## Building a Curve25519 Hardware Accelerator

July 15th, 2021

Note: this post uses a $\LaTeX$ plugin to render math symbols. It may require a desktop browser for the full experience…

The “double ratchet” algorithm is integral to modern end-to-end-encrypted chat apps, such as Signal, WhatsApp, and Matrix. It gives encrypted conversations the properties of resilience, forward secrecy, and break-in recovery; basically, even if an adversary can manipulate or observe portions of an exchange, including certain secret materials, the damage is limited with each turn of the double ratchet.

The double-ratchet algorithm is a soup of cryptographic components, but one of the most computationally expensive portions is the “Diffie-Hellman (DH) key exchange”, using Elliptic Curve Diffie-Hellman (ECDH) with Curve25519. How expensive? This post from 2020 claims a speed record of 3.2 million cycles on a Cortex-M0 for just one of the core mathematical operations: fairly hefty. A benchmark of the x25519-dalek Rust crate on a 100 MHz RV32-IMAC implementation clocks in at 100ms per DH key exchange, of which several are involved in a double-ratchet. Thus, any chat client implementation on a small embedded CPU would suffer from significant UI lag.

There are a few strategies to rectify this, ranging from adding a second CPU core to off-load the crypto, to making a full-custom hardware accelerator. Adding a second RISC-V CPU core is expedient, but it wouldn’t do much to force me to understand what I was doing as far as the crypto goes; and there’s already a strong contingent of folks working on multi-core RISC-V on FPGA implementations. The last time I implemented a crypto algorithm was for RSA on a low-end STM32 back in the mid 2000’s. I really enjoyed getting into the guts of the algorithm and stretching my understanding of the underlying mathematical primitives. So, I decided to indulge my urge to tinker, and make a custom hardware accelerator for Curve25519 using Litex/Migen and Rust bindings.

### What the Heck Is $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$?

I wanted the accelerator primitive to plug directly into the Rust Dalek Cryptography crates, so that I would be as light-fingered as possible with respect to changes that could lead to fatal flaws in the broader cryptosystem. I built some simple benchmarks and profiled where the most time was being spent doing a double-ratchet using the Dalek Crypto crates, and the vast majority of the time for a DH key exchange was burned in the Montgomery Multiply operation. I won’t pretend to understand all the fancy math-terms, but people smarter than me call it a “scalar multiply”, and it actually consists of thousands of “regular” multiplies on 255-bit numbers, with modular reductions in the prime field $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$, among other things.

Wait, what? So many articles and journals I read on the topic just talk about prime fields, modular reduction and blah blah blah like it’s asking someone to buy milk and eggs at the grocery store. It was a real struggle to try and make sense of all the tricks used to do a modular multiply, much less to say even elliptic curve point transformations.

Well, that’s the point of trying to implement it myself — by building an engine, maybe you won’t understand all the hydrodynamic nuances around fuel injection, but you’ll at least gain an appreciation for what a fuel injector is. Likewise, I can’t say I deeply understand the number theory, but from what I can tell multiplication in $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$ is basically the same as a “regular 255-bit multiply” except you do a modulo (you know, the % operator) against the number $2^{{255}}-19$ when you’re done. There’s a dense paper by Daniel J. Bernstein, the inventor of Curve25519, which I tried to read several times. One fact I gleaned is part of the brilliance of Curve25519 is that arithmetic modulo $2^{{255}}-19$ is quite close to regular binary arithmetic except you just need to special-case 19 outcomes out of $2^{{255}}$. Also, I learned that the prime modulus, $2^{{255}}-19$, is where the “25519” comes from in the name of the algorithm (I know, I know, I’m slow!).

After a whole lot more research I stumbled on a paper by Furkan Turan and Ingrid Verbauwhede that broke down the algorithm into terms I could better understand. Attempts to reach out to them to get a copy of the source code was, of course, fruitless. It’s a weird thing about academics — they like to write papers and “share ideas”, but it’s very hard to get source code from them. I guess that’s yet another reason why I never made it in academia — I hate writing papers, but I like sharing source code. Anyways, the key insight from their paper is that you can break a 255-bit multiply down into operations on smaller, 17-bit units called “limbs” — get it? digits are your fingers, limbs (like your arms) hold multiple digits. These 17-bit limbs map neatly into 15 Xilinx 7-Series DSP48E block (which has a fast 27×18-bit multiplier): 17 * 15 = 255. Using this trick we can decompose multiplication in $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$ into the following steps:

1. Schoolbook multiplication with 17-bit “limbs”
2. Collapse partial sums
3. Propagate carries
4. Is the sum $\geq$ $2^{{255}}-19$?
6. Propagate carries again, in case the addition by 19 causes overflows

The multiplier would run about 30% faster if step (6) were skipped. This step happens in a fairly small minority of cases, maybe a fraction of 1%, and the worst-case carry propagate through every 17-bit limb is diminishingly rare. The test for whether or not to propagate carries is fairly straightforward. However, short-circuiting the carry propagate step based upon the properties of the data creates a timing side-channel. Therefore, we prefer a slower but safer implementation, even if we are spending a bunch of cycles propagating zeros most of the time.

Buckle up, because I’m about to go through each step in the algorithm in a lot of detail. One motivation of the exercise was to try to understand the math a bit better, after all!

#### TL;DR: Magic Numbers

If you’re skimming this article, you’ll see the numbers “19” and “17” come up over and over again. Here’s a quick TL;DR:

• 19 is the tiny amount subtracted from $2^{{255}}$ to arrive at a prime number which is the largest number allowed in Curve25519. Overflowing this number causes things to wrap back to 0. Powers of 2 map efficiently onto binary hardware, and thus this representation is quite nice for hardware people like me.
• 17 is a number of bits that conveniently divides 255, and fits into a single DSP hardware primitive (DSP48E) that is available on the target device (Xilinx 7-series FPGA) that we’re using in this post. Thus the name of the game is to split up this 255-bit number into 17-bit chunks (called “limbs”).

