How physical addresses map to rows and banks in DRAM

In my previous blog post, I discussed how Intel Sandy Bridge CPUs map physical addresses to locations in the L3 cache.

Now I'll discuss how these CPUs' memory controllers map physical addresses to locations in DRAM -- specifically, to row, bank and column numbers in DRAM modules. Let's call this the DRAM address mapping. I'll use one test machine as a case study.

Motivation: the rowhammer bug

I am interested in the DRAM address mapping because it is relevant to the "rowhammer" bug.

Rowhammer is a problem with some DRAM modules whereby certain pessimal memory access patterns can cause memory corruption. In these DRAMs, repeatedly activating a row of memory (termed "row hammering") can produce electrical disturbances that produce bit flips in vulnerable cells in adjacent rows of memory.

These repeated row activations can be caused by repeatedly accessing a pair of DRAM locations that are in different rows of the same bank of DRAM. Knowing the DRAM address mapping is useful because it tells us which pairs of addresses satisfy this "same bank, different row" (SBDR) property.

Guessing and checking an address mapping

For my case study, I have a test machine containing DRAM that is vulnerable to the rowhammer problem. Running rowhammer_test on this machine demonstrates bit flips.

I'd like to know what the DRAM address mapping is for this machine, but apparently it isn't publicly documented: This machine has a Sandy Bridge CPU, but Intel don't document the address mapping used by these CPUs' memory controllers.

rowhammer_test does not actually need to identify SBDR address pairs. rowhammer_test just repeatedly tries hammering randomly chosen address pairs. Typically 1/8 or 1/16 of these pairs will be SBDR pairs, because our machine has 8 banks per DIMM (and 16 banks in total). So, while we don't need to know the DRAM address mapping to cause bit flips on this machine, knowing it would help us be more targeted in our testing.

Though the address mapping isn't documented, I found that I can make an educated guess at what the mapping is, based on the DRAM's geometry, and then verify the guess based on the physical addresses that rowhammer_test reports. rowhammer_test can report the physical addresses where bit flips occur ("victims") and the pairs of physical addresses that produce those bit flips ("aggressors"). Since these pairs must be SBDR pairs, we can check a hypothesised address mapping against this empirical data.

Memory geometry

The first step in hypothesising an address mapping for a machine is to check how many DIMMs the machine has and how these DIMMs are organised internally.

I can query information about the DIMMs using the decode-dimms tool on Linux. (In Ubuntu, decode-dimms is in the i2c-tools package.) This tool decodes the DIMMs' SPD (Serial Presence Detect) metadata.

My test machine has 2 * 4GB SO-DIMMs, giving 8GB of memory in total.

decode-dimms reports the following information for both of the DIMMs:

Size                                            4096 MB
Banks x Rows x Columns x Bits                   8 x 15 x 10 x 64
Ranks                                           2

This means that, for each DIMM:

• Each of the DIMM's banks contains 2^15 rows (32768 rows).
• Each row contains 2^10 * 64 bits = 2^16 bits = 2^13 bytes = 8 kbytes.

Each DIMM has 2 ranks and 8 banks. Cross checking the capacity of the DIMM gives us the reported size, as expected:

8 kbytes per row * 32768 rows * 2 ranks * 8 banks = 4096 MB = 4 GB

The DRAM address mapping

On my test machine, it appears that the bits of physical addresses are used as follows:

• Bits 0-5: These are the lower 6 bits of the byte index within a row (i.e. the 6-bit index into a 64-byte cache line).
• Bit 6: This is a 1-bit channel number, which selects between the 2 DIMMs.
• Bits 7-13: These are the upper 7 bits of the index within a row (i.e. the upper bits of the column number).
• Bits 14-16: These are XOR'd with the bottom 3 bits of the row number to give the 3-bit bank number.
• Bit 17: This is a 1-bit rank number, which selects between the 2 ranks of a DIMM (which are typically the two sides of the DIMM's circuit board).
• Bits 18-32: These are the 15-bit row number.
• Bits 33+: These may be set because physical memory starts at physical addresses greater than 0.

Why is the mapping like that?

