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## Archive for the ‘precursor’ Category

### Building a Curve25519 Hardware Accelerator

Thursday, 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!

### Adding a ChaCha Cipher to Precursor’s TRNG

Tuesday, 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!

### Monitoring the Health of Precursor’s TRNGs

Monday, June 14th, 2021

This post is an abridged version of a longer-form narrative on implementing health monitoring for Precursor/Betrusted’s TRNGs. It’s the first of a series of two posts; the second, on implementing a CSPRNG conditioner for the TRNG, will go up later.

### Background

Online health monitors are simple statistical tests that give users an indication of the quality of the entropy being produced. On-line health tests are like a tachometer on an engine: they give an indication of overall health, and can detect when something fails spectacularly; but they can’t tell you if an engine is designed correctly. Thus, they are complimentary to longer-running, rigorous diagnostic tests. We cover some of these tests at our wiki, plus we have a CI bench which generates gigabytes of raw entropy, over runtimes measured in months, that is run through a series of proofing tools, such as the Dieharder test suite and the NIST STS test suite.

It’s important that the health monitoring happens before any conditioning or mixing of the raw data happens, and significantly, there is no one-size-fits-all health monitor for a TRNG: it’s even advised by the (NIST SP 800-90B sec 4.4) specification to have tests that are tailored to the noise source.

### Thinking

The first thing I do before doing any changes is some book research. As my graduate advisor, Tom Knight, used to say, “Did you know you could save a whole afternoon in the library by spending two weeks at the lab bench?”

NIST SP 800-90B section 4.4 specifies some health tests. The NIST spec seems to be fairly well regarded, so I’ll use this as a starting point for our tests. The tests come with the caveat that they only detect catastrophic failures in the TRNGs; they are no substitute for a very detailed, long-run statistical analysis of the TRNG outputs at the design phase (which we have already done). NIST recommends two tests: a repetition count test, and an adaptive proportion test.

This is the repetition count test:

With the following criteria for failure:

And this is the adaptive proportion test:

With the following example cutoff values:

OK, so now we know the tests. What do these even mean in the context of our TRNGs?

### Ring Oscillator Implementations

Our Ring Oscillator architecture consists of 33 independent ring
oscillators that operate in two phases. In the first phase, the rings oscillate independently of each other to collect phase noise; the state of each ring is XOR’d into a state bit through snapshots taken at regular intervals. In a second phase, they are merged into a single large ring oscillator, where they then have to reach a consensus. A single bit is taken from the consensus oscillator, and progressively shifted into the state of the ring oscillator.

Above is a simplified “two bit” version of the generator, where instead of 33 oscillators we have 3, and instead of 32 bits of entropy we get 2. The red arrow is the flow of entropy in the first phase, and the green arrow is the flow of entropy in the second phase. The two phases are repeated N+1 bit times (33 times in the full implementation, 3 times in the simplified diagram above).

We can see from the diagram that entropy comes from the following
sources:

• Random phase accumulated in the smaller ring oscillators due to the accumulation of phase noise during a “dwell” phase that can be set by software (nominally 1000 ns)
• Decision noise associated with the sampling jitter of the smaller ring oscillators with an initial sampling flip-flop. Note that the ring oscillators operate at a higher frequency (~300-500MHz) than the sampling rate of the flip flops (100 MHz).
• Global phase accumulated during the consensus process of the larger ring oscillator. The time to achieve consensus is set by a “delay” parameter that is set by software (nominally 40ns)
• Cross-element mixing through the continuous shifting of bits to the right, and further XOR’ing of phase

The global phase consensus and cross-element mixing is quite important because ring oscillators have a tendency to couple and phase-lock due to crosstalk side-channels on both signal and power. In this architecture, each ring’s local noise conditions, including its crosstalk, is applied across each of the 32 output bits; and each ring’s oscillation is “reset” with an arbitrary starting value between each cross-element phase.

A higher rate of aggregate entropy is achieved by running four instances of the core described above in parallel, and XOR’ing their result together. In addition, the actual delay/dwell parameters are dynamically adjusted at run-time by picking some of the generated entropy and adding it to the base dwell/delay parameters.

