Question: XORification is the technique to make a Boolean function or formula harder by replacing every variable $x$ by the XOR of $k\geq 2$ distinct variables $x\_1 \oplus \ldots \oplus x\_k$. 

I am aware of uses of this technique in proof complexity, mainly to obtain space lower bounds for resolution-based proof systems, e.g., in the papers:

* Eli Ben-Sasson. Size space tradeoffs for resolution. STOC 2002, 457-464.
* Eli Ben-Sasson and Jakob Nordström. Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions. ICS 2011, 401-416.

Are there other uses of this technique in other areas?

Answer: 
Here's an somewhat relevant example we are currently covering in my class.

The "storage access function" is defined on $2^k+k$ bits as:

$SA(x\_1,...,x\_{2^k}, a\_1,...,a\_k) = x\_{bin(a\_1 \cdots a\_k)}$ 

where $bin(a\_1 \cdots a\_k)$ is the unique integer in $\{1,\ldots,2^k\}$ corresponding to the string $a\_1 \cdots a\_k$.

$SA$ has formulas of size about $O(k \cdot 2^k)$ over AND/OR/NOT: have $2^k$ groups of all possible $k$-ANDs over the $a\_i$ variables, so that exactly one group outputs $1$ on every input. Then AND each bit $x\_i$ with the output of the corresponding group, then OR all of these outputs. 

However, the following "SA of XOR" function, on $2^{k+1}$ inputs, requires about $2^{3k}$-size formulas over AND/OR/NOT:

$SA(x\_1,...,x\_{2^k}, \bigoplus\_{j=1}^{2^k/k} a\_{1,j},..., \bigoplus\_{j=1}^{2^k/k} a\_{k,j}) = x\_{bin(a\_1 \cdots a\_k)}$.

This is often called "Andreev's function" in the literature. Hastad [proved](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.9886) (improving a component of Andreev's argument) that cubic-size formulas are essentially necessary. (It is not hard to find nearly cubic-size formulas for it, too.)
This might be a slight reach, but the idea of XOR'ing a bunch of things to make a task "harder" shows up in cryptography. It first appeared in the guise of [Yao's XOR lemma](http://theory.csail.mit.edu/ftp-data/pub/ftp/tcryptol/old-05.ps). If $X$ is a slightly unpredictable random variable, then $Y = X\_1 \oplus X\_2 \oplus \cdots \oplus X\_k$ is extremely unpredictable if $k$ is large enough, where the $X\_i$'s are independent draws of $X$.

Nowadays, this technique is quite standard in crypto, typically for amplifying a weak construction (commitment scheme, oblivious transfer protocol, etc) into a strong one.