#### Schoolbook Multiplication

The first step in the algorithm is called “schoolbook multiplication”. It’s like the kind of multiplication you learned in elementary or primary school, where you write out the digits in two lines, multiply each digit in the top line by successive digits in the bottom line to create successively shifted partial sums that are then added to create the final result. Of course, there is a twist. Below is what actual schoolbook multiplication would be like, if you had a pair of numbers that were split into three “limbs”, denoted as A[2:0] and B[2:0]. Recall that a limb is not necessarily a single bit; in our case a limb is 17 bits.

                   |    A2        A1       A0
x              |    B2        B1       B0
------------------------------------------
| A2*B0     A1*B0    A0*B0
A2*B1  | A1*B1     A0*B1
A2*B2    A1*B2  | A0*B2
(overflow)         (not overflowing)


The result of schoolbook multiplication is a result that potentially has 2x the number of limbs than the either multiplicand.

Mapping the overflow back into the prime field (e.g. wrapping the overflow around) is a process called reduction. It turns out that for a prime field like $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$, reduction works out to taking the limbs that extend beyond the base number of limbs in the field, shifting them right by the number of limbs, multiplying it by 19, and adding it back in; and if the result isn’t a member of the field, add 19 one last time, and take the result as just the bottom 255 bits (ignore any carry overflow). If this seems magical to you, you’re not alone. I had to draw it out before I could understand it.

This trick works because the form of the field is $2^{{n}}-p$: it is a power of 2, reduced by some small amount $p$. By starting from a power of 2, most of the binary numbers representable in an n-bit word are valid members of the field. The only ones that are not valid field members are the numbers that are equal to $2^{{n}}-p$ but less than $2^{{n}}-1$ (the biggest number that fits in n bits). To turn these invalid binary numbers into members of the field, you just need to add $p$, and the reduction is complete.

A diagram illustrating modular reduction

The diagram above draws out the number lines for both a simple binary number line, and for some field $\mathbf{{F}}_{{{{2^{{n}}}}-p}}$. Both lines start at 0 on the left, and increment until they roll over. The point at which $\mathbf{{F}}_{{{{2^{{n}}}}-p}}$ rolls over is a distance $p$ from the end of the binary number line: thus, we can observe that $2^{{n}}-1$ reduces to $p-1$. Adding 1 results in $2^{{n}}$, which reduces to $p$: that is, the top bit, wrapped around, and multiplied by $p$.

As we continue toward the right, the numbers continue to go up and wrap around, and for each wrap the distance between the “plain old binary” wrap point and the $\mathbf{{F}}_{{{{2^{{n}}}}-p}}$ wrap point increases by a factor of $p$, such that $2^{{n+1}}$ reduces to $2*p$. Thus modular reduction of natural binary numbers that are larger than our field $2^{{n}}-p$ consists of taking the bits that overflow an $n$-bit representation, shifting them to the right by $n$, and multiplying by $p$.

In order to convince myself this is true, I tried out a more computationally tractable example than $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$: the prime field $\mathbf{{F}}_{{{{2^{{6}}}}-5}} = 59$. The members of the field are from 0-58, and reduction is done by taking any number modulo 59. Thus, the number 59 reduces to 0; 60 reduces to 1; 61 reduces to 2, and so forth, until we get to 64, which reduces to 5 — the value of the overflowed bits (1) times $p$.

Let’s look at some more examples. First, recall that the biggest member of the field, 58, in binary is 0b00_11_1010.

Let’s consider a simple case where we are presented a partial sum that overflows the field by one bit, say, the number 0b01_11_0000, which is decimal 112. In this case, we take the overflowed bit, shift it to the right, multiply by 5:

0b01_11_0000
^ move this bit to the right multiply by 0b101 (5)

0b00_11_0000 + 0b101 = 0b00_11_0101 = 53


And we can confirm using a calculator that 112 % 59 = 53. Now let’s overflow by yet another bit, say, the number 0b11_11_0000. Let’s try the math again:

0b11_11_0000
^ move to the right and multiply by 0b101: 0b101 * 0b11 = 0b1111

0b00_11_0000 + 0b1111 = 0b00_11_1111


This result is still not a member of the field, as the maximum value is 0b0011_1010. In this case, we need to add the number 5 once again to resolve this “special-case” overflow where we have a binary number that fits in $n$ bits but is in that sliver between $2^{{n}}-p$ and $2^{{n}}-1$:

0b00_11_1111 + 0b101 = 0b01_00_0100


At this step, we can discard the MSB overflow, and the result is 0b0100 = 4; and we can check with a calculator that 240 % 59 = 4.

Therefore, when doing schoolbook multiplication, the partial products that start to overflow to the left can be brought back around to the right hand side, after multiplying by $p$, in this case, the number 19. This magical property is one of the reasons why $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$ is quite amenable to math on binary machines.

Let’s use this finding to rewrite the straight schoolbook multiplication form from above, but now with the modular reduction applied to the partial sums, so it all wraps around into this compact form:

                   |    A2        A1       A0
x              |    B2        B1       B0
------------------------------------------
| A2*B0     A1*B0    A0*B0
| A1*B1     A0*B1 19*A2*B1
+ | A0*B2  19*A2*B2 19*A1*B2
----------------------------
S2        S1       S0


As discussed above, each overflowed limb is wrapped around and multiplied by 19, creating a number of partial sums S[2:0] that now has as many terms as there are limbs, but with each partial sum still potentially overflowing the native width of the limb. Thus, the inputs to a limb are 17 bits wide, but we retain precision up to 48 bits during the partial sum stage, and then do a subsequent condensation of partial sums to reduce things back down to 17 bits again. The condensation is done in the next three steps, “collapse partial sums”, “propagate carries”, and finally “normalize”.

However, before moving on to those sections, there is an additional trick we need to apply for an efficient implementation of this multiplication step in hardware.

