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+p256-m is a minimalistic implementation of ECDH and ECDSA on NIST P-256,
+especially suited to constrained 32-bit environments. It's written in standard
+C, with optional bits of assembly for Arm Cortex-M and Cortex-A CPUs.
+
+Its design is guided by the following goals in this order:
+
+1. correctness & security;
+2. low code size & RAM usage;
+3. runtime performance.
+
+Most cryptographic implementations care more about speed than footprint, and
+some might even risk weakening security for more speed. p256-m was written
+because I wanted to see what happened when reversing the usual emphasis.
+
+The result is a full implementation of ECDH and ECDSA in **less than 3KiB of
+code**, using **less than 768 bytes of RAM**, with comparable performance
+to existing implementations (see below) - in less than 700 LOC.
+
+_Contents of this Readme:_
+
+- [Correctness](#correctness)
+- [Security](#security)
+- [Code size](#code-size)
+- [RAM usage](#ram-usage)
+- [Runtime performance](#runtime-performance)
+- [Comparison with other implementations](#comparison-with-other-implementations)
+- [Design overview](#design-overview)
+- [Notes about other curves](#notes-about-other-curves)
+- [Notes about other platforms](#notes-about-other-platforms)
+
+## Correctness
+
+**API design:**
+
+- The API is minimal: only 4 public functions.
+- Each public function fully validates its inputs and returns specific errors.
+- The API uses arrays of octets for all input and output.
+
+**Testing:**
+
+- p256-m is validated against multiple test vectors from various RFCs and
+  NIST.
+- In addition, crafted inputs are used for negative testing and to reach
+  corner cases.
+- Two test suites are provided: one for closed-box testing (using only the
+  public API), one for open-box testing (for unit-testing internal functions,
+and reaching more error cases by exploiting knowledge of how the RNG is used).
+- The resulting branch coverage is maximal: closed-box testing reaches all
+  branches except four; three of them are reached by open-box testing using a
+rigged RNG; the last branch could only be reached by computing a discrete log
+on P-256... See `coverage.sh`.
+- Testing also uses dynamic analysis: valgrind, ASan, MemSan, UBSan.
+
+**Code quality:**
+
+- The code is standard C99; it builds without warnings with `clang
+  -Weverything` and `gcc -Wall -Wextra -pedantic`.
+- The code is small and well documented, including internal APIs: with the
+  header file, it's less than 700 lines of code, and more lines of comments
+than of code.
+- However it _has not been reviewed_ independently so far, as this is a
+  personal project.
+
+**Short Weierstrass pitfalls:**
+
+Its has been [pointed out](https://safecurves.cr.yp.to/) that the NIST curves,
+and indeed all Short Weierstrass curves, have a number of pitfalls including
+risk for the implementation to:
+
+- "produce incorrect results for some rare curve points" - this is avoided by
+  carefully checking the validity domain of formulas used throughout the code;
+- "leak secret data when the input isn't a curve point" - this is avoided by
+  validating that points lie on the curve every time a point is deserialized.
+
+## Security
+
+In addition to the above correctness claims, p256-m has the following
+properties:
+
+- it has no branch depending (even indirectly) on secret data;
+- it has no memory access depending (even indirectly) on secret data.
+
+These properties are checked using valgrind and MemSan with the ideas
+behind [ctgrind](https://github.com/agl/ctgrind), see `consttime.sh`.
+
+In addition to avoiding branches and memory accesses depending on secret data,
+p256-m also avoid instructions (or library functions) whose execution time
+depends on the value of operands on cores of interest. Namely, it never uses
+integer division, and for multiplication by default it only uses 16x16->32 bit
+unsigned multiplication. On cores which have a constant-time 32x32->64 bit
+unsigned multiplication instruction, the symbol `MUL64_IS_CONSTANT_TIME` can
+be defined by the user at compile-time to take advantage of it in order to
+improve performance and code size. (On Cortex-M and Cortex-A cores wtih GCC or
+Clang this is not necessary, since inline assembly is used instead.)
+
+As a result, p256-m should be secure against the following classes of attackers:
+
+1. attackers who can only manipulate the input and observe the output;
+2. attackers who can also measure the total computation time of the operation;
+3. attackers who can also observe and manipulate micro-architectural features
+   such as the cache or branch predictor with arbitrary precision.
+
+However, p256-m makes no attempt to protect against:
+
+4. passive physical attackers who can record traces of physical emissions
+   (power, EM, sound) of the CPU while it manipulates secrets;
+5. active physical attackers who can also inject faults in the computation.
+
+(Note: p256-m should actually be secure against SPA, by virtue of being fully
+constant-flow, but is not expected to resist any other physical attack.)
