Sagemath

SageMath

Sage is the go-to environment for mathematical cryptanalysis and attack modeling. Use it when standard scripting is not enough for algebraic structures and lattice workflows.

Typical use cases

• Finite field computations (GF, extension fields).
• Polynomial rings and elimination/resultants.
• Elliptic curve arithmetic and discrete-log style tasks.
• Lattice modeling (matrix, LLL) for partial leakage recovery.
• Fast prototyping of attack math before operationalizing in Python.

Invocation modes

# REPL
sage

# Run .sage script
sage solver.sage

# Run Python file with Sage runtime
sage -python solver.py

# Jupyter with Sage kernel
sage -n jupyter

Practical workflow

1. Parse input values (files, network output, pubkeys).
2. Model algebra in Sage-native structures (GF, rings, curves, matrices).
3. Validate with small synthetic tests.
4. Execute full solve, export recovered secrets to plain Python bytes/ints.
5. Integrate with pwntools or service scripts for secret recovery.

Common cryptanalysis patterns

From real-world cryptanalysis work, common patterns include:

• from sage.all import * alongside pwntools remote interaction.
• Factoring and formatting factorization answers for oracle protocols.
• Elliptic curve order / point lifting / discrete-log solves.
• Lattice assembly via identity/augment/stack + LLL().
• Finite-field polynomial construction and resultant-based elimination.
• .sage solvers combining bitwise transforms, polynomial operations over GF(2^N), and iterative remote interaction rounds.

Output hygiene

• Convert Sage integers to Python ints before serialization.
• Convert numeric secrets to bytes with explicit endianness/length handling.
• Keep deterministic seeds/options when random components appear.

Resources

• references/python-sage-scripts.md — authoritative patterns for .sage and sage -python workflows, imports, interoperability, and pitfalls.
• references/cryptanalysis-patterns.md — practical cryptanalysis templates (GF, EC, lattice, polynomial elimination) grounded in real solver style.