We consider random instances of the MAX-2-XORSAT optimization problem. A 2-XOR formula is a conjunction of Boolean equations (or clauses) of the form x ⊕ y = 0 or x ⊕ y = 1. The MAX-2-XORSAT problem asks for the maximum number of clauses which can be satisfied by any assignment of the variables in a 2-XOR formula. In this work, formula of size m on n Boolean variables are chosen uniformly at random from among all $\binom{n(n-1) }{ m}$ possible choices. Denote by X n,m the minimum number of clauses that can not be satisfied in a formula with n variables and m clauses. We give precise characterizations of the r.v. X n,m around the critical density of random 2-XOR formula. We prove that for random formulas with m clauses X n,m converges to a Poisson r.v. with mean $-\frac{1}{4}\log(1-2c)-\frac{c}{2}$ when m = cn, c ∈ ]0,1/2[ constant. If , μ and n are both large but μ = o(n 1/3), with is normal. If $m = \frac{n}{2} + O(1)n^{2/3}$ , is normal. If with 1 ≪ μ = o(n 1/3) then $\frac{ 12X_{n,m}}{2\mu^3+\log{n}-3\log(\mu)} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1$ . For any absolute constant ε> 0, if μ = εn 1/3 then $\frac{8(1+\varepsilon)}{n( \varepsilon^2 - \sigma^2)} X_{n,m} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1$ where σ ∈ (0,1) is the solution of (1 + ε)e − ε = (1 − σ)e σ . Thus, our findings describe phase transitions in the optimization context similar to those encountered in decision problems.