Thus, when looking at this architecture and comparing it against the NIST spec, the question is, how do we apply the Repetition Count test and the Adaptive Proportion tests? The Repetition Count test is probably not sensitive enough to apply on the 32-bit aggregate output. It’s probably best to apply the Repetition Count and Adaptive Proportion test a bit upstream of the final generated number, at the sampled output of the ring oscillators, just to confirm that no constituent ring oscillator is “stuck” for any reason. However, the amount of logic resources consumed by adding this must be considered, since we have (33 * 4) = 132 separate oscillators to consider. Thus, for practical reasons, it’s only feasible to instrument one output from each of the four cores that is indicative of the health of the entire bank of oscillators.

Picking the right spot to instrument is tricky. The “large” ring oscillator is actually low-quality entropy, because it has a period of about 30MHz but is oversampled at 100MHz. Thus the majority of the entropy is contributed from the repeated undersampling of the “small” rings. The final sampling point chosen is the output of the sampling register after it’s “soaked up” enough entropy from the combination of a small ring and a large ring to result in a useful measurement.

Originally, I had tried looking at the “large” oscillator only to try to find something more “raw”, under the hypothesis that we would be more likely to catch problems in the system at a less refined stage; the problem is that it was so “raw” that all we caught was problems. However, we do use this tap as a “true negative” test, to ensure that the health tests are capable of flagging an entropy source that is less than perfect.

I’m also going to introduce an extra test that’s inspired by the Runs Test in the STS suite, that I call “MiniRuns”. This test records the frequency of continuous runs of bits: 0/1, 00/11, 000/111, 0000/1111, etc. This test will offer more insight into the dominant projected failure mode of the ring oscillator, namely, it oscillating as a perfectly synchronized square wave — a condition that neither of the recommended NIST tests are capable of capturing. However, if the oscillator becomes too deterministic, we should see a shift in the distribution of run lengths out of the MiniRuns test.

### Avalanche Generator Implementation

The avalanche generator consists of two avalanche diodes biased from a shared power supply, sampled by a pair of op-amps with a slight bit of gain; see our page on its theory of
operation
for details on the physics and electronic design. Here, we focus on its system integration. The outputs of each of the op-amps are sampled with a 12-bit ADC at a rate of ~1MSPS, and XOR’d together. As this sampling rate is close to the effective noise bandwidth of the diodes, we reduce the sampling rate by repeatedly shifting-by-5 and XOR’ing the results a number of times that can be set by software, nominally, 32 times into a 32-bit holding register, which forms the final entropy output. This 32x oversampling reduces the rate of the system to 31.25kHz.

Thus in this scheme, entropy comes from the following sources:

• The avalanche properties of two individual diodes. These are considered to be high-quality properties derived from the amplification of true thermal noise.
• The sampling interval of the ADC versus the avalanche waveform
• Noise inherent in the ADC itself
• Note that the two diodes do share a bias supply, so there is an opportunity for some cross-correlation from supply noise, but we have not seen this in practice.

Because we are oversampling the avalanche waveform and folding it onto itself, what we are typically measuring is the projected slope of the avalanche waveform plus the noise of the ADC. Significantly, the SNR of the Xilinx 7-series “12-bit” ADC integrated into our FPGA is 60dB. This means we actually have only 10 “good” bits, implying that the bottom two bits are typically too noisy to be used for signal measurements. The XADC primitive compensates for this noise by offering automatic averaging over 16 samples; we turn this off when sampling the avalanche noise generators, because we actually *want* this noise, but turn it on for all the other duties of the XADC.

It’s also important to consider the nature of sampling this analog
waveform with an ADC. The actual waveform itself can have a DC offset, or some total amplitude variation, so naturally the LSBs will be dense in entropy, while the MSBs may be virtually constant. By focusing on the bottom 5 bits out of 12 with the 5-bit sliding window, we are effectively ignoring the top 7 bits. What does this do to the effective waveform? It’s a bit easier to show graphically:

Below is a waveform at full resolution.

