Construct own Elliptic Curve

[a.a]

For some research I would like to use a secp256k1 like Elliptic Curve, but instead of 256 bit length it should be 32 bit length. Does somebody have something ready?

I mean, it is important to get the right values for G, n and P. How do you construct them for 32 bit?

Thx

[1714702810]

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[1714702810]

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[gmaxwell]

To be analogous to secp256k1, your prime should be congruent to 3 mod 4, should have a primitive cube root of unity, and the generated group should be prime.

The curve created by a=0 b=11 over field P=2^32-116325 has these properties. N is 2^32-4443.

There are many such choices that work.  I chose the option with the largest group size subject to the restriction that the order was still under 2^32, that the twist had cofactor of 4 or less, and that the embedding degree was 'large' (in this case almost 2^30, though there are somewhat smaller groups where the embedding degree is close to 2^32). I chose the smallest b among the isomorphic alternatives.

For a generator,  you could use G = {0x02, 0x20c2c3af}  (x=0x01 isn't on the curve). All points on the curve are equally good as a generator.


results=[]
for x in [xf for xf in range(2^32,2^32-300000,-1) if xf%4==3 and Integer(xf).is_prime() and FiniteField(xf)(1).nth_root(3)!=1]:
  for b in range(1,15):
    order=EllipticCurve([FiniteField(x)(0), FiniteField(x)(b)]).order()
    if order<2^32 and order.is_prime()==True:
      ordert=EllipticCurve([FiniteField(x)(0), FiniteField(x)(-b)]).order()
      ordertp=factor(ordert)[-1][0]
      if ordert/ordertp<=4:
        results.append((order,RR(log(FiniteField(order)(x).multiplicative_order(),2)),ordert,-x,-b))
sorted(results)

[a.a]

Hmm. Can you please provide me a 2^16 curve with starting point 1 please?

It is easier to generate all points for 2^16 than for 2^32 )

[gmaxwell]

P=64231 N=64633 a=0 b=3  G={0x01,0x02}