexev.gitbook.io/rednote/pentesting-process/crypto-attacks/dsa-dss.md?ask=
```
The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.
Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.