If we were to only consider 11 bits out of the 12, we effectively take half the graph and “wrap it over itself”, as shown below:

Down to 10 bits, it looks like this:

And down to 9 bits, like this:

And so forth. By the time we are down to just considering 5 bits, we’ve now taken the effective DC offset and amplitude variations and turned them into just another random variable that helps add to the entropy pool. Now take two of these, XOR them together, and add in the effective noise of the ADC itself, and you’ve arrived at the starting point for the ADC entropy pool.

In terms of on-line entropy tests, it probably makes the most sense to apply the Repetition Count test and the Adaptive Proportion tests to the bottom 5 bits of the raw ADC feed from each avalanche diode (as opposed to the full 12-bit output of the ADC). We don’t expect to hit “perfect entropy” with the raw ADC feed, but these tests should be able to at least isolate situations where e.g. the bias voltage goes too low and the avalanche effect ceases to work.

In addition to these tests, it’s probably good to have an “absolute
excursion” test, where the min/max of the raw avalanche waveforms are recorded over a time window, to detect a diode that is flat-lining due to aging effects, or a bias voltage source that is otherwise malfunctioning. This test is not suitable for catching if an attacker is maliciously injecting a deterministic waveform on top of the avalanche diodes, but is well-suited as a basic health check of the TRNG’s core mechanisms under nominal environmental conditions.

### Developing

After installing the tooling necessary to build a Precursor/Betrusted SoC, 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 the first section of this
article.
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 Vex 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.

### Demo

This is the point in the cooking show at which we put the turkey into an oven, say something to the effect of “…and in about five hours, your bird should be done…” yet somehow magically pull out a finished turkey for carving and presentation by the time we finish the sentence.

So: after a bunch more driver-writing and breaking out signals to gain visibility into the various metrics and failure modes, we can see the on-line health tests in action.

Above is an example of the trng test data output on Precursor, set where RO 0 is connected to the “large” oscillator that runs too slowly (serving as a “true negative” test), and the others are connected to the final output tap for the test (serving as a “true positive”). In the output, you can see the four ring oscillators (numbered 0-3) with the frequency of each of five run lengths printed out. RO 0 has a significant depression in the count for the run-length 1 bin, compared to the other oscillators (440 vs 515, 540, and 508).

One final detail is implementing an automated decision mechanism for the MiniRuns test. Since the MiniRuns test wasn’t from the NIST suite, I couldn’t simply read a manual to derive a threshold. Instead, I had to consult with my perlfriend, who also happens to be an expert at statistics, to help me understand what I was doing and derive a model that could help me set limits. Originally, she suggested a chi-square test. This would be great, but the math for it would be too complicated for an automated quick power-on test. So, we downgraded the test to simple max/min thresholds on the counts for each “bin” of runs. I used a similar criteria to that suggested in the NIST test, that is, 𝛂 = 2^-20, to set the thresholds, and baked that into the hardware code. Here’s a link to the original spreadsheet that she used to compute both the chi-square and the final, simpler min/max tests. One future upgrade could be to implement some recurring process in Xous that collects updated results from the MiniRuns test and does the more sophisticated chi-square tests on it; but that’s definitely a “one for the road” feature.

### Closing Thoughts

The upshot is that we now have all the mandatory NIST tests plus one each “tailored” tests for each type of TRNG. Adding the MiniRuns automated criteria increased utilization to 56.5% — raising the total space used by the tests from about 2% of the FPGA to a bit over 4%. The MiniRuns test is expensive because it is currently configured to check for runs ranging from length 1 to 5 over 4 banks of ring oscillators — so that’s 5 * 4 * (registers/run ~30?) = ~600 registers just for the core logic, not counting the status readout or config inputs.

Later on, if I start running out of space, cutting back on some of the instrumentation or the depth of the runs measured might be a reasonable thing to do. I would suggest disposing of some of the less effective NIST tests entirely in favor of the home-grown tests, but in the end I may have to kick out the more effective supplemental tests. The reality is that it’s much easier to defend keeping inferior-but-spec-compliant tests in the system, rather than opting for superior tests at the expense of the specification tests.

That’s it for part 1. If you’re super-eager to read more, you can read the full wiki entry on data conditioning for the TRNG at the Precursor/Betrusted documentation wiki. Or, you can just wait until I get around to chopping the page down to size and repackaging it into a more bite-side blog entry.