+
+**Warning:** p256-m requires an externally-provided RNG function. If that
+function is not cryptographically secure, then neither is p256-m's key
+generation or ECDSA signature generation.
+
+_Note:_ p256-m also follows best practices such as securely erasing secret
+data on the stack before returning.
+
+## Code size
+
+Compiled with
+[ARM-GCC 9](https://developer.arm.com/tools-and-software/open-source-software/developer-tools/gnu-toolchain/gnu-rm/downloads),
+with `-mthumb -Os`, here are samples of code sizes reached on selected cores:
+
+- Cortex-M0: 2988 bytes
+- Cortex-M4: 2900 bytes
+- Cortex-A7: 2924 bytes
+
+Clang was also tried but tends to generate larger code (by about 10%). For
+details, see `sizes.sh`.
+
+**What's included:**
+
+- Full input validation and (de)serialisation of input/outputs to/from bytes.
+- Cleaning up secret values from the stack before returning from a function.
+- The code has no dependency on libc functions or the toolchain's runtime
+  library (such as helpers for long multiply); this can be checked for the
+Arm-GCC toolchain with the `deps.sh` script.
+
+**What's excluded:**
+
+- A secure RNG function needs to be provided externally, see
+  `p256_generate_random()` in `p256-m.h`.
+
+## RAM usage
+
+p256-m doesn't use any dynamic memory (on the heap), only the stack. Here's
+how much stack is used by each of its 4 public functions on selected cores:
+
+| Function                  | Cortex-M0 | Cortex-M4 | Cortex-A7 |
+| ------------------------- | --------: | --------: | --------: |
+| `p256_gen_keypair`        |       608 |       564 |       564 |
+| `p256_ecdh_shared_secret` |       640 |       596 |       596 |
+| `p256_ecdsa_sign`         |       664 |       604 |       604 |
+| `p256_ecdsa_verify`       |       752 |       700 |       700 |
+
+For details, see `stack.sh`, `wcs.py` and `libc.msu` (the above figures assume
+that the externally-provided RNG function uses at most 384 bytes of stack).
+
+## Runtime performance
+
+Here are the timings of each public function in milliseconds measured on
+platforms based on a selection of cores:
+
+- Cortex-M0 at  48 MHz: STM32F091 board running Mbed OS 6
+- Cortex-M4 at 100 MHz: STM32F411 board running Mbed OS 6
+- Cortex-A7 at 900 MHz: Raspberry Pi 2B running Raspbian Buster
+
+| Function                  | Cortex-M0 | Cortex-M4 | Cortex-A7 |
+| ------------------------- | --------: | --------: | --------: |
+| `p256_gen_keypair`        |       921 |       145 |        11 |
+| `p256_ecdh_shared_secret` |       922 |       144 |        11 |
+| `p256_ecdsa_sign`         |       990 |       155 |        12 |
+| `p256_ecdsa_verify`       |      1976 |       309 |        24 |
+| Sum of the above          |      4809 |       753 |        59 |
+
+The sum of these operations corresponds to a TLS handshake using ECDHE-ECDSA
+with mutual authentication based on raw public keys or directly-trusted
+certificates (otherwise, add one 'verify' for each link in the peer's
+certificate chain).
+
+_Note_: the above figures where obtained by compiling with GCC, which is able
+to use inline assembly. Without that inline assembly (22 lines for Cortex-M0,
+1 line for Cortex-M4), the code would be roughly 2 times slower on those
+platforms. (The effect is much less important on the Cortex-A7 core.)
+
+For details, see `bench.sh`, `benchmark.c` and `on-target-benchmark/`.
+
+## Comparison with other implementations
+
+The most relevant/convenient implementation for comparisons is
+[TinyCrypt](https://github.com/intel/tinycrypt), as it's also a standalone
+implementation of ECDH and ECDSA on P-256 only, that also targets constrained
+devices. Other implementations tend to implement many curves and build on a
+shared bignum/MPI module (possibly also supporting RSA), which makes fair
+comparisons less convenient.
+
+The scripts used for TinyCrypt measurements are available in [this
+branch](https://github.com/mpg/tinycrypt/tree/measurements), based on version
+0.2.8.