In order to minimize the amount of data movement, we observe that for each row, the “B” values are shared between all the multipliers, and the “A” values are constant along the diagonals. Thus we can avoid re-loading the “A” values every cycle by shifting the partial sums diagonally through the computation, allowing the “A” values to be loaded as “A” and “A*19” into holding register once before the computations starts, and selecting between the two options based on the step number during the computation.

Mapping schoolbook multiply onto the hardware array to minimize data movement

The diagram above illustrates how the schoolbook multiply is mapped onto the hardware array. The top diagram is an exact redrawing of the previous text box, where the partial sums that would extend to the left have been multiplied by 19 and wrapped around. Each colored block corresponds to a given DSP48E1 block, which you may recall is a fast 27×18 multiplier hardware primitive built into our Xilinx FPGAs. The red arrow illustrates the path of a partial sum in both the schoolbook form and the unwrapped form for hardware implementation. In the bottom diagram, one can clearly see that the Ax coefficients are constant for each column, and that for each row, the Bx values are identical across all blocks in each step. Thus each column corresponds to a single DSP48E1 block. We take advantage of the ability of the DSP48E1 block to hold two selectable A values to pre-load Ax and Ax*19 before the computation starts, and we bus together the Bx values and change them in sequence with each round. The partial sums are then routed to the “down and right” to complete the mapping. The final result is one cycle shifted from the canonical mapping.

We have a one-cycle structural pipeline delay going from this step to the next one, so we use this pipeline delay to do a shift with no add by setting the opmode from C+M to C+0 (in other words, instead of adding to the current multiplication output for the last step, we squash that input and set it to 0).

The fact that we pipeline the data also gives us an opportunity to pick up the upper limb of the partial sum collapse “for free” by copying it into the “D” register of the DSP48E1 during the shift step.

In C, the equivalent code basically looks like this:

   // initialize the a_bar set of data
for( int i = 0; i < DSP17_ARRAY_LEN; i++ ) {
a_bar_dsp[i] = a_dsp[i] * 19;
}
operand p;
for( int i = 0; i < DSP17_ARRAY_LEN; i++ ) {
p[i] = 0;
}

// core multiply
for( int col = 0; col < 15; col++ ) {
for( int row = 0; row < 15; row++ ) {
if( row >= col ) {
p[row] += a_dsp[row-col] * b_dsp[col];
} else {
p[row] += a_bar_dsp[15+row-col] * b_dsp[col];
}
}
}


By leveraging the special features of the DSP48E1 blocks, in hardware this loop completes in just 15 clock cycles.

#### Collapse Partial Sums

At this point, the potential width of the partial sum is up to 43 bits wide. This next step divides the partial sums up into 17-bit words, and then shifts the higher to the next limbs over, allowing them to collapse into a smaller sum that overflows less.

... P2[16:0]   P1[16:0]      P0[16:0]
... P1[33:17]  P0[33:17]     P14[33:17]*19
... P0[50:34]  P14[50:34]*19 P13[50:34]*19


Again, the magic number 19 shows up to allow sums which “wrapped around” to add back in. Note that in the timing diagram you will find below, we refer to the mid- and upper- words of the shifted partial sums as “Q” and “R” respectively, because the timing diagram lacks the width within a data bubble to write out the full notation: so Q0,1 is P14[33:17] and R0,2 is P13[50:34] for P0[16:0].

Here’s the C code equivalent for this operation:

     // the lowest limb has to handle two upper limbs wrapping around (Q/R)
prop[0] = (p[0] & 0x1ffff) +
(((p[14] * 1) >> 17) & 0x1ffff) * 19 +
(((p[13] * 1) >> 34) & 0x1ffff) * 19;
// the second lowest limb has to handle just one limb wrapping around (Q)
prop[1] = (p[1] & 0x1ffff) +
((p[0] >> 17) & 0x1ffff) +
(((p[14] * 1) >> 34) & 0x1ffff) * 19;
// the rest are just shift-and-add without the modular wrap-around
for(int bitslice = 2; bitslice < 15; bitslice += 1) {
prop[bitslice] = (p[bitslice] & 0x1ffff) + ((p[bitslice - 1] >> 17) & 0x1ffff) + ((p[bitslice - 2] >> 34));
}


This completes in 2 cycles after a one-cycle pipeline stall delay penalty to retrieve the partial sum result from the previous step.

#### Propagate Carries

The partial sums will generate carries, which need to be propagated down the chain. The C-code equivalent of this looks as follows:

   for(int i = 0; i < 15; i++) {
if ( i+1 < 15 ) {
prop[i+1] = (prop[i] >> 17) + prop[i+1];
prop[i] = prop[i] & 0x1ffff;
}
}


The carry-propagate completes in 14 cycles. Carry-propagates are expensive!

#### Normalize

We’re almost there! Except that $0 \leq result \leq 2^{{256}}-1$, which is slightly larger than the range of $\mathbf{{F}}_{{{{2^{{255}}}}-19}}$.

Thus we need to check if number is somewhere in between 0x7ff….ffed and 0x7ff….ffff, or if the 256th bit will be set. In these cases, we need to add 19 to the result, so that the result is a member of the field $2^{{255}}-19$ (the 256th bit is dropped automatically when concatenating the fifteen 17-bit limbs together into the final 255-bit result).

We use another special feature of the DSP48E1 block to help accelerate the test for this case, so that it can complete in a single cycle without slowing down the machine. We use the “pattern detect” (PD) feature of the DSP48E1 to check for all “1’s” in bit positions 255-5, and a single LUT to compare the final 5 bits to check for numbers between {prime_string} and $2^{{255}}-1$. We then OR this result with the 256th bit.

With the help of the special DSP48E1 features, this operation completes in just a single cycle.

After adding the number 19, we have to once again propagate carries. Even if we add the number 0, we also have to “propagate carries” for constant-time operation, to avoid leaking information in the form of a timing side-channel. This is done by running the carry propagate operation described above a second time.