Monday, June 14th, 2021

It was pointed out that we’re missing a step-by-step guide on how to go from an idea, to hardware, to a fully implemented feature in Xous for Precursor. So, here it is.

Because Precursor uses an FPGA for its SoC, we can add new features to the hardware “on the fly”. In this case, we’re going to add some improvements to our basic managed TRNG block. To review, the existing TRNG consists of an avalanche noise source and a ring oscillator noise source as hardware-based sources of “true” entropy. Two generators are used for the following reasons:

• An external discrete generator is easy to check (just put an oscilloscope on the avalanche noise source), but harder to protect against physical access attacks
• An integrated, on-chip generator is harder to hack (more robust against a pair of tweezers executing a short-to-ground attack, or RF interference attacks), but harder to check (is the data from the TRNG, or merely a decoy CSPRNG implant?)
• All hardware mechanisms are fallible; having two sources improves robustness against transient drop-outs or aging failures

We’ve already done extensive, months-long characterization of both of the TRNG sources and found them to produce passable raw entropy. 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. It’s important that the health monitoring happens before any conditioning or mixing of the raw data happens, and significantly, there is no one-size-fits-all health monitor for a TRNG: it’s advised (NIST SP 800-90B sec 4.4) to have tests that are tailored to the noise source.
2. Conditioning of the raw data. 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 simultaneously protects against minor flaws in the TRNG while improving the availability of random numbers.

The following lengthy posts walk step-by-step through the thought process, implementation and debugging process of adding these features. Few people would even notice these features, and if everything is doing its job right (that is, the TRNG’s raw data is working correctly) is indistinguishable from the state before all this effort. However, we take TRNGs seriously here; so much rides on the quality of these random numbers that it’s probably worth the effort to harden them against failures, be they unintentional, malicious, or just design bugs.

I have to be honest, I spent a lot of time to check a box that few people care about, but I’ve come to realize that’s mainly what writing OS code and firmware is about. You can get more fame and dopamine from creating a cool UI theme with an afternoon of work. It’s also really hard to explain to everyday people what you’re doing exactly with all this time and effort; but without the underlying frameworks that make things durable and reliable, we all might as well be drawing chalk pictures on the side walk.

Without further ado, here are the two guides for adding features, there’s some repetition between the posts so they can be read independently.

I’ll also take highlights from these wiki articles and repost them to the blog here, creating a “TL;DR” version that is also neatly delivered to the inboxes of my blog’s email subscribers.

### What’s the Value of Hackable Hardware, Anyway?

Friday, December 11th, 2020

There is plenty of skepticism around the value of hackable products. Significantly, hackability is different from openness: cars are closed-source, yet support vibrant modding communities; gcc is one of the “real OG”s of open source, but few users find it easy to extend or enhance. Is it better to have a garden planted by the most knowledgeable botanists and maintained by experienced gardeners, or an open plot of land maintained by whoever has the interest and time?

Above left: Walled garden of Edzell Castle; above right: Thorncliffe Park community garden.

In the case of hardware products, consumer behavior consistently affirms a strong preference for well-curated gardens. Hardware is hard – not only is it difficult to design and validate, supply chains benefit from economies of scale and predictable user demand. The larger a captive audience, the more up-front money one can invest into developing a better hardware product. However, every decision to optimize comes with inherent trade-offs. For example, anytime symmetry is broken, one must optimize for either a right-handed or a left-handed version.

Above: touching the spot indicated by the red arrow would degrade antenna performance on an iPhone 4. This spot would naturally rest on the palm of a left-handed user. Image adapted from “iPhone 4” by marc.flores, licensed under CC BY 2.0.

Some may recall a decade ago when the iPhone 4 was launched, left-handed people noticed the phone would frequently drop calls. It turned out the iPhone 4 was designed with a critical antenna element that would fail when held naturally by a left-handed person. The late Steve Jobs responded to this problem by telling users to “just avoid holding it that way”. Even if he didn’t mean it, I couldn’t help but feel like he was saying the iPhone 4 was perfect and left-handers like me were just defective humans who should be sent to re-education camps on how to hold things.

