[curves] Torsion-safe representatives (was: Ed25519 "clamping" and its effect on hierarchical key derivation)

[Henry de Valence]

On Mon, Mar 06, 2017 at 11:36:14AM -0800, Tony Arcieri wrote:
> Ed25519 performs the following operations on private scalars immediately
> prior to use:
>
> scalar[0] &= 248;
> scalar[31] &= 63;
> scalar[31] |= 64;
>
> I've heard this referred to as "clamping" although that may not be the best
> term.
>
> These operations are not applied to the canonical scalar, i.e. the one
> which is serialized and persisted as part of the keypair. Instead Ed25519
> implementations generally flip these bits immediately prior to use, either
> for signing or deriving the public key.
>
> [...]
>
> In these schemes, it's not possible to "clamp" the derived scalar
> immediately prior to signing ("post-clamping" I guess?), as this would
> result in a different public key (i.e. the math simply does not work out as
> the groups are no longer commutative). Instead, if any clamping is to be
> performed it must happen immediately to the parent scalar, and/or to any
> scalars derived by both the public and private key holders in such a scheme.

Hi all,

this subject came up today at the Tor meeting in a discussion with Ian
Goldberg, George Kadianakis, Isis Lovecruft, and myself.

As Tony noted, using bit-twiddling to mask away the low bits of a scalar is
problematic because the bit-twiddling is not well-defined `(mod l)`.  (Here `l`
is the order of the basepoint, so the full group has order `8*l`).  This means
that the "clamping" is not compatible with any arithmetic operations on
scalars.

We (Ian, George, Isis, and myself) have the following suggestion.

Define a "torsion-safe representative" of `a \in Z` to be an integer `a' \in
Z` such that `a ≡ a' (mod l)` and `a' ≡ 0 (mod 8)`.

This means that `a B = a' B` for the basepoint `B`, but `a' T = id` for `T` a
torsion point, so accidentally multiplying by a torsion point can't leak
information.  However, unlike "clamping", this operation is actually
well-defined, and leaves the scalar unchanged, in the sense that scalar
multiplication by `a` and by `a'` are the same on the prime-order subgroup.

How do we compute such a representative?  Since (using the CRT)

    k := 3l + 1 ≡ 1 (mod l)
                ≡ 0 (mod 8),

`k*a mod 8l` is a torsion-safe representative of `a`.  Computing `k*a mod 8l`
directly is a problem because an implementation may only have arithmetic modulo
`l`, not `8l`.  However,

    k*a ≡ (3l+1)*a ≡ 3al + a (mod 8l),

so it's sufficient to compute `3al (mod 8l)` and add it to `a`.  Since

     3a mod 8 = 3a + 8n,
    (3a mod 8)*l = 3al + 8ln ≡ 3al (mod 8l),

so it's sufficient to compute `3a mod 8` and use a lookup table of

    [0, 1l, 2l, 3l, 4l, 5l, 6l, 7l]

to get `3al (mod 8l)`.  Arranging the lookup table as

    [0, 3l, 6l, 1l, 4l, 7l, 2l, 5l]

means that `a mod 8` indexes `3al (mod 8l)`.  Therefore, computing a
torsion-safe representative for a scalar `a` just amounts to computing

    a + permuted_lookup_table[a & 7]

in constant time.  If the input `a` is reduced, so that `a < l`, then the
torsion-safe representative is bounded by `8l < 2^256` and therefore fits in 32
bytes.

This ends up being slightly more work than just bit-twiddling, but not by much,
and it's certainly insignificant compared to the cost of a scalar
multiplication.  There's a prototype implementation here [0] in case anyone is
curious to see what it looks like.

Cheers,
Henry de Valence

[0]: https://github.com/hdevalence/curve25519-dalek/commit/2ae0bdb6df26a74ef46d4332b635c9f6290126c7
(subject to rebasing...)

The permuted lookup table is, explicitly:

sage: l = 2^252 + 27742317777372353535851937790883648493
sage: lookup = [((3 * i) % 8)*l for i in range(8)]
sage: lookup
[0,
 21711016731996786641919559689128982722571349078139722818005852814856362752967,
 43422033463993573283839119378257965445142698156279445636011705629712725505934,
 7237005577332262213973186563042994240857116359379907606001950938285454250989,
 28948022309329048855892746252171976963428465437519630424007803753141817003956,
 50659039041325835497812305941300959685999814515659353242013656567998179756923,
 14474011154664524427946373126085988481714232718759815212003901876570908501978,
 36185027886661311069865932815214971204285581796899538030009754691427271254945]