siec

README

siec

Super-Isolated Elliptic Curve Implementation in Go

This package exports a super-isolated elliptic curve. Over the base field 𝔽ₚ, the curve E does not admit any isogenies to other curves.

We can verify the curve properties in Sage.

K.<isqrt3> = QuadraticField(-3)
pi = 2^127 + 2^25 + 2^12 + 2^6 + (1 - isqrt3)/2
p = ZZ(pi.norm())
N = ZZ((pi-1).norm())
E = EllipticCurve(GF(p),[0,19]) # E: y^2 = x^3 + 19
G = E([5,12])
# p is a 255 bit prime with hamming weight 14
assert p.is_prime()
assert len(p.bits()) == 255
assert sum(p.bits()) == 14
# N is a 255 bit prime
assert N.is_prime()
assert len(N.bits()) == 255
# E has N points
assert E.count_points() == N
# The Frobenius endomorphism on E satisfies the same minimal polynomial as pi
assert E.frobenius_polynomial() == pi.minpoly()
# pi generates a maximal order
assert K.order([pi]).is_maximal()
# K has class number 1
assert K.class_number() == 1
# Examples of order 6 endomorphisms
assert E([28948022309329048855892746252183396360433790236562615305258360005404201062400*G[0],-G[1]]) == 170141183460469231731687303715917664320*G
assert 170141183460469231731687303715917664320^3 % N == N-1

Documentation

Index

• type SIEC255Params
  • func SIEC255() *SIEC255Params
  • func (curve *SIEC255Params) Add(x1, y1, x2, y2 *big.Int) (x3, y3 *big.Int)
  • func (curve *SIEC255Params) Compress(x, y *big.Int) (c [32]byte)
  • func (curve *SIEC255Params) Decompress(c [32]byte) (x, y *big.Int)
  • func (curve *SIEC255Params) Double(x1, y1 *big.Int) (x3, y3 *big.Int)
  • func (curve *SIEC255Params) GenerateKey(rand io.Reader) (k []byte, x, y *big.Int, err error)
  • func (curve *SIEC255Params) IsOnCurve(x, y *big.Int) bool
  • func (curve *SIEC255Params) Params() *SIEC255Params
  • func (curve *SIEC255Params) ScalarBaseMult(k []byte) (x, y *big.Int)
  • func (curve *SIEC255Params) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int)

Types

type SIEC255Params

type SIEC255Params struct {
	P       *big.Int // the order of the underlying field
	N       *big.Int // the order of the base point
	A       *big.Int // the other constant of the curve equation
	B       *big.Int // the constant of the curve equation
	Gx, Gy  *big.Int // (x,y) of the base point
	BitSize int      // the size of the underlying field
	Name    string   // the canonical name of the curve
}

SIEC255Params contains the parameters of an elliptic curve and also provides a generic, non-constant time implementation of Curve.

func SIEC255

func SIEC255() *SIEC255Params

SIEC255 returns a Curve which implements SIEC255.

func (*SIEC255Params) Add

func (curve *SIEC255Params) Add(x1, y1, x2, y2 *big.Int) (x3, y3 *big.Int)

Add returns the sum of (x1,y1) and (x2,y2)

func (*SIEC255Params) Compress

func (curve *SIEC255Params) Compress(x, y *big.Int) (c [32]byte)

Compress compresses a point down to 32 bytes. The first 31 bytes (index 0,1,2, and most of 3) represent the x coordinate (little endian). The last bit of the fourth byte is the parity of the y coordinate.

func (*SIEC255Params) Decompress

func (curve *SIEC255Params) Decompress(c [32]byte) (x, y *big.Int)

Decompress uncompresses a point and returns an actual (x,y) pair.

func (*SIEC255Params) Double

func (curve *SIEC255Params) Double(x1, y1 *big.Int) (x3, y3 *big.Int)

Double returns 2*(x,y)

func (*SIEC255Params) GenerateKey

func (curve *SIEC255Params) GenerateKey(rand io.Reader) (k []byte, x, y *big.Int, err error)

GenerateKey returns a public/private key pair. The private key is generated using the given reader, which must return random data. This is copied from https://golang.org/src/crypto/elliptic/elliptic.go?s=7368:7453#L266

func (*SIEC255Params) IsOnCurve

func (curve *SIEC255Params) IsOnCurve(x, y *big.Int) bool

IsOnCurve reports whether the given (x,y) lies on the curve.

func (*SIEC255Params) Params

func (curve *SIEC255Params) Params() *SIEC255Params

Params returns the parameters for the curve.

func (*SIEC255Params) ScalarBaseMult