Of course, as a hardware engineer, I can also sympathize with why Steve Jobs might have felt this way – clearly, a huge amount of effort and thought went into designing a technical masterpiece that was also of museum-quality construction. It’s frustrating to be told, after spending years and billions of dollars trying to build “the perfect product” that they somehow got it wrong because humans aren’t a homogeneous population. Rumors have it Apple spent tens of millions of dollars building micron-precision production jigs out of injection-molding grade tooling to ensure the iPhone4 was simply perfect in terms of production tolerances; duplicating all of those to make a mirror-image version for left-handers that make up 10% of the market size just made no business sense. It proved to be cheaper and easier, ultimately, to take full refunds or to give out rubber bumpers to the users who requested them.

I do think there is such a thing as “over-designing” a product. For example, contemporary “high concept” smartphone design is minimalist – phone companies have removed headphone jacks, hidden the front camera, and removed physical buttons. There is clearly no place for screws in this world; the love affair of smartphones and adhesives has proven to be … sticky. Adhesives, used in place of screws in modern smartphones, are so aggressive that removing them either requires additional equipment, such as a hot plate and solvents, or simply destroying the outer bezel by breaking the outer case off in pieces and replacing it with an entirely new bezel upon re-assembly. In other words, hacking a modern smartphone necessarily implies the destruction or damage of adhesive-bound parts.

With Precursor, I’m bringing screws back.

Precursor’s screws are unapologetic – I make no attempt to hide them or cover them with bits of tape or rubber inserts. Instead, I’ve sourced custom-made Torx T3 metric screws with a black oxide finish that compliments the overall color scheme of Precursor. Six of them line the front, as a direct invitation for users to remove them and see what’s inside. I’ve already received ample criticism for the decision to show screws as “primitive”, “ugly”, “out of touch with modern trends” — but in the end, I feel the visual clutter of these six screws is a small price to pay for the gain in hackability.

Of course, the anti-screw critics question the value of hackability. Surely, I am sacrificing mass-market appeal to enable a fringe market; if hackability was so important, wouldn’t Apple and Google already have incorporated it into their phones? Wouldn’t we see more good examples of hackability already?

This line of questioning is circular: you can’t get good examples of hacks until you have made hackable products wide-spread. However, the critics are correct, in a way: in order to bootstrap an ecosystem, we’re going to need some good examples of why hackability matters.

In the run-up to crowdfunding Precursor, I was contemplating a good demo hack for Precursor. Fortuitously, a fellow named Matt Campbell opened a GitHub issue requesting a text-to-speech option for blind users. This led me to ask what might be helpful in terms of a physical keyboard design to assist blind people. You can read the thread for yourself, but I’ll highlight that even the blind community itself is divided on whether or not there is such a thing as the “blind ghetto” — an epithet applied by some users who feel that blindness-specific products tend to lag behind modern smartphones, tablets, and laptops. However, given that most modern gadgets struggle to consider the needs of 10% of the population that’s left-handed, I’m readily sympathetic to the notion that gadgets make little to no concession to accommodate the even smaller number of blind users.

Matt was articulate in specifying his preferred design for a pocketable keyboard. He referred me to the “Braille ‘n Speak” (shown above) as an example of an existing braille keyboard. Basically, it takes the six dots that make up braille, and lines them up horizontally into three left and three right sets of buttons, adding a single button in the middle that functions as a space bar. Characters are entered by typing chords that correspond to the patterns of the six dots in the braille alphabet. Not being a braille user myself, I had to look up what the alphabet looked like. I made the guide below based on a snippet from Wikipedia to better understand how such a keyboard might be used.

Ironically, even though Matt had linked me to the picture of the Braille n’ Speak, it still took a while to sink in that a braille variant of Precursor did not need a display. I’m a bit ashamed to admit my first sketches involved trying to cram this set of switches into the existing keyboard area of the Precursor, without first removing the display entirely. I had to overcome my own prejudice about how the world should look and it took me time and empathy to understand this new perspective.

