Sagemath

OpenCROW Runner - SageMath

Use this skill to execute SageMath reliably from the local conda environment sage.

Quick Start

Use the bundled runner:

python ~/.codex/skills/sagemath/scripts/run_sage.py --code 'print(factor(2^20 - 1))'

Or execute an existing .sage file:

python ~/.codex/skills/sagemath/scripts/run_sage.py --file /absolute/path/to/script.sage

Workflow

1. Decide whether the task is best expressed as inline code or a .sage file.
2. For small one-off computations, pass the code with --code.
3. For longer programs or reusable work, create a .sage file and pass it with --file.
4. For CTF crypto tasks, prefer Sage when the problem involves polynomial rings, finite fields, matrices over modular domains, lattices, or symbolic number theory operations.
5. Review stdout/stderr from SageMath and report the meaningful result back to the user.

Runner Notes

• The runner always invokes conda run -n sage sage.
• For --code, the runner writes the code into a temporary .sage file first. This avoids quoting issues and keeps behavior aligned with file execution.
• Pass --timeout SECONDS when a computation may hang or run too long.
• Pass --keep-temp only when debugging generated Sage code.

Patterns

For inline calculations:

python ~/.codex/skills/sagemath/scripts/run_sage.py --code '
R.<x> = QQ[]
f = x^4 - 1
print(f.factor())
'

For file-based workflows:

1. Create a .sage file in the workspace.
2. Run it with the bundled runner.
3. If the script generates files, verify the outputs before replying.

For CTF cryptography workflows:

• Use Sage integer and modular arithmetic for RSA attacks, CRT reconstruction, inverses, and exponent relations.
• Use polynomial rings and small_roots() for Coppersmith-style attacks when the instance is suitable.
• Use GF(p) or extension fields for ECC, finite-field equations, and structured algebraic recovery.
• Use Matrix, vector, and LLL for lattice-based recovery problems and hidden-number style attacks.
• Use recurrence solving, modular equations, and matrix lifting for LCG, xorshift, and related PRNG state recovery when Sage algebra is useful.

References

• For CTF-oriented patterns and starter snippets, read references/ctf-crypto.md when the task involves RSA, ECC, lattices, or PRNG cryptanalysis.

Templates

• Copy or adapt rsa-starter.sage for modular arithmetic, CRT, and private-exponent recovery tasks.
• Copy or adapt ecc-starter.sage for finite-field and elliptic-curve exploration.
• Copy or adapt lattice-lll-starter.sage for integer-lattice and hidden-structure attacks.
• Copy or adapt lcg-state-recovery.sage for linear congruential generator recovery.
• Copy or adapt xorshift-linear-model.sage for GF(2)-linear xorshift modeling.
• Copy or adapt mersenne-twister-state-tools.sage for MT19937 tempering and state-word recovery work.
• Copy or adapt mt19937-partial-state-starter.sage for partial-state or masked-bit MT analysis.
• Copy or adapt mt19937-full-state-recovery.sage for full 624-word MT state reconstruction from outputs.
• Copy or adapt rsa-small-root-starter.sage for Coppersmith-style experimentation with small_roots().
• Copy or adapt rsa-boneh-durfee-starter.sage for low-private-exponent setup and polynomial construction.
• Copy or adapt hidden-number-lattice-starter.sage for nonce-bias and hidden-number lattice setups.
• Copy or adapt ecdsa-nonce-reuse-starter.sage for same-nonce ECDSA private-key recovery.
• Copy or adapt ecdsa-partial-nonce-lattice-starter.sage for biased or partially leaked nonce lattice attacks.
• Copy or adapt lcg-truncated-output-starter.sage for recovering LCG state from high-bit leaks.

Resource

scripts/run_sage.py

Use this script instead of calling SageMath manually unless there is a specific reason not to. It standardizes environment selection, timeout handling, and inline-code execution.