+
+**Code size**
+
+| Core      | p256-m | TinyCrypt |
+| --------- | -----: | --------: |
+| Cortex-M0 |   2988 |      6134 |
+| Cortex-M4 |   2900 |      5934 |
+| Cortex-A7 |   2924 |      5934 |
+
+**RAM usage**
+
+TinyCrypto also uses no heap, only the stack. Here's the RAM used by each
+operation on a Cortex-M0 core:
+
+| operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| key generation     |    608 |       824 |
+| ECDH shared secret |    640 |       728 |
+| ECDSA sign         |    664 |       880 |
+| ECDSA verify       |    752 |       824 |
+
+On a Cortex-M4 or Cortex-A7 core (identical numbers):
+
+| operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| key generation     |    564 |       796 |
+| ECDH shared secret |    596 |       700 |
+| ECDSA sign         |    604 |       844 |
+| ECDSA verify       |    700 |       808 |
+
+**Runtime performance**
+
+Here are the timings of each operation in milliseconds measured on
+platforms based on a selection of cores:
+
+_Cortex-M0_ at  48 MHz: STM32F091 board running Mbed OS 6
+
+| Operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| Key generation     |    921 |       979 |
+| ECDH shared secret |    922 |       975 |
+| ECDSA sign         |    990 |      1009 |
+| ECDSA verify       |   1976 |      1130 |
+| Sum of those 4     |   4809 |      4093 |
+
+_Cortex-M4_ at 100 MHz: STM32F411 board running Mbed OS 6
+
+| Operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| Key generation     |    145 |       178 |
+| ECDH shared secret |    144 |       177 |
+| ECDSA sign         |    155 |       188 |
+| ECDSA verify       |    309 |       210 |
+| Sum of those 4     |    753 |       753 |
+
+_Cortex-A7_ at 900 MHz: Raspberry Pi 2B running Raspbian Buster
+
+| Operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| Key generation     |     11 |        13 |
+| ECDH shared secret |     11 |        13 |
+| ECDSA sign         |     12 |        14 |
+| ECDSA verify       |     24 |        15 |
+| Sum of those 4     |     59 |        55 |
+
+_64-bit Intel_ (i7-6500U at 2.50GHz) laptop running Ubuntu 20.04
+
+Note: results in microseconds (previous benchmarks in milliseconds)
+
+| Operation          | p256-m | TinyCrypt |
+| ------------------ | -----: | --------: |
+| Key generation     |   1060 |      1627 |
+| ECDH shared secret |   1060 |      1611 |
+| ECDSA sign         |   1136 |      1712 |
+| ECDSA verify       |   2279 |      1888 |
+| Sum of those 4     |   5535 |      6838 |
+
+**Other differences**
+
+- While p256-m fully validates all inputs, Tinycrypt's ECDH shared secret
+  function doesn't include validation of the peer's public key, which should be
+done separately by the user for static ECDH (there are attacks [when users
+forget](https://link.springer.com/chapter/10.1007/978-3-319-24174-6_21)).
+- The two implementations have slightly different security characteristics:
+  p256-m is fully constant-time from the ground up so should be more robust
+than TinyCrypt against powerful local attackers (such as an untrusted OS
+attacking a secure enclave); on the other hand TinyCrypt includes coordinate
+randomisation which protects against some passive physical attacks (such as
+DPA, see Table 3, column C9 of [this
+paper](https://www.esat.kuleuven.be/cosic/publications/article-2293.pdf#page=12)),
+which p256-m completely ignores.
+- TinyCrypt's code looks like it could easily be expanded to support other
+  curves, while p256-m has much more hard-coded to minimize code size (see
+"Notes about other curves" below).
+- TinyCrypt uses a specialised routine for reduction modulo the curve prime,
+  exploiting its structure as a Solinas prime, which should be faster than the
+generic Montgomery reduction used by p256-m, but other factors appear to
+compensate for that.
+- TinyCrypt uses Co-Z Jacobian formulas for point operation, which should be
+  faster (though a bit larger) than the mixed affine-Jacobian formulas
+used by p256-m, but again other factors appear to compensate for that.
+- p256-m uses bits of inline assembly for 64-bit multiplication on the
+  platforms used for benchmarking, while TinyCrypt uses only C (and the
+compiler's runtime library).
+- TinyCrypt uses a specialised routine based on Shamir's trick for
+  ECDSA verification, which gives much better performance than the generic
+code that p256-m uses in order to minimize code size.
+
+## Design overview
+
+The implementation is contained in a single file to keep most functions static
+and allow for more optimisations. It is organized in multiple layers:
+
+- Fixed-width multi-precision arithmetic
+- Fixed-width modular arithmetic
+- Operations on curve points
+- Operations with scalars
+- The public API
+
+**Multi-precision arithmetic.**
+
+Large integers are represented as arrays of `uint32_t` limbs. When carries may
+occur, casts to `uint64_t` are used to nudge the compiler towards using the
+CPU's carry flag. When overflow may occur, functions return a carry flag.