Once I had a better grasp of Matt’s request, I set about designing a customized braille variant. Precursor was designed for this style of hacking: the keyboard is a simple 2-layer PCB that’s cheap and easy to re-design, and the front bezel is also a PCB, which is a bit more expensive to redesign. Fortunately, I was able to amortize setup costs by bundling the braille front bezel variant with another variant that I had to fabricate anyways for the crowdfunding campaign. Beyond that, I also had to come up with some custom key caps to complement the switches.

The major challenge in designing any type of mobile-friendly keyboard is always a trade-off between the hand feel of the switches, versus thinness of the overall design. On one side of the spectrum, full-travel mechanical switches have a wonderful hand feel, but are thicker than a sausage. On the other side of the spectrum, dome switches and printed carbon ink patterns are thinner than a credit card, but can feel mushy and have a limited “sweet spot” — the region of a key switch with optimal tactile feedback and operational force curves. The generally well-regarded Thinkpad keyboards go with a middle-ground solution that’s a few millimeters thick, using a “scissor” mechanism to stabilize the key caps over a silicone dome switch, giving individual keys a bit of travel while ensuring that the “sweet spot” covers the entire key cap. Optimizing key switch mechanisms is hard: some may recall the controversy over Apple’s re-design of the MacBook’s keyboard to use a “butterfly” mechanism, which shaved a couple mm of thickness, but led to lawsuits over a defect where the keyboard allegedly stopped working when small bits of dust or other particles got trapped under it.

Given the short time frame and a shoestring budget, I decided to use an ultra-thin (0.35mm) tactile switch that I could buy off-the-shelf from Digikey and create custom key caps with small dimples to assist users in finding the relatively small sweet spots typical of such switches. I have sincere hopes this is a pretty good final solution; while it lacks a scissor mechanism to spread off-centered force, the simple mechanism meant I didn’t have to stick with a square key cap and could do something more comfortable and ergonomic to focus forces into the sweet spot. At the very least, the mechanism would be no worse than the current mechanism used in Precursor’s existing keyboard for sighted users, which is similarly a dome switch plus a hybrid-polymer key film.

Next, I had to figure out where to place the switches. To assist with this, I printed a 1:1 scale version of the Precursor case, dipped my fingertips in ink, and proceeded to tap on the printout in what felt like a natural fashion.

I then took the resulting ink spots and dimensioned their centers, to place the centroid of each key cap. I also asked my partner, who has smaller hands, to place her fingers over the spots and noted the differences in where her fingers lay to help shape the final key caps for different-sized hands.

Next, using the “master profile” discussed in the previous post on Precursor’s mechanical design, I translated this into a sketch to help create a set of key caps based on splines that matched the natural angle of fingers.

Above, you can see an early sketch of the key caps, showing the initial shape with dimples for centering the fingers.

Before moving ahead and spending a few hundred dollars to build a functional prototype, I decided to get Matt’s feedback on the design. We submitted the design to Shapeways and had a 3D print sent to Matt, which he graciously paid for. After receiving the plastic dummy, his feedback was that the center space bar should be a single key, instead of two separate keys, so I merged the two separate key caps of the space bar together into a single piece, while retaining two separate switches wired in parallel under the space bar. I felt this was a reasonable compromise that would allow for a “sweet spot” that serviced lefties as well as righties.

I then re-designed the keyboard PCB, which was a fairly simple task, because the braille keyboard consists of only eight switches. I just had to be careful to pick row/column pairs that would not conflict during chording and be sure to include the row/column pairs necessary to turn Precursor on after being put to sleep. I also redesigned the bezel; eliminating the display actually makes manufacturing a little bit easier because it also removes a beveling step in the manufacturing process. I kept the RF antenna in exactly the same location, as its design was already well-characterized and it takes a lot of effort to tune the antenna. Finally, I decided to manufacture the key switches out of aluminum. The switches have a lot of fine features and I needed a stiff material that could translate off-target force to the key switches to enlarge the sweet spot as much as possible.

Above: The prototype of Precursor with braille keyboard.

