computational complexity - Is #k-XORSAT #P-complete?

k-XORSAT is the problem of deciding whether a Boolean formula $$bigwedge_{i in I} oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $oplus$ denotes the binary XOR operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.

Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $sum_{j=1}^k x_{s_{ij}}equiv 1; (mod 2)$, has a solution over $mathbb{Z}_2 = mathbb{Z}/2mathbb{Z}$.

Every decision problem Q has an associated counting problem #Q, which (informally speaking) requires establishing the number of distinct solutions. Such counting problems form the complexity class #P. The "hardest" problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.

The counting problem associated with any NP-complete decision problem is #P-complete. However, many "easy" decision problems (some even solvable in linear time) also lead to #P-complete counting problems. For instance, Leslie Valiant's original 1979 paper The Complexity of Computing the Permanent shows that computing the permanent of a 0-1 matrix is #P-complete. As a second example, the list of #P-complete problems in the companion paper The Complexity of Enumeration and Reliability Problems includes #MONOTONE 2-SAT; this problem requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables and no negated variables are allowed. (MONOTONE 2-SAT is of course rather trivial as a decision problem.)

Andrea Montanari has written about the partition function of $k$-XORSAT in some lecture notes, and his book with Marc Mézard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).

These considerations lead to the following question:

Is #$k$-XORSAT #P-complete?

Note that the formula in Montanari's notes does not obviously appear to answer this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the Tutte polynomial.

Some difficult counting problems in #P can still be approximated in a certain sense, by means of a scheme called an FPRAS. Jerrum, Sinclair, and collaborators have linked the existence of an FPRAS for #P problems to the question of whether $NP = RP$. I would therefore also be interested in the subsidiary question

Does #$k$-XORSAT have an FPRAS?

Edit: clarified second question as per comment by Tsuyoshi Ito. Note that Peter Shor's answer renders this part of the question moot.

• Next na.numerical analysis - What is the time complexity of computing sin(x) to t bits of precision?
• Previous stochastic calculus - Moment Generating Function: Pulling a term out of k-times differentiation