+
+This layer contains optional assembly for Cortex-M and Cortex-A cores, for the
+internal `u32_muladd64()` function, as well as two pure C versions of this
+function, depending on whether `MUL64_IS_CONSTANT_TIME`.
+
+This layer's API consists of:
+
+- addition, subtraction;
+- multiply-and-add, shift by one limb (for Montgomery multiplication);
+- conditional assignment, assignment of a small value;
+- comparison of two values for equality, comparison to 0 for equality;
+- (de)serialization as big-endian arrays of bytes.
+
+**Modular arithmetic.**
+
+All modular operations are done in the Montgomery domain, that is x is
+represented by `x * 2^256 mod m`; integers need to be converted to that domain
+before computations, and back from it afterwards. Montgomery constants
+associated to the curve's p and n are pre-computed and stored in static
+structures.
+
+Modular inversion is computed using Fermat's little theorem to get
+constant-time behaviour with respect to the value being inverted.
+
+This layer's API consists of:
+
+- the curve's constants p and n (and associated Montgomery constants);
+- modular addition, subtraction, multiplication, and inversion;
+- assignment of a small value;
+- conversion to/from Montgomery domain;
+- (de)serialization to/from bytes with integrated range checking and
+  Montgomery domain conversion.
+
+**Operations on curve points.**
+
+Curve points are represented using either affine or Jacobian coordinates;
+affine coordinates are extended to represent 0 as (0,0). Individual
+coordinates are always in the Montgomery domain.
+
+Not all formulas associated with affine or Jacobian coordinates are complete;
+great care is taken to document and satisfy each function's pre-conditions.
+
+This layer's API consists of:
+
+- curve constants: b from the equation, the base point's coordinates;
+- point validity check (on the curve and not 0);
+- Jacobian to affine coordinate conversion;
+- point doubling in Jacobian coordinates (complete formulas);
+- point addition in mixed affine-Jacobian coordinates (P not in {0, Q, -Q});
+- point addition-or-doubling in affine coordinates (leaky version, only used
+  for ECDSA verify where all data is public);
+- (de)serialization to/from bytes with integrated validity checking
+
+**Scalar operations.**
+
+The crucial function here is scalar multiplication. It uses a signed binary
+ladder, which is a variant of the good old double-and-add algorithm where an
+addition/subtraction is performed at each step. Again, care is taken to make
+sure the pre-conditions for the addition formulas are always satisfied. The
+signed binary ladder only works if the scalar is odd; this is ensured by
+negating both the scalar (mod n) and the input point if necessary.
+
+This layer's API consists of:
+
+- scalar multiplication
+- de-serialization from bytes with integrated range checking
+- generation of a scalar and its associated public key
+
+**Public API.**
+
+This layer builds on the others, but unlike them, all inputs and outputs are
+byte arrays. Key generation and ECDH shared secret computation are thin
+wrappers around internal functions, just taking care of format conversions and
+errors. The ECDSA functions have more non-trivial logic.
+
+This layer's API consists of:
+
+- key-pair generation
+- ECDH shared secret computation
+- ECDSA signature creation
+- ECDSA signature verification
+
+**Testing.**
+
+A self-contained, straightforward, pure-Python implementation was first
+produced as a warm-up and to help check intermediate values. Test vectors from
+various sources are embedded and used to validate the implementation.
+
+This implementation, `p256.py`, is used by a second Python script,
+`gen-test-data.py`, to generate additional data for both positive and negative
+testing, available from a C header file, that is then used by the closed-box
+and open-box test programs.
+
+p256-m can be compiled with extra instrumentation to mark secret data and
+allow either valgrind or MemSan to check that no branch or memory access
+depends on it (even indirectly). Macros are defined for this purpose near the
+top of the file.
+
+**Tested platforms.**
+
+There are 4 versions of the internal function `u32_muladd64`: two assembly
+versions, for Cortex-M/A cores with or without the DSP extension, and two
+pure-C versions, depending on whether `MUL64_IS_CONSTANT_TIME`.
+
+Tests are run on the following platforms:
+
+- `make` on x64 tests the pure-C version without `MUL64_IS_CONSTANT_TIME`
+  (with Clang).
+- `./consttime.sh` on x64 tests both pure-C versions (with Clang).
+- `make` on Arm v7-A (Raspberry Pi 2) tests the Arm-DSP assembly version (with
+  Clang).
+- `on-target-*box` on boards based on Cortex-M0 and M4 cores test both
+  assembly versions (with GCC).
+
+In addition:
+
+- `sizes.sh` builds the code for three Arm cores with GCC and Clang.
+- `deps.sh` checks for external dependencies with GCC.