About three weeks later, all the parts for the braille keyboard had arrived. I decided to use purple anodization for the key switches which, combined with the organic key shapes, gives the final design a bit of a Wakanda-esque “Black Panther” aesthetic, especially when mounted in a brass case. The key switch feel is about in line with what I imagined, with the caveat that one of the switches feels a little less nice than the rest, but I think that’s due to a bad solder job on the switch itself. I haven’t had a chance to trace it down because…well, I’ve had to write a bunch of posts like this to fund Precursor. I have also been working with Xobs to refactor Xous in hopes of putting together enough code to send Matt a prototype he can evaluate without having to write gobs of embedded hardware driver code himself.

Above is a quick photo showing the alignment of fingers to keys. Naturally, it’s perfect for my hand because it was designed around it. I’m looking forward to hearing Matt’s opinion about the feel of the keys.

Above is a photo of the custom parts for the braille keyboard. At the top, you can see the custom bezel with key caps and the RF antenna matching circuitry on the top right. On the bottom, you can see the custom keyboard PCB mounted onto a Precursor motherboard. The keyboard PCB is mostly blank and, because of the small number of keys and the flexibility of the FPGA, there’s an option to mount more peripherals on the PCB.

Despite not being yet finalized, I hope this exercise is sufficient to demonstrate the potential value of hackable products. The original design scope for Precursor (née Betrusted) did not explicitly include a braille keyboard option, but thanks to modular design principles and the use of accessible construction materials, I was able to produce a prototype in about a month that has a similar fit and finish as the mainstream product.

As long as humans choose to embrace diversity, I think hackability will have value. A notional “perfect” product implies there’s such a thing as a “perfect” user. However, in reality, even the simple conundrum of left- or right-handedness challenges the existence of a singular “perfect” product for all of humanity. Fortunately, accommodating the wonderfully diverse, quirky, and interesting range of humanity implicates just a few simple engineering principles, such as embracing screws over adhesives, openness, and modularity. That we can’t hack our products isn’t a limitation of physics and engineering. Precursor demonstrates one can build something simultaneously secure and hackable, while being compact and pocketable. This suggests the relative lack of hackable products on the market isn’t a fundamental limitation. Maybe we just need a little more imagination, maybe we need to be a little more open-minded about aesthetics, and maybe companies need to be willing to take brave steps toward openness and inclusivity.

For Apple, true “courage to move on and do something new that betters all of us” was to remove the headphone jack, which resulted in locking users deeper into a walled-garden ecosystem. For hackers like myself, our “courage” is facing blunt criticisms for making “ugly” products with screws in order to facilitate mods, such as braille keyboards, in order to expand the definition of “all of us” beyond a set of privileged, “perfect” users.

I hope this braille keyboard is just the first example of many mods for Precursor that adapt the product for unique end-users, bucking the trend of gaslighting users to mold their behavior and preferences to fit the product. If you’ve got an itch to develop your own yet-to-be-seen feature in a mobile device, please visit our crowdfunding campaign page to learn more about Precursor. We’re close to being funded, but we’ve only a few days left in the campaign. After the campaign concludes on December 15th, the limited edition will no longer be available, and pricing of the standard model goes up. If you like what you see, please consider helping us to bring Precursor to life!


text:
# The function is given by the formula f(x)=2x^{2}-5x+3 a) find f(-1).b) at what values x, f(x)=1? c)Does the point A(1;0) belong to the graph of the function

The function is given by the formula $f\left(x\right)=2{x}^{2}-5x+3$
a) find f(-1).
b) at what values x, f(x)=1?
c)Does the point A(1;0) belong to the graph of the function
You can still ask an expert for help

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Illuddybopylu
$a\right)f\left(-1\right)=2\ast \left(-1{\right)}^{2}-5\ast \left(-1\right)+3=2+5+3=10$
$b\right)2{x}^{2}-5x+3=1⇔2{x}^{2}-5x+2=0$
$D={5}^{2}-4\ast 2\ast 2=25-16=9$
x1=5+3/4=8/4=2
x2=5-3/4=2/4=0.5.
$c\right)2\ast {1}^{2}-5\ast 1+3=0$
2-5+3=0
-3+3=0
0=0
This point belongs to the graph.
###### Not exactly what you’re looking for?
mercuross8
$F\left(x\right)=2{x}^{2}-5x+3$
$1\right)F\left(-1\right)=2\ast \left(-1{\right)}^{2}-5\ast \left(-1\right)+3=2+5+3=10$
2)F(x)=1 find x
$2{x}^{2}-5x+3=1$
$2{x}^{2}-5x+2=0$
${x}^{2}-2.5x+1=0$
x1+x2=2.5
x1*x2=1
x1=0.5
x2=2
3)A(1.0)
x=1
$2\ast {1}^{2}-5\ast 1+3=0$

