> For the complete documentation index, see [llms.txt](https://alham-rizvi.gitbook.io/alhamrizvi/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://alham-rizvi.gitbook.io/alhamrizvi/cryptography/rsa/14.-generalized-hastad-attack.md). # Generalized Håstad Attack ## 1. Overview This is an extension of the CRT attack. Earlier case: ``` Same message M Same exponent e Different moduli ``` Now generalized case: ``` Messages are different but related by known polynomials ``` *** ## 2. Problem Setup We are given: ``` Ci = (fi(M))^e mod ni ``` Where: ``` fi(x) = known polynomial (like x + k, a·x + b, etc.) ``` Example: ``` C1 = M^e mod n1 C2 = (M + 1)^e mod n2 C3 = (M + 2)^e mod n3 ``` Goal: ``` Recover M ``` *** ## 3. Core Idea Instead of same message: ``` We have structured variations of M ``` We combine all equations using CRT to get: ``` F(M)^e = combined_value ``` Then solve using: ``` small root techniques (Coppersmith) ``` *** ## 4. Key Insight ``` Even if messages differ, structure leaks information ``` Because: ``` fi(M) is known ``` So we can build equations in `M`. *** ## 5. Special Case (Linear) If: ``` fi(x) = x + ki ``` Then: ``` Ci = (M + ki)^e mod ni ``` This becomes: ``` multi-equation small root problem ``` *** ## 6. Attack Strategy 1. Define polynomials: ``` fi(x)^e − Ci ≡ 0 mod ni ``` 2. Combine using CRT 3. Construct global polynomial 4. Solve for small root `M` *** ## 7. SageMath Implementation (Concept) ```python # SageMath n1 = ... n2 = ... n3 = ... C1 = ... C2 = ... C3 = ... e = 3 R. = PolynomialRing(ZZ) f1 = x^e - C1 f2 = (x + 1)^e - C2 f3 = (x + 2)^e - C3 # Combine via CRT logic (conceptual) # Then solve small root # In practice: use lattice-based method ``` *** ## 8. Practical Approach Most CTFs simplify this: ``` Small e Simple relations (x + k) Few equations ``` Then: ``` You can brute-force or reduce to Franklin-Reiter ``` *** ## 9. When to Recognize Look for: ``` Multiple ciphertexts Different moduli Same exponent Messages slightly different but structured ``` Examples: ``` M, M+1, M+2 aM+b patterns ``` *** ## 10. Why This Breaks RSA Because: ``` RSA preserves algebraic structure ``` Multiple structured encryptions give: ``` enough equations to solve for M ``` *** ## 11. Difference from Basic Håstad | Type | Condition | | ----------- | ---------------- | | Basic | Same message | | Generalized | Related messages | *** ## 12. Limitations ``` Needs known structure Requires enough equations Requires small exponent ``` *** ## 13. Summary ``` Related messages + same exponent → build equations → combine via CRT → solve for M ``` *** ## 14. Practice Problem ``` e = 3 n1 = 10177 n2 = 10193 n3 = 10211 C1 = M^3 mod n1 C2 = (M+1)^3 mod n2 C3 = (M+2)^3 mod n3 ``` *** ## Task ``` Recover M ``` --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## 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, and the optional `goal` query parameter: ``` GET https://alham-rizvi.gitbook.io/alhamrizvi/cryptography/rsa/14.-generalized-hastad-attack.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. 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.