Symbolic addresses (and array indices) are a problem for symbolic execution that could lead to a large explosion in SMT terms due to the need to add an ‘ite’ expression for every potentially aliasing write. Many symbolic execution tools deal with this by selectively concretizing addresses used for reads, for writes or for both – leading to lower coverage of the program (i.e., unsound results).
This paper proposes an efficient structure for tracking writes to symbolic addresses that does not limit their range (as in angr) and can support state merging (as in Veritesting avgerinos:icse:2014).
The base implementation represents memory as a set of tuples (e, v, t, delta) representing a store to address e of value v at time t under condition delta. (e, v and delta are all symbolic, t is concrete virtual time).
- Stores add (e,v,t,true) and increment virtual time.
- Loads construct an ite expression starting with the oldest stores.
- Merges conjoin deltas and increment virtual time to the max of the two paths.
They apply many optimizations to make this scale.
The ite-construction during a load can be terminated if an exact match is found.
Range calculations are used to limit the number of stores considered during a load.
Multiple stores to the same address can be garbage collected.
A trick for uninitialized memory involving negative virtual time.
The memory is organized as bytes and multi-byte accesses are split/joined as required.
An interval tree is used to efficiently find stored intervals that overlap with a read/write.
The interval tree is paged to support efficient copy-on-write operations and each tuple is contained in exactly one page. Page size is determined empirically.
Writes to concrete addresses (the majority af accesses) are stored in a separate structure that maps concrete addresses to a pair of a symbolic value and a timestamp. (This is a paged hashmap to support efficient copy-on-write.)
This was implemented as a plugin to angr (which already had a modular structure).
This was evaluated on the Cromulence (CROMU) group from the Cyber Grand Challenge by comparing with three different configurations of angr: fully concrete, symbolic within bounded address ranges (default angr) and fully symbolic but without all the tricks mentioned above. They run for 2 hours with 32GB RAM in what sounds like a breadth-first exploration. They are interested in how many paths are discarded due to concretization (in the standard angr) and whether tools timeout due to excessive size of symbolic expressions.
Results show that
- the more symbolic the treatment of addresses the more paths found
- their optimizations avoid timeouts