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[Frama-c-discuss] WP failing to prove a simple absence of RTE unsigned overflow

On Mon, 3 Nov 2014 23:14:32 +0000
Gregory Maxwell <gmaxwell at gmail.com> wrote:

> Greetings, I haven't used frama-c in some time, it's changed quite a
> bit (and I've likely forgotten even more...) and I'm having a hard
> time making progress with it at all this time around.
> 
> I'm working on libsecp256k1 a free software cryptographic signature
> library which will be used by the reference Bitcoin software;
> verifying that optimized implementations are exactly correct is
> critical for this application
> 
> I'm trying to prove some simple theorems about some inner kernels used
> for field arithmetic.
> 
> To start with, I need to first prove that the code is overflow free,
> under assumptions. (I don't just need to do this to make WP happy, its
> required for correctness of the code.
> 
> Here is a reduced testcase that I'm trying:
> 
> $ cat a.c
> #include <stdint.h>
> /*@ requires \valid(a + (0..9));
>     requires \valid(b + (0..9));
>     requires (a[0] < 1073741824);
>     requires (a[1] < 1073741824);
>     requires (a[2] < 1073741824);
>     requires (a[3] < 1073741824);
>     requires (a[4] < 1073741824);
>     requires (a[5] < 1073741824);
>     requires (a[6] < 1073741824);
>     requires (a[7] < 1073741824);
>     requires (a[8] < 1073741824);
>     requires (a[9] < 67108864);
>     requires (b[0] < 1073741824);
>     requires (b[1] < 1073741824);
>     requires (b[2] < 1073741824);
>     requires (b[3] < 1073741824);
>     requires (b[4] < 1073741824);
>     requires (b[5] < 1073741824);
>     requires (b[6] < 1073741824);
>     requires (b[7] < 1073741824);
>     requires (b[8] < 1073741824);
>     requires (b[9] < 67108864);
> */
> void static inline secp256k1_fe_mul_inner(const uint32_t *a, const
> uint32_t *b, uint32_t *r) {
>     uint64_t d;
>     d  = (uint64_t)a[0] * b[9]
>        + (uint64_t)a[1] * b[8]
>        + (uint64_t)a[2] * b[7]
>        + (uint64_t)a[3] * b[6]
>        + (uint64_t)a[4] * b[5]
>        + (uint64_t)a[5] * b[4]
>        + (uint64_t)a[6] * b[3]
>        + (uint64_t)a[7] * b[2]
>        + (uint64_t)a[8] * b[1]
>        + (uint64_t)a[9] * b[0];
> }

I have attempted to do something similar with the reference Curve25519
implementation.

The strategy I found works well with the WP plugin is to break down
complex arithmetic expressions into basic functions, kind of like you
would write assembly.

So your example code would look like this:

    #include <stdint.h>
    /*@ 
        ensures \result == a * b;
     */
    static inline uint64_t
    _uint32_mul(const uint32_t a, const uint32_t b)
    {
        return (uint64_t) a * (uint64_t) b;
    }
    /*@ 
        requires a <= UINT64_MAX / 2;
        requires b <= UINT64_MAX / 2;
        ensures \result == a + b;
     */
    static inline uint64_t
    _uint64_sum(const uint64_t a, const uint64_t b)
    {
        return a + b;
    }
    /*@ 
        requires a <= UINT64_MAX / 10;
        requires b <= UINT64_MAX / 10;
        requires c <= UINT64_MAX / 10;
        requires d <= UINT64_MAX / 10;
        requires e <= UINT64_MAX / 10;
        requires f <= UINT64_MAX / 10;
        requires g <= UINT64_MAX / 10;
        requires h <= UINT64_MAX / 10;
        requires i <= UINT64_MAX / 10;
        requires j <= UINT64_MAX / 10;
        ensures \result == a + b + c + d + e + f + g + h + i + j;
     */
    static inline uint64_t
    _uint64_sum_10(const uint64_t a, const uint64_t b,
        const uint64_t c, const uint64_t d,
        const uint64_t e, const uint64_t f,
        const uint64_t g, const uint64_t h,
        const uint64_t i, const uint64_t j)
    {
        return (_uint64_sum(a,
                _uint64_sum(b,
                _uint64_sum(c,
                _uint64_sum(d,
                _uint64_sum(e,
                _uint64_sum(f,
                _uint64_sum(g,
                _uint64_sum(h,
                _uint64_sum(i,j))))))))));
    }
    /*@ 
        requires \valid(a + (0..9));
        requires \valid(b + (0..9));
        requires (a[0] < 1073741824);
        requires (a[1] < 1073741824);
        requires (a[2] < 1073741824);
        requires (a[3] < 1073741824);
        requires (a[4] < 1073741824);
        requires (a[5] < 1073741824);
        requires (a[6] < 1073741824);
        requires (a[7] < 1073741824);
        requires (a[8] < 1073741824);
        requires (a[9] < 67108864);
        requires (b[0] < 1073741824);
        requires (b[1] < 1073741824);
        requires (b[2] < 1073741824);
        requires (b[3] < 1073741824);
        requires (b[4] < 1073741824);
        requires (b[5] < 1073741824);
        requires (b[6] < 1073741824);
        requires (b[7] < 1073741824);
        requires (b[8] < 1073741824);
        requires (b[9] < 67108864);
    */
    static inline void
    secp256k1_fe_mul_inner(const uint32_t a[10], const uint32_t b[10])
    {
        uint64_t d;
        d = _uint64_sum_10(
                _uint32_mul(a[0], b[9]),
                _uint32_mul(a[1], b[8]),
                _uint32_mul(a[2], b[7]),
                _uint32_mul(a[3], b[6]),
                _uint32_mul(a[4], b[5]),
                _uint32_mul(a[5], b[4]),
                _uint32_mul(a[6], b[3]),
                _uint32_mul(a[7], b[2]),
                _uint32_mul(a[8], b[1]),
                _uint32_mul(a[9], b[0])
            );
    }

Check it with the WP and RTE plugins:

    $ frama-c -machdep $(uname -m) -pp-annot -wp -wp-rte \
        -wp-prop @ensures,unsigned_overflow a.c 
    ...
    [wp] Proved goals:    7 / 7
         Qed:             3  (4ms-32ms)
         Alt-Ergo:        4  (76ms-80ms) (13)

It works, but it's very verbose.  You may find that the value analysis
plugin is less tedious, I don't know.

Best of luck.

Mansour