Once the second carry propagate is finished, we have the final result.

#### Potential corner case

There is a potential corner case where if the carry-propagated result going into “normalize” is between

0xFFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFDA and
0xFFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFEC


In this case, the top bit would be wrapped around, multiplied by 19, and added to the LSB, but the result would not be a member of $2^{{255}}-19$ (it would be one of the 19 numbers just short of $2^{{255}}-1$), and the multiplier would pass it on as if it were a valid result.

In some cases, this isn’t even a problem, because if the subsequent result goes through any operation that also includes a reduce operation, the result will still reduce correctly.

However, I do not think this corner case is possible, because the overflow path to set the high bit is from the top limb going from 0x1_FFFF -> 0x2_0000 (that is, 0x7FFFC -> 0x80000 when written MSB-aligned) due to a carry coming in from the lower limb, and it would require the carry to be very large, not just +1 as shown in the simple rollover case, but a value from 0x1_FFED-0x1_FFDB.

I don’t have a formal mathematical proof of this, but I strongly suspect that carry values going into the top limb cannot approach these large numbers, and therefore it is not possible to hit this corner case. Consider that the biggest value of a partial sum is 0x53_FFAC_0015 (0x1_FFFF * 0x1_FFFF * 15). This means the biggest value of the third overflowed 17-bit limb is 0x14. Therefore the biggest value resulting from the “collapse partial sums” stage is 0x1_FFFF + 0x1_FFFF + 0x14 = 0x4_0012. Thus the largest carry term that has to propagate is 0x4_0012 >> 17 = 2. 2 is much smaller than the amount required to trigger this condition, that is, a value in the range of 0x1_FFED-0x1_FFDB. Thus, perhaps this condition simply can’t happen? It’d be great to have a real mathematician comment if this is a real corner case…

#### Real Hardware

You can jump to the actual code that implements the above algorithm, but I prefer to think about implementations visually. Thus, I created this timing diagram that fully encapsulates all of the above steps, and the data movements between each part (click on the image for an editable, larger version; works best on desktop):

Block diagrams of the multiplier and even more detailed descriptions of its function can be found in our datasheet documentation. There’s actually a lot to talk about there, but the discussion rapidly veers into trade-offs on timing closure and coding technique, and farther away from the core topic of the Curve25519 algorithm itself.

### Didn’t You Say We Needed Thousands of These…?

So, that was the modular multiply. We’re done right? Nope! This is just one core op in a sequence of thousands of these to do a scalar multiply. One potentially valid strategy could be to try to hang the modular multiplier off of a Wishbone bus peripheral and shove numbers at it, and come back and retrieve results some time later. However, the cost of pushing 256-bit numbers around is pretty high, and any gains from accelerating the multiply will quickly be lost in the overhead of marshaling data. After all, a recurring theme in modern computer architecture is that data movement is more expensive than the computation itself. Damn you, speed of light!

Thus, in order to achieve the performance I was hoping to get, I decided to wrap this inside a microcoded “CPU” of sorts. Really more of an “engine” than a car — if a RISC-V CPU is your every-day four-door sedan optimized for versatility and efficiency, the microcoded Curve25519 engine I created is more of a drag racer: a turbocharged engine block on wheels that’s designed to drive long flat stretches of road as fast as possible. While you could use this to drive your kids to school, you’ll have a hard time turning corners, and you’ll need to restart the engine after every red light.

Above is a block diagram of the engine’s microcoded architecture. It’s a simple “three-stage” pipeline (FETCH/EXEC/RETIRE) that runs at 50MHz with no bypassing (that would be extremely expensive with 256-bit wide datapaths). I was originally hoping we could close timing at 100MHz, but our power-optimized -1L FPGA just wouldn’t have it; so the code sequencer runs at 50MHz; the core multiplier at 100MHz; and the register file uses four phases at 200MHz to access a simple RAM block to create a space-efficient virtual register file that runs at 50MHz.

The engine has just 13 opcodes:

There’s no compiler for it; instead, we adapted the most complicated Rust macro I’ve ever seen from johnas-schievink’s rustasm6502 crate to create the abomination that is engine25519-as. Here’s a snippet of what the assembly code looks like, in-lined as a Rust macro:

let mcode = assemble_engine25519!(
start:
// from FieldElement.invert()
// let (t19, t3) = self.pow22501();   // t19: 249..0 ; t3: 3,1,0
// let t0  = self.square();           // 1         e_0 = 2^1
mul %0, %30, %30  // self is W, e.g. %30
// let t1  = t0.square().square();    // 3         e_1 = 2^3
mul %1, %0, %0
mul %1, %1, %1
// let t2  = self * &t1;              // 3,0       e_2 = 2^3 + 2^0
mul %2, %30, %1
// let t3  = &t0 * &t2;               // 3,1,0
mul %3, %0, %2
// let t4  = t3.square();             // 4,2,1
mul %4, %3, %3
// let t5  = &t2 * &t4;               // 4,3,2,1,0
mul %5, %2, %4

// let t6  = t5.pow2k(5);             // 9,8,7,6,5
psa %28, #5       // coincidentally, constant #5 is the number 5
mul %6, %5, %5
pow2k_5:
sub %28, %28, #1  // %28 = %28 - 1
brz pow2k_5_exit, %28
mul %6, %6, %6
brz pow2k_5, #0
pow2k_5_exit:
// let t7  = &t6 * &t5;               // 9,8,7,6,5,4,3,2,1,0
mul %7, %6, %5
);


The mcode variable is a [i32] fixed-length array, which is quite friendly to our no_std Rust environment that is Xous.

Fortunately, the coders of the curve25519-dalek crate did an amazing job, and the comments that surround their Rust code map directly onto our macro language, register numbers and all. So translating the entire scalar multiply inside the Montgomery structure was a fairly straightforward process, including the final affine transform.