This mapping fits the results from rowhammer_test (see below), but we can also explain that the address bits are mapped this way to give good performance for typical memory access patterns, such as sequential accesses and strided accesses:

• Channel parallelism: Placing the channel number at bit 6 means that cache lines will alternate between the two channels (i.e. the two DIMMs), which can be accessed in parallel. This means that if we're accessing addresses sequentially, the load will be spread across the two channels.

  As an aside, Ivy Bridge (the successor to Sandy Bridge) apparently makes the mapping of the channel number more complex. An Intel presentation mentions "Channel hashing" and says that this "Allows channel selection to be made based on multiple address bits. Historically, it had been "A[6]". Allows more even distribution of memory accesses across channels."

• Bank thrashing: Generally, column, bank and row numbers are arranged to minimise "bank thrashing" (frequently changing a bank's currently activated row).

  Some background: DRAM modules are organised into banks, which in turn are organised into rows. Each bank has a "currently activated row" whose contents are copied into a row buffer which acts as a cache that can be accessed quickly. Accessing a different row takes longer because that row must be activated first. So, the DRAM address mapping places SBDR pairs as far apart as possible in physical address space.

  Row hammering is a special case of bank thrashing where two particular rows are repeatedly activated (perhaps deliberately).

• Bank parallelism: Banks can be accessed in parallel (though to a lesser degree than channels), so the bank number changes before the row number as the address is increased.

• XOR scheme: XORing the row number's lower bits into the bank number is a trick to avoid bank thrashing when accessing arrays with large strides. For example, in the mapping above, the XORing causes addresses X and X+256k to be placed in different banks instead of being an SBDR pair.

  Bank/row XORing schemes are described in various places, such as:

  • David Tawei Wang's PhD thesis, "Modern DRAM memory systems: Performance analysis and scheduling algorithms", 2005. See section 5.3.5, "Bank Address Aliasing (stride collision)". This thesis has some good background info on DRAM in general.
  • The paper "Reducing DRAM Latencies with an Integrated Memory Hierarchy Design", 2001. See Figure 3.

Checking against rowhammer_test's output

I ran rowhammer_test_ext (the extended version of rowhammer_test) on my test machine for 6 hours, and it found repeatable bit flips at 22 locations. (See the raw data and analysis code.)

The row hammering test generates a set of (A1, A2, V) triples, where:

• V is the victim address where we see the bit flip.
• A1 and A2 are the aggressor addresses that we hammer.
• We sort A1 and A2 so that A1 is closer to V than A2 is. We tentatively assume that the closer address, A1, is the one that actually causes the bit flips (though this wouldn't necessarily be true if the DRAM address mapping were more complicated).

There are three properties we expect to hold for all of these results:

• Row: A1 and V's row numbers should differ by 1 -- i.e. they should be in adjacent rows. (A2 can have any row number.)

  This property makes it easy to work out where the bottom bits of the row number are in the physical address.

  We find this property holds for all but 2 of the results. In those 2 results, the row numbers differ by 3 rather than 1.

• Bank: V, A1 and A2 should have the same bank number. Indeed, we find this property holds for all 22 results. This only holds when applying the row/bank XORing scheme.

• Channel: V, A1 and A2 should have the same channel number. This holds for all the results. It happens that all of our results have channel=0, because rowhammer_test only selects 4k-aligned addresses and so only tests one channel. (Maybe this could be considered a bug.)

Possible further testing

There are two further experiments we could run to check whether the DRAM address mapping evaluates the SBDR property correctly, which I haven't tried yet:

• Timing tests: Accessing SBDR address pairs repeatedly should be slower than accessing non-SBDR pairs repeatedly, because the former cause row activations and the latter don't.

• Exhaustive rowhammer testing: Once we've found an aggressor address, A1, that causes a repeatable bit flip, we can test this against many values of address A2. Hammering (A1, A2) can produce bit flips only if this is an SBDR pair.

Furthermore, taking out one DIMM from our test machine should remove the channel bit from the DRAM address mapping and change the aggressor/victim addresses accordingly. We could check whether this is the case.