A modulus is the positive integer m that bounds values produced by modular arithmetic. In a linear congruential generator, each state is reduced modulo m: Xₙ₊₁ = (aXₙ + c) mod m. section 2.2.2
The modulus is part of the generator’s parameters, alongside the multiplier, increment, and seed. Recovering it from observed outputs can expose the generator’s full state and enable prediction. section 2.2.2
Recovering an LCG modulus
An exploit collects successive differences between outputs, forms values of the form ΔᵢΔᵢ₋₂ − Δᵢ₋₁², and computes their greatest common divisor. The resulting modulus is then used to recover the multiplier and increment before cloning the generator. section 2.2.2
Equation 2.3
\[X_{n+1} = \left( a X_n + c \right)\bmod m \label{equation:lcg}\]Shows the modulus operation inside the LCG recurrence. section 2.2.2