### How Well Does It Run?

The fully accelerated Montgomery multiply operation was integrated into a fork of the curve25519-dalek crate, and wrapped into some benchmarking primitives inside Xous, a small embedded operating system written by Xobs. A software-only implementation of curve25519 would take about 100ms per DH operation on a 100MHz RV32-IMAC CPU, while our hardware-accelerated version completes in about 6.7ms — about a 15x speedup. Significantly, the software-only operation does not incur the context-switch to the sandboxed hardware driver, whereas our benchmark includes the overhead of the syscall to set up and run the code; thus the actual engine itself runs a bit faster per-op than the benchmark might hint at. However, what I’m most interested in is in-application performance, and therefore I always include the overhead of swapping to the hardware driver context to give an apples-to-apples comparison of end-user application performance. More importantly, the CPU is free to do other things while the engine does it’s thing, such as servicing the network stack or updating the UX.

I think the curve25519 accelerator engine hit its goals — it strapped enough of a rocket on our little turtle of a CPU so that it’ll be able render a chat UX while doing double-ratchets as a background task. I also definitely learned more about the algorithm, although admittedly I still have a lot more to learn if I’m to say I truly understand elliptic curve cryptography. So far I’ve just shaken hands with the fuzzy monsters hiding inside the curve25519 closet; they seem like decent chaps — they’re not so scary, I just understand them poorly. A couple more interactions like this and we might even become friends. However, if I were to be honest, it probably wouldn’t be worth it to port the curve25519 accelerator engine from its current FPGA format to an ASIC form. Mask-defined silicon would run at least 5x faster, and if we needed the compute power, we’d probably find more overall system-level benefit from a second CPU core than a domain-specific accelerator (and hopefully by then the multi-core stuff in Litex will have sufficiently stabilized that it’d be a relatively low-risk proposition to throw a second CPU into a chip tape-out).

That being said, I learned a lot, and I hope that by sharing my experience, someone else will find Curve25519 a little more approachable, too!

## Name that Ware June 2021

June 30th, 2021

The Ware for June 2021 is shown below.

This is, I think, the last from the Don Straney collection (thank you so much for helping to bridge me through the pandemic!). Maybe I’m showing a bit too much here to make it hard to guess, but, honestly, I had never seen the inside of one of these types of things before, and I probably never will in person; so I wanted to share more than less of its construction details. It’s kind of crazy what a rat’s nest it is on the inside; I particularly love the tiny loops of wire which I’m pretty sure are meant to serve as inductive elements, and not just jumpers.

## Winner, Name that Ware May 2021

June 30th, 2021

For email subscribers, apologies in advance if you get two copies of the Name That Ware post series this month — I’m still trying to sort out some new email client configurations. I’m guessing it’ll take a few tries before everything works right.

On to the Ware — looks like I left a little too much text in on this one! Taylan Ayken got the exact model and make, a 256-channel Net Amps 410 from Electrical Geodesics. Congrats, email me for your prize!

Whenever you hear things like “256 channels” these days it’s easy to assume it has to be all integrated into a single chip. But, then you come across gems like these and you’re re-assured that the technique of the board-level integration of hundreds of channels is still alive and kicking, despite the best efforts of Moore’s Law to integrate everything into a single epoxy blob.

## Email Subscription Plugin Migration

June 26th, 2021

Sorry for this bit of administrivia — apparently, Feedburner, which I have been using for years to serve email subscriptions, is ceasing the email service in a couple of weeks. My first instinct was to just trash the service altogether, but I was pleasantly surprised to see I had a not-small number of email subscribers. So, instead of just trashing the feature I’m attempting to do a graceful migration.

Unfortunately, I’m not well-qualified to configure an email plugin for WordPress — social media is about fifty abstraction layers above my comfort zone of solder and assembly language. It doesn’t help that if you try to Google “email subscription plugin WordPress” you are assaulted by a long list of low-quality but highly SEO-optimized and/or promoted suggestions, many of which seem downright sketchy. So, based on nothing more than the number of installs and the appearance of an active maintainer/developer, I picked the Icegram plugin to try and manage email subscriptions from here forward.

I’m hoping existing subscribers barely notice any change, and of course, feel free to unsubscribe if you find anything annoying — the last thing I want to do is to spam anyone.

But, it seems email plugins have gotten far too crafty about by far too many things. If anyone has any advice on how I could handle this transition better or if anyone notices this plugin doing something strange or privacy-invasive, drop a comment here and I’ll try to figure things out. I’m not going to learn PHP and CSS to fix cosmetic annoyances like the improperly sized email sign-up field on Firefox…but I’d rather trash email updates altogether than invade a not-small number of people’s privacy.

Running a blog was so much simpler twenty years ago…

## Adding a ChaCha Cipher to Precursor’s TRNG

June 22nd, 2021

This is the second post of a two-part series on Betrusted/Precursor’s True Random Number Generator (TRNG).

A bulletproof source of random numbers is a key component of any cryptosystem, so we’ve done an extensive, months-long characterization of Precursor’s dual, redundant TRNG sources, which consists of an avalanche noise generator and a ring oscillator. We’ve found them to produce passable raw entropy, but you don’t have to take my word for it. You can download our raw data and run your on analysis on it if you like (at least until our ISP cuts us off for serving multiple 10GiB files filled with random data).

However, the system is still missing two features that are generally considered to be best practice:

1. Independent, on-line health monitors of the raw TRNG outputs, discussed in our previous post.
2. Conditioning of the raw data.

Because Precursor uses an FPGA for its SoC, we can add new features to the hardware “on the fly”, and in this post will focus on adding the hardware for the conditioning of raw data. The post is a bit of a slog to read through; I’ll try not to do too many in this style. But occasionally, I think it’s good to have a “reality check” that conveys the depth of difficulties encountered while implementing a feature, instead of making everything look easy or cramming all the details into a tweet. This post will hopefully also serve as a detailed reference for the handful of future developers who want to implement similar features, and would like to avoid some of the mistakes I’ve made.

