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NTRUFatigue

Experiments and Predictions for attacks on NTRU in the overstretched Regime

This repository contains the artifacts associated to the article

[DvW21] NTRU Fatigue: How Stretched is Overstretched ? by Léo Ducas and Wessel van Woerden https://eprint.iacr.org/2021/999

Contributers

  • Léo Ducas
  • Wessel van Woerden

Requirements

Some older versions of SageMath contain a faulty FPLLL version which contains a bug that prevents our experiments from running successfully. Either use SageMath 9.3+ or manually install fpylll.

Description of files

Short description of the files:

  • bkz2_callback.py (BKZ2.0 including a hook to detect SKR and DSD events)
  • estimator.sage (Estimator for SKR and DSD events)
  • experiment.py (To run progressive BKZ until a SKR or DSD event is detected)
  • find_fatigue.py (A soft binary search to find the fatigue point)
  • ntru_keygen.py (NTRU instance generator)
  • cli.py (helper file)
  • paper/lucky_lift.sage (Preliminary analysis of an anecdotal Lucky Lift event)
  • paper/claim3_5.ipynb (Symbolic algebra proof of Claim 3.5)

Estimator

The estimator estimator.sage requires sage. An example with parameters q=257, n=73, sigma^2=2/3 on a matrix NTRU instance, assuming progressive BKZ with 8 tours.

load("estimator.sage")
res=combined_attack_prob(257, 73, 2/3., ntru="matrix", fixed_tours=8)
print(res[0]) # average beta
print(res[1]) # probability of SKR event
print(res[2]) # probability of DSD event
print(res[3]) # Distribution of detection position kappa

Experiments

The experiments use fpylll, and can be ran using sage or after installing fpylll manually. For manual installation follow these instructions. In addition to fpylll dependencies, the package scipy is also required. And don't forget to active the fpylll environment by running `source ./activate' in the fpylll dir.

Additionally make sure that the following environment variables are set to 1, to prevent numpy from taking over all threads.

export MKL_NUM_THREADS=1
export NUMEXPR_NUM_THREADS=1
export OMP_NUM_THREADS=1

Parameters:

--n X / -n X         # length of ntru secret, ntru lattice has dimension 2*n
--q X / -q X         # modulus
--circulant X        # keys have circulant structure (default 1)
--trials X / -t X    # do X trials per parameter set (default 1)
--workers X / -w X   # use X parallel threads (default 1)
--float_type X       # use floating point type X in {double, ld, dd, qd} for GSO in fplll. (Default "double")
--full_data X / -f X # show full data in CSV format (default 0)

Precision needs to be increased with n and q. Increase it if you encounter the infamous "infinte loop in Babai" error message. Using "dd" and "qd" requires the library libqd before compilation and installation of fplll/fpylll.

Parameters n and q can be given a single value (-n 51) , a list of values (e.g. -n 73 89), or an interval of prime integers (-q 1300~1400p). Key generation may be extremely slow if q or n isn't prime.

Example of an NTRU attack with n=127, and q ranging over all primes from 1300 to 1400

[sage/python] experiments.py --n 127 --q 1300~1400p --trials 2 --workers 2 --float_type dd --circulant 1 --tours 8 -f 1

Explanation of results:

DSD                # DSD (1) or SKR (0) event
DSD_lf             # Squared length ratio between detected dense vector and secret key
kappa              # Detection position in basis
beta               # Successful blocksize
DS_vol             # Log-volume of dense sublattice
foundbyLLL         # If the dense vector was inserted by intermediate LLL calls
slope              # Slope of log-profile on block [n-30:n+30) (or smaller for n<30) at moment of detection
sqproj_rel         # Squared length ratio between projection pi_kappa(v) and the detected dense vector v.

Example of finding the fatigue point for n=73 and 89

[sage/python] find_fatigue.py --n 73 89 --trials 2 --workers 2 --tours 8 --circulant 0 --float_type dd -f 1