To be clear, the requirements from our legal department are: * I need to ask them before releasing code, and * I need to comply with US export control law, which (as Legal interprets it) requires preventing IPs associated with a few specific countries from downloading the software.
I don’t believe that GitHub can filter by country (please tell me if that’s changed), but SourceForge or some other service might be able to. Anyway, I haven’t gotten the ball rolling yet, so it might take a while to get permission, especially since the software isn’t done yet. But I’ll see what I can do. Cheers, — Mike On Feb 12, 2014, at 12:26 PM, Michael Hamburg <[email protected]> wrote: > Hi Trevor, > > Not a github initially, because of Rambus legal and export control and all > that. I’ll see if I can set up something more private and get back to you. > > Cheers, > — Mike > > On Feb 12, 2014, at 11:22 AM, Trevor Perrin <[email protected]> wrote: > >> Could we expect a github? I'd love to see this! >> >> Trevor >> >> >> On Tue, Feb 11, 2014 at 12:31 AM, Mike Hamburg <[email protected]> wrote: >>> Hello curves, >>> >>> I've been working on implementation for the new curves, and I'd like to >>> report status and some formulas and issues I found. >>> >>> I'm aiming for a fairly generic C/intrinsics implementation which should >>> support any curves with minimal extra effort, but I'm starting with >>> Ed448-Goldilocks because it's mine. I have Haswell and Sandy Bridge test >>> machines. I also have a vectorless Cortex A9, but it doesn't work yet >>> because I'm using 64x64->128-bit multiply intrinsics. Here's what I've >>> found so far. >>> >>> If you have any suggestions on the formulas or algorithms, I'd definitely >>> appreciate it. >>> >>> Field arithmetic: >>> * Karatsuba is beneficial for Ed448. >>> * Radix 2^56 in a 64-bit limb, 8 limbs. >>> * M ~ 153cy on Sandy Bridge, 125cy on Haswell >>> * square ~ 0.75M >>> * small fixed mul ~ 0.25M >>> * add/sub (unreduced) ~ 0.04M, a little cheaper on Haswell because of AVX >>> >>> I'm using the 1/sqrt(x) point encoding for now, basically because I already >>> have code for that from an earlier project. I'm not yet counting the time >>> to serialize and deserialize field elements, which is maybe 100 cyles at >>> most (counting the full reduce / checking that input is fully reduced). I'm >>> not yet counting hashing or RNG times. >>> >>> My earlier email about 1/sqrt(x) was slightly off: it encodes even points on >>> the curve, but odd points on the twist. >>> >>> I haven't tried blind+EGCD for inverses or Legendre symbol checks. It might >>> well be a win. One inverse square root is 56k Sandy cycles (I don't >>> remember the Haswell number). >>> >>> Full Montgomery ladder: >>> * Decompress. >>> * Constant-time ladder by 448-bit scalar. The scalar should be even for >>> security. It actually could be 447 bits. >>> * Recompress. Reject points on the twist. This is basically free, but >>> important because they can't be encoded with the 1/sqrt(x) encoding. >>> >>> This takes about 571kcy on Haswell, and 688kcy on Sandy, corrected for >>> TurboBoost. >>> >>> I'm using the formula from the thread on efficient laddering with the >>> isomorphic curve, but twisted. Let (xd,zd) be the point to de doubled, and >>> (xa,za) be the point to be added. >>> A = (xd+zd) >>> B = (xd-zd) >>> DA = (xa-za)*A >>> BC = (xa+za)*B >>> >>> oxa = (DA+BC)^2 >>> oza = (DA-BC)^2 * xbase >>> >>> AA = A^2 >>> BB = B^2 >>> AAod = AA*(1-d) >>> E = AA-BB >>> >>> oxd = AAod*BB >>> ozd = E*(AAod-E) >>> >>> return (oxd,ozd,oxa,oza) >>> >>> Except I'm actually using zbase instead of xbase, because of the 1/sqrt(x) >>> format. >>> >>> Twisted Edwards (a=-1) windowed algorithm: >>> * Assumes that cofactor is canceled somehow. >>> * Recode scalar in signed form, because it's easy and I'm lazy. >>> * Compute 8 odd multiples of P. >>> * Constant-time add/sub chain with a 4-bit window, 448 bits. Could be 446 >>> bits, except that 446 isn't divisible by 4. >>> * No compress or decompress. >>> >>> This takes slightly less time than the Montgomery ladder, some 530kcy on >>> Haswell and 636 kcy on Sandy. A 5-bit window makes things maybe 1-2% >>> faster, but uses extra complexity and memory so I didn't think it was >>> worthwhile. >>> >>> I'm using readdition coordinates: >>> "Projective