Despite best efforts to make TRNGs unbiased and flawless, they are really hard to get right. Furthermore, they are only capable of producing high-quality entropy at a limited data rate. Thus, most practical systems take a TRNG output and run it through a cryptographic stream cipher to generate a final datastream; this conditioning of the raw data simultaneously protects against minor flaws in the TRNG while improving the availability of entropy.

So in Precursor, we’d like to add some conditioning on the collected entropy. To be clear, we will always make the raw entropy pool available for inspection — this is absolutely necessary for testing and CI coverage — but it’s a good idea to throw the output of your TRNG into a cryptographically sound stream cipher of some type, so that in the case that your TRNG suffers a sporadic drop-out, you don’t have a disastrous, instantaneous collapse of entropy. Depending on how you configure the conditioning, you can also increase the effective availability of random numbers.

After poking around a bit on the Internet, it seems popular to feed a seed of entropy into the ChaCha20 stream cipher (I refer to it as a “cipher” in this post, but I think more technically because of the way I’m using it, it should be referred to as a CSPRNG – Cryptographically Secure Pseudo Random Number Generator). This is done in the Linux kernel as well as by cryptech.is’s HSM and a few other implementations. The devil, of course, is always in the details. Cryptech.is takes the output of their TRNGs, hashes them with a SHA2 block, and then feeds it into a ChaCha20 stream cipher. Linux maintains an entropy pool that is collected from a variety of low-and-high-quality sources, blends them using some fast and slow techniques based upon the nature of the source, runs it through SHA1, and then into ChaCha20 to create the final stream.

In our implementation, we’re skipping the SHA pre-hash of the entropy pool, for the following reasons:

1. We don’t have the space (as in FPGA gates) or computational resources (as in CPU cycles for software emulation) to do this continuously
2. The raw entropy sources themselves are fairly high-quality (as opposed to in the case of Linux they are taking things like time stamps and network traffic patterns, so using SHA-1 to blind that is quite reasonable); and, we have two entropy sources of different architectures, each source consisting of multiple raw entropy generation elements. In other words, we’re starting from a pretty good place already.
3. The ChaCha20 cipher is mainly to protect against transient drop-out or unpredicted aging issues in the TRNG.
4. The ChaCha20 cipher itself is considered to be cryptographically secure; augmenting it with a SHA pre-hash does not seem to lend any clear benefit in our case.
5. We will prefer to reseed the ChaCha20 cipher at a fairly high rate, rather than seed it occasionally through a more expensive SHA operation.

Also in reading some of the docs out there, it seem that 20 is the minimum number of rounds considered to be cryptographically secure for ChaCha, but “more rounds is better”. Thus, while tweaking the Secworks open-source Chacha implementation to fit in Precursor, I think I’ll also create an option to let the ChaCha rounds function run to at least a minimum number of rounds but otherwise “free run” when the system is idle. This makes it useless as a cipher, but if the power impact is minimal, this should help the diffusion of entropy across all the bits of the cipher and make it more robust against attempts to back-track its state.

The diagram above gives an “artist’s impression” of the architecture of the system. It’s an early sketch, before I fully understood everything going on. The final details came out a little different, but the big ideas are there. Not shown is logic that ensures the ChaCha20 cipher has completed its minimum amount of rounds between each output sample, and automatic re-seeding logic that pulls in 32 bits of entropy from the TRNGs on a regular basis at some re-seeding interval that can be programmed by software.

After installing the tooling necessary to build a Precursor/Betrusted SoC and run simulations, I started writing the code.

Here’s the general method I use to develop code:

1. Think about what I’m trying to do. See above.
2. Write the smaller submodules.
3. Wrap the smaller modules into a simulation framework that shakes most of the skeletons out of the closet.
4. Repeat 1-3, working your way up the chain until you arrive at your full solution.
5. Write drivers for your new feature
6. Slot your feature into the actual hardware
7. Test in real hardware
8. Continuously integrate, if possible, either by re-running your sim against every repo change or better yet recompiling and re-running your test on actual hardware.

The key to this loop is the simulation. The better your simulation, the better your outcome. By “better simulation”, I mean, the less assumptions and approximations made in the test bench. For example, one could simulate a module by hooking it up to a hand-rolled set of Verilog vectors that exercises a couple read and write cycles and verifies nothing explodes; or, one could simulate a module by hooking it up to a fully simulated CPU, complete with power-on reset and multiple clock phases, and using a Rust-based framework to exercise the same reads and writes. The two test benches ostensibly achieve the same outcome, but the latter checks much more of the hairy corner cases.

For Betrusted/Precursor, we developed a comprehensive simulation framework that achieves the latter scenario as much as possible. We simulate a full, gate-level VexRISCV CPU, running a Rust-built BIOS, employing as many of the Xilinx-provided hardware models as we can for things like the PLL and global power-on reset. You can make a new testbench by running the “new_sim.py” script in the deps/gateware/sim directory, and it will automatically copy over all the basics so you can just instantiate your core module and start running simulation code.

Another thing I learned perhaps too late was to pick a good IDE. Python sucks to develop. It sucks even more without an IDE. For a long time I was writing in vanilla emacs. The game-changer for me was being able to control-click through modules and find their definitions; most IDEs support this.

### Writing The Block

I noted in Cryptech.is’s presentations that they were using a 24-round variant with a potential to go to 32; and even though cryptanalysis shows its solid with 20 rounds, perhaps there is still a potential that maybe some knowledge of the output could be used to infer something about the next set of outputs.

So, I still made a very light-fingered change, where the core round function could be advanced by one during idle periods based on a force_round input. In any case, the output of the ChaCha cipher is not used until at least 20 rounds in any case, but on average it will go many more rounds (thousands of rounds) before the output is used, and the number of rounds advanced depends upon the exact timing of the code that accesses the TRNG.

