Quantum.Synthesis.Diophantine

This module provides some number-theoretic functions, particularly functions for solving the Diophantine equation

t^†t = ξ,

where ξ ∈ ℤ[√2] and t ∈ ℤ[ω], or ξ ∈ D[√2] and t ∈ D[ω].

In general, solving this equation can be hard, as it depends on the ability to factor the integer n = ξ^•ξ into primes. We formulate the solution as a step computation (see Quantum.Synthesis.StepComp), so that the caller can dynamically determine how much time to spend on solving the equation, or can attempt to solve several such equations in parallel.

In many cases, even a partial factorization of n is sufficient to determine that no solution exists. This implementation is written to take advantage of such cases.

Diophantine solvers

diophantine :: RandomGen g => g -> ZRootTwo -> StepComp (Maybe ZOmega)

Given ξ ∈ ℤ[√2], find t ∈ ℤ[ω] such that t^†t = ξ, if such t exists, or return Nothing otherwise.

diophantine_dyadic :: RandomGen g => g -> DRootTwo -> StepComp (Maybe DOmega)

Given an element ξ ∈ D[√2], find t ∈ D[ω] such that t^†t = ξ, if such t exists, or return Nothing otherwise.

diophantine_associate :: RandomGen g => g -> ZRootTwo -> StepComp (Maybe ZOmega)

Given ξ ∈ ℤ[√2], find t ∈ ℤ[ω] such that t^†t ~ ξ, if such t exists, or Nothing otherwise. Unlike diophantine, the equation is only solved up to associates, i.e., up to a unit of the ring.

Factoring

find_factor :: RandomGen g => g -> Integer -> StepComp Integer

Given a positive composite integer n, find a non-trivial factor of n using a simple Pollard-rho method. The expected runtime is O(√p), where p is the size of the smallest prime factor. If n is not composite (i.e., if n is prime or 1), this function diverges.

relatively_prime_factors :: EuclideanDomain a => a -> a -> (a, [(a, Integer)])

Given a factorization n = ab of some element of a Euclidean domain, find a factorization of n into relatively prime factors,

n = u c₁^k₁ ⋅ … ⋅ c_m^k_m,

where m ≥ 2, u is a unit, and c₁, …, c_m are pairwise relatively prime.

While this is not quite a prime factorization of n, it can be a useful intermediate step for computations that proceed by recursion on relatively prime factors (such as Euler's φ-function, the solution of Diophantine equations, etc.).

Computations in ℤ_n

power_mod :: Integer -> Integer -> Integer -> Integer

Modular exponentiation, using the method of repeated squaring. power_mod a k n computes a^k (mod n).

root_of_negative_one :: RandomGen g => g -> Integer -> StepComp Integer

Compute a root of −1 in ℤ_n, where n > 0. If n is a positive prime satisfying n ≡ 1 (mod 4), this succeeds within an expected number of 2 ticks. Otherwise, it probably diverges.

As a special case, if this function notices that n is not prime, then it diverges without doing any additional work.

root_mod :: RandomGen g => g -> Integer -> Integer -> StepComp Integer

Compute a root of a in ℤ_n, where n > 0. If n is an odd prime and a is a non-zero square in ℤ_n, then this succeeds in an expected number of 2 ticks. Otherwise, it probably diverges.