Henry, the technique you showed is fantastic, I love it! I have a lame question, though. You mention that `a*B = a'*B` holds for the base point. But is it also true for any point in the B's subgroup? The reason I ask is that I need to have not just regular EdDSA signatures, but also DLEQs (discrete log equality proofs) with random generator points.
Thank you! > > 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] > _______________________________________________ > Curves mailing list > Curves@moderncrypto.org > https://moderncrypto.org/mailman/listinfo/curves _______________________________________________ Curves mailing list Curves@moderncrypto.org https://moderncrypto.org/mailman/listinfo/curves