Thus, the overall flow of events for the conditioning block are as follows:

1. On boot, we pull in 384 bits of “key” state (256 bits key + 64 bits iv + 64 bits nonce) from the TRNG, and then initialize the cipher.
2. We also pull in 512 bits of “input” state. This is a fixed quantity that is XOR’d with the ChaCha block state. I worry that tying this to a fixed value could lead to leakage of the ChaCha state to the output, thus, on every boot we load in a new random input value.
3. Once the cipher is initialized, a counter keeps track of how many blocks we’ve generated. Once we have passed a threshold, the conditioner automatically requests another 32-bit word of entropy from the TRNG and does a rotate-and-XOR of the data into the “key” state. The key state is re-incorporated on the next “advance” call of the block.
4. Every “selfmix” interval, the cipher will also advance its rounds counter by one (this is the supplemental entropy/scrambling as noted previously)
5. The user can also supply seeding data at any time, which is folded in with a rotate-and-XOR function. Note for every seed provided, it’s best practice to read 16 words of TRNG output, to ensure that the seed state is incorporated into the TRNG as a whole. It doesn’t harm things to not do that, just, if you kept on applying the same seed over and over again it’ll eventually just XOR into and out of the key pool without doing anything useful.
6. The output of the cipher is latched into a 512-bit register, and urandom data outputs are multiplexed to the kernel and userspace through the CSR. A valid bit is provided, and it should be checked prior to every read of data from the Conditioned TRNG output.

### Building, and Failing

So your code is all writ, now time to try a quick “smoke test” to see if the build fails. And, sure enough, even though the build completes, I’m not meeting timing. We’re using a -1L variant of the Spartan 7 part, which is the lowest power model (important for battery life) but also the slowest performer in the group. So, even though Cryptech.is advertised hitting 100MHz performance on the block, I did note they were using a -3 variant (one of the fastest models) of a fairly high-end Artix FPGA.

So, back to the drawing board — it’s time to drop the clock frequency on the core to 50MHz. This is probably fine from a performance standpoint, as it could still yield 1.6Gbps burst rate of random numbers, but it does introduce some potentially thorny issues with clock domain crossings. In particular, I don’t get to be lazy and just use a bunch of Migen’s BusSynchronizer primitives to bridge 512-bit data between domains. That would be…expensive, in terms of logic resources. Instead, I’m going to have to be careful and use multi-cycle paths with explicit timing exceptions to make sure this works and without consuming undue amounts of resources.

What — you mean you thought getting code to compile was the end of the story? In hardware, every flip flop is a race condition against a clock — just think of it that way. Every time you make a register, you now have a potential concurrency problem. Now scale that up to the tens of thousands of registers in an FPGA, and you get an idea of why correct compilation is only a small portion of the whole story.

### Failing More, in Order to Get Better

After reducing the frequency of the core to 50MHz and fighting for a half day with the Vivado constraints syntax to convince it not to spend half an hour routing cross-clock-domain that are actually relaxed, I finally have the design building with timing closure. The next step is going back into the simulation bench and checking all the corner cases.

I modified the existing trng_managed simulation bench to add a series of tests that would exercise all the new features — including checking the rate of read, all the dual port options, and deliberately forcing some reads to fail by bypassing certain checks on validity of data to really push the engine to its limit. In the process of running these benches, I found more new corner cases, for example, the ChaCha engine would stall if the raw entropy FIFO ran out of seeding data; it would just wait for reboot of the avalanche+ring oscillator TRNGs, a process that can take milliseconds. This shouldn’t happen on the urandom engine — it is the “unlimited” random data engine after all, so some fixes were made to keep the block state evolving even if we ran out of seeding data. However, I have to emphasize this really is a corner case, you have to basically configure the system to constantly reseed and pull entropy out at a rate much higher than you can do anything useful with it to hit that limit. But, that’s the point of testing — to break your system before users do.

### Rinse, Lather, Repeat

Now that the simulation runs, we repeat the loop and see how badly timing is broken. It turns out with all the tweaks in place, I’ve introduced an accidental critical path. While the design can meet timing with this, it takes almost 3x the time to build, which means we’re probably going to start failing builds pretty soon. I should probably fix this problem now, while the design details are fresh in my head.

I open up the design in the Vivado tools and have it plot a timing histogram for me, below is an example of the output:

Above: Example of a slack histogram generated by Vivado. Green bars toward the left indicate paths with less timing slack, to the right have more. So for example, the left-most bin has 206 paths with 0.041ns-0.227ns of slack. The more “left-heavy” the histogram is, the harder the design is to place & route, and the more likely the process will not complete with perfect timing closure.

The left-most bin are the most over-constrained elements, and I can see that the trngmanaged_holding_buf registers are just barely passing.

I generate a schematic view like this inside the Vivado design tool, and backtracing the inputs to the holding_buf registers I can see the signal eventually goes through an enormous combinational path that includes the VexRiscV’s address decoders. This means that we have a critical path that extends from inside the VexRiscV load/store unit all the way to the 512-bit shifter of the ChaCha block.

If it were the case that we actually needed to pull data cycle-by-cycle out of the ChaCha block and somehow change the behavior of the system within a single cycle, then this path would make sense. But in practice, the actual rate is far lower than that — the tightest, most unrolled read loops take at least 2 cycles to execute; but more typically 10-12 cycles per iteration if you’re doing any sort of useful load or store with the data and running it from a rolled-up loop. Furthermore, the ChaCha block is basically a “read-only” block — there’s nothing the CPU is going to do to reconfigure it on the fly that would prevent us from
pipelining this.

So, I go back and add a “depth 1 FIFO” to the output, to break the shift register out of the critical path while still preserving the read-invalidate semantics required by the CSR interface (that is, the contract we have with the CSR interface is the cycle after a random number is read, either the next value should be a valid, new random number; or it can be an invalid number, but the “valid” bit must be cleared that very cycle). Using a simple pipeline register won’t work, because it introduces an extra cycle delay on the read-invalidate semantics and it’s possible, although very unlikely, that a code loop could have back-to-back loads and it will grab a stale “valid” state.