format: plain


text:
# Closed geodesic loops around points in compact manifolds

Since in a compact Riemannian manifold $M$ the only totally convex subset is the whole manifold itself, see Closed manifold has no nontrivial totally convex subset?, it should follow that for every point $p\in M$ there exists a geodesic loop starting and ending at $p$ (not necessarily "closing" smoothly). I was looking for a (possibly simple) direct proof of this fact without passing by the general theorem (whose proof is not easy, in my opinion). Thanks.

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(I take it you don't count the constant loop.) Of course, if $M$ has nontrivial $\pi_1$, then you can find a geodesic with fixed endpoints representing any nontrivial homotopy class by minimizing the energy. In the simply-connected case I don't see an immediate proof. –  Theo Johnson-Freyd May 20 '13 at 0:32
Suppose $p$ is totally convex, so there is no geodesic loop based at $p$. By the main theorem of the Morse theory the loop space $\Omega_p M$ has a homotopy type of a CW complex with cells correponding to geodesic loops at $p$, so $\Omega_p M$ is contractible, i.e. $\pi_i(M)=0$ for $i>0$. Since $M$ is connected, it is contractible. No closed manifold is contractible by Poincare diality. I do not think there is a proof without Morse theory in disguise. –  Igor Belegradek May 20 '13 at 2:59
In fact, one needs very little from the Morse theory because there is no critical points (e.g. the index theorem is not needed); basically, replace the loop space by the finite dimensional manifold of piecewise geodesic loops, and note that the energy functional has no critical points, so the finite dimensional manifold is contractible. The rest is elementary algebraic topology. What could be easier? –  Igor Belegradek May 20 '13 at 14:44
@Igor, the manifold of broken geodesics is not complete. So you have to be bit more careful, to make sure that there is a critical point. In other words, read my answer. –  Anton Petrunin May 20 '13 at 18:29
Anton: the argument in my first comment surely works. The argument in the second comment does not give complete detail, but I think the point is that the gradient flow of the energy functional gives a deformation retraction of the manifold of broken geodesics loops onto a small neighborhood of a constant loop at $p$, and that small neighborhood is contractible. –  Igor Belegradek May 21 '13 at 3:56

Here is a standard argument, I learned it from "Comparision Theorems in Riemannian Geometry" by Cheeger and Ebin. It avoids use of infinite dimensional spaces.

Choose the smallest $k>0$ such that $\pi_kM\ne 0$. Choose a spheroid which represents a nontricial element of $\pi_kM$. We can assume that the spheroid is swapped by an $\mathbb S^{k-1}$-parameter family of broken geodesics such that the length of each edge is smaller than the injectivity radius of $M$; denote it by $i_M$.

Start a natural curve-shortening process --- velocity vector of every vertex should depend on the directions and lengths of the two edges coming from it. You have to choose one which keep the lengths of the edges below $i_M$ and such that the rate of length (or energy) decay is estimated through rate of change of the broken geodesic.

Note that there is a lower bound for the maximal length of broken geodesics; "maximal" means "maximal in the $\mathbb S^{k-1}$-family after spending arbitrary time in the process". Say, this value can not go below $i_M$.

It follows that after long time in the process, one broken line in the family almost does not change the length. Hence this it almost does not move; the later implies that and all the angles almost $\pi$. Pass to the limit and you get the loop.

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Anton: You need to use minimax argument here since you are looking for an unstable critical point of energy. –  Misha May 20 '13 at 2:08
@Misha, Yes, I do use minimax. I made an update, it should be more clear now. –  Anton Petrunin May 20 '13 at 3:30