half-niels" for the tables, ((y-x)/2 : (y+x)/2 : dxy : 1) * z. >>> "Lazy extended coordinates" for the accumulator, (x : y : z : t : u) where >>> xy = tuz. >>> >>> I might replace the lazy extended coordinates with Hisil et al's lookahead >>> extended-or-not coordinates, which use less memory but require more care. >>> >>> Full constant-time scalarmul using twisted Edwards: >>> * Decompress points, rejecting those on the twist. >>> * Isogenize to the twisted curve, canceling the cofactor. >>> * Above windowed algorithm. >>> * Isogenize back to the main curve, effectively multiplying by 4. >>> * Recompress. >>> >>> This takes slightly longer than the Montgomery ladder: something like 633kcy >>> on Haswell and 750kcy on Sandy. So Edwards or twisted Edwards is best for >>> points you've already got in projective form, and Montgomery is best for >>> ECDH. Unsurprising. >>> >>> The total executable code size to test and bench the arithmetic and curve >>> routines is currently around 41k under clang -O4 -fPIC. That'll get bigger >>> once there are precomputed tables. >>> >>> Formulas: >>> I'm making use of the "inverse square root trick": >>> >>> def trick(a,b,i): >>> # assumes p==3 mod 4; similar trick exists for 1 mod 4 >>> # returns sqrt(+-a/b), 1/i, is_square(a/b) >>> # assumes a,b,i are nonzero >>> ai = a*i >>> abi = b*ai >>> s = 1/sqrt(+- abi*i) # using a powering ladder >>> output sqrt(+-a/b) = s*ai >>> s2abi = s^2*abi >>> issquare = s2abi * i # = Legendre symbol >>> if you care about the result of 1/i when a/b is nonsquare: >>> output 1/i = s2abi*issquare >>> else: >>> output 1/i = s2abi >>> >>> You can tweak the trick to change the Legendre symbol of the output >>> according to some other variable as well; this depends on the residue of p >>> mod 8. >>> >>> The formula I'm using for point compression with Montgomery form is: >>> >>> Let P1 + P2 = P3 and (u1,v1) = P1 etc. Then >>> 4*v1*v2*u3 = (u1*u2-1)^2 - u3^2*(u1-u2)^2 >>> >>> To compute the numerator of the RHS, do: >>> sa = (z2*z1 - x2*x1) * z3 >>> sb = (x2*z1 - z2*x1) * x3 >>> numerator = (sa + sb) * (sa - sb) >>> This is good enough to get the Legendre symbol. It shouldn't be too hard to >>> convert this into a formula with some other sign bit using the inverse >>> square root trick. >>> >>> This is on an untwisted (B=1) curve, but the same "ought" to be true of >>> 4*B*v1*v2*u3 on a twisted one. >>> >>> To serialize an Edwards point, we have to deal with the fact that the >>> isomorphic curve you'd get from Wikipedia is twisted, because it sets B = >>> 4/(1-d) which isn't square, at least when p==3 mod 4. So I'm negating x to >>> get to the curve: >>> 4y^2/(d-1) = x^3 + 2(d+1)/(d-1) * x^2 + x >>> where you can then scale y by sqrt(4/(d-1)) to get the standard curve. >>> >>> To deserialize an Edwards point, compute >>> denominator = (u+1)^2 * (d-1) + 4u >>> x = 2 sqrt(u/denominator) >>> y = (1+u)/(1-u) >>> using the inverse square root trick. This lands you on E_(1,d), because it >>> scales the x-coordinate to get rid of the twisting that the obvious >>> decompression would give you. >>> >>> You have to check if u=0 or u=1. The latter isn't on the curve, but you >>> have to make sure it doesn't slip past the check due to the zero divide. >>> The former works in the 1/sqrt(x) encoding without any checks. >>> >>> To do: >>> I'm planning to use WNAF for variable time scalar mul, WNAF for signature >>> verification, and a precomputed signed comb for key generation and Schnorr >>> signing. >>> >>> I'm experimenting with the best way to implement Elligator. I currently >>> only have the map to the curve done, and I might change the signs. My >>> implementation maps directly to affine using the inverse square root trick. >>> I'll report the formula once I'm done messing around with it. >>> >>> And of course there's API packaging, testing on ARM, etc. >>> >>> Cheers, >>> -- Mike >>> >>> >>> >>> >>> _______________________________________________ >>> Curves mailing list >>> [email protected] >>> https://moderncrypto.org/mailman/listinfo/curves >>> > > _______________________________________________ > Curves mailing list > [email protected] > https://moderncrypto.org/mailman/listinfo/curves _______________________________________________ Curves mailing list [email protected] https://moderncrypto.org/mailman/listinfo/curves