After running another simulation just to make sure we haven’t introduced any new bugs, I run a compilation again and hope this greatly relaxes the timing. It didn’t, but long story short, I had accidentally leaked one path in the FIFO through as a combinational element. After spending a couple hours tracing that issue out, I finally met with some success.

After applying these fixes, the compilation time is back down to its normal value, and we can see that the timing bins at the very far left of the histogram have a fraction of paths as before — 69, down from 206. It looks like the holding_buf register is still causing some troubles, but at least we’ve taken pressure off of the actually most difficult paths, which are inside the Curve 25519 Engine, and not some made-up timing emergencies inside the shift register at the output stage of the TRNG.

The price of all the timing fixes is increased resource utilization. Initial estimates showed about a 6% area cost for the ChaCha block without the timing fixes, but we ended up with around 10.5% area cost once all is said and done. The SoC uses around 68.2% of the FPGA now, which is still fine for routability and gives some space for a few more smaller blocks, but it’s probably at the upper limit of what we can reasonably expect to fit while having short (~8-9 minute) compilation cycles. I’m also making a mental note to myself that as part of the histogram analysis it looks like the next critical path to knock back is in the CSR address decoders. Adding the dozens of registers to read out the results from the TRNG health tests has probably forced an extra layer of LUTs to be instantiated inside the core arbitration logic; thus, pruning some of these registers if they end up being superfluous could be a way to win back some timing margin if I end up in a jam later.

Well, that took some time in simulation. But we’re not done yet! It’s time to run this code on real hardware, and then upgrade the kernel and services of Xous to use it — and let’s not forget CI integration as well.

First things first, let’s just take the design, as-is, with no mods to Xous and see if it boots, and what the power draw change is. The good news is that things boot, and the system draws about 5 mA extra, or about 5% more idle power than it did before. The ChaCha whitener is in the always-on domain, because it has to be able to run while the kernel is idling to produce random data. In particular, the core can “wake up” the avalanche generator and ring oscillators while the CPU is idle to summon more TRNG data even when the CPU is asleep; as such its clock can’t be gated off, but it is better than the alternative of running the CPU to manage these processes. A quick test tweaking the knobs on the duty cycle of the self-mixing for the ChaCha core shows it doesn’t have a measurable impact on the power, which means that probably most of the extra power is just the fixed cost of clocking that much more logic on the fabric.

The next step is to switch over the kernel and core services from using raw random numbers to the new ChaCha conditioned “urandom” source. This should have been easy in theory since everything worked in simulation, but of course — it didn’t work. I was greeted with a full system hang on the first go. This kind of bug is dreaded, because it’s some issue in the full system integration that exists outside of the testbenches, so you have to resort to long compile cycles and “printf” debugging.

Some further poking revealed that even worse yet, on some boots, the system worked, on others, it didn’t. However, the pattern was enough for me to suspect an issue in the reset logic. Going back into my code, I did find a subtle bug where I forgot to assign the reset line for the ChaCha core to the 50MHz clock domain (it was still tied to the reset for the 100MHz clock domain). For good measure, I also added a reset pulse extender, just to make sure it saw a long, solid reset. The polarity of the ChaCha core reset is opposite from what the rest of the system uses, so it’s also possible there is some glitches due to that. Anyways, after another four hours of tweaking and testing these fixes, re-running simulations, and cross-checking documentation about the reset process to really make sure we found a root cause, the system seems to come up as expected from the “urandom” source.

### For TRNGs, The Work is Never Done

The final phase of modifying any TRNG is continuous integration. This is extremely important because you can only get a hint if a TRNG is mis-behaving in subtle ways after collecting dozens of gigabytes of data from it over dozens of runs; and due to the limited rate of entropy, gathering such quantity of data can take weeks. Thus, we have a Precursor unit dedicated to doing nothing but rotating through its entropy sources and feeding its data into a set of test suites to try and uncover any subtle biases in the generators. You can browse the raw data and results of our CI bench here.

Now that we have this conditioned output, we need to integrate that into the test bench. It’s also going to be important to do a few “cold boot with recorded data” runs as well, so we can compare boot-to-boot to make sure that we have actually seeded the generator with random data and we’re not just accidentally running a ChaCha cipher (which should have excellent statistical randomness) with a fixed set of keys (which would have an identical stream boot-to-boot despite the excellent statistics).

In the process of re-integrating these tests into the CI bench, I discovered a number of problems with our original tool for shuttling random numbers to the host for analysis. For a number of reasons, the original CI tool relied on a fairly complicated Rust program with multiple threads, which was getting hard to maintain and the Rust wasn’t adding a lot of value; in fact I found a subtle integration bug that was causing all the tests for the past month to be run only on the ring oscillator!

Since we had a pretty decent Python script for doing USB updates over USB, I modified that into a script currently living in xous-core/tools/trng_test.py. This script is matched to a thread that runs inside the trng server on the Xous side when it is built with one of the TRNG tester options. On the Precursor side, a pair of 512kiB buffers are filled and the host is told which of the two to read from. The host-side Python script then reads the buffers and echos its contents out as binary data to stdout, which can then by piped into an analysis program (such as dieharder) or recorded to disk for later analysis. This whole mess is then wrapped in some other shell-script Frankenstein that’s fired off by a cron job which manages the rebuild, reflashing, and logging of data via a Raspberry Pi that can keep an eye on the serial console for errors during the CI process. It feels like a house made of playing cards, but it works!

### Closing Thoughts

If you made it this far, congratulations. Hopefully it conveys, with some accuracy, the range of difficulties one might encounter when implementing a “typical” feature like this for an FPGA-based SoC, and it may even be useful to the handful of developers who want to actually attempt something like this!