+
+## Notes about other curves
+
+It should be clear that minimal code size can only be reached by specializing
+the implementation to the curve at hand. Here's a list of things in the
+implementation that are specific to the NIST P-256 curve, and how the
+implementation could be changed to expand to other curves, layer by layer (see
+"Design Overview" above).
+
+**Fixed-width multi-precision arithmetic:**
+
+- The number of limbs is hard-coded to 8. For other 256-bit curves, nothing to
+  change. For a curve of another size, hard-code to another value. For multiple
+curves of various sizes, add a parameter to each function specifying the
+number of limbs; when declaring arrays, always use the maximum number of
+limbs.
+
+**Fixed-width modular arithmetic:**
+
+- The values of the curve's constant p and n, and their associated Montgomery
+  constants, are hard-coded. For another curve, just hard-code the new constants.
+For multiple other curves, define all the constants, and from this layer's API
+only keep the functions that already accept a `mod` parameter (that is, remove
+convenience functions `m256_xxx_p()`).
+- The number of limbs is again hard-coded to 8. See above, but it order to
+  support multiple sizes there is no need to add a new parameter to functions
+in this layer: the existing `mod` parameter can include the number of limbs as
+well.
+
+**Operations on curve points:**
+
+- The values of the curve's constants b (constant term from the equation) and
+  gx, gy (coordinates of the base point) are hard-coded. For another curve,
+  hard-code the other values. For multiple curves, define each curve's value and
+add a "curve id" parameter to all functions in this layer.
+- The value of the curve's constant a is implicitly hard-coded to `-3` by using
+  a standard optimisation to save one multiplication in the first step of
+`point_double()`. For curves that don't have a == -3, replace that with the
+normal computation.
+- The fact that b != 0 in the curve equation is used indirectly, to ensure
+  that (0, 0) is not a point on the curve and re-use that value to represent
+the point 0. As far as I know, all Short Weierstrass curves standardized so
+far have b != 0.
+- The shape of the curve is assumed to be Short Weierstrass. For other curve
+  shapes (Montgomery, (twisted) Edwards), this layer would probably look very
+different (both implementation and API).
+
+**Scalar operations:**
+
+- If multiple curves are to be supported, all function in this layer need to
+  gain a new "curve id" parameter.
+- This layer assumes that the bit size of the curve's order n is the same as
+  that of the modulus p. This is true of most curves standardized so far, the
+only exception being secp224k1. If that curve were to be supported, the
+representation of `n` and scalars would need adapting to allow for an extra
+limb.
+- The bit size of the curve's order is hard-coded in `scalar_mult()`. For
+  multiple curves, this should be deduced from the "curve id" parameter.
+- The `scalar_mult()` function exploits the fact that the second least
+  significant bit of the curve's order n is set in order to avoid a special
+case. For curve orders that don't meet this criterion, we can just handle that
+special case (multiplication by +-2) separately (always compute that and
+conditionally assign it to the result).
+- The shape of the curve is again assumed to be Short Weierstrass. For other curve
+  shapes (Montgomery, (twisted) Edwards), this layer would probably have a
+very different implementation.
+
+**Public API:**
+
+- For multiple curves, all functions in this layer would need to gain a "curve
+  id" parameter and handle variable-sized input/output.
+- The shape of the curve is again assumed to be Short Weierstrass. For other curve
+  shapes (Montgomery, (twisted) Edwards), the ECDH API would probably look
+quite similar (with differences in the size of public keys), but the ECDSA API
+wouldn't apply and an EdDSA API would look pretty different.
+
+## Notes about other platforms
+
+While p256-m is standard C99, it is written with constrained 32-bit platforms
+in mind and makes a few assumptions about the platform:
+
+- The types `uint8_t`, `uint16_t`, `uint32_t` and `uint64_t` exist.
+- 32-bit unsigned addition and subtraction with carry are constant time.
+- 16x16->32-bit unsigned multiplication is available and constant time.
+
+Also, on platforms on which 64-bit addition and subtraction with carry, or
+even 64x64->128-bit multiplication, are available, p256-m makes no use of
+them, though they could significantly improve performance.
+
+This could be improved by replacing uses of arrays of `uint32_t` with a
+defined type throughout the internal APIs, and then on 64-bit platforms define
+that type to be an array of `uint64_t` instead, and making the obvious
+adaptations in the multi-precision arithmetic layer.
+
+Finally, the optional assembly code (which boosts performance by a factor 2 on
+tested Cortex-M CPUs, while slightly reducing code size and stack usage) is
+currently only available with compilers that support GCC's extended asm
+syntax (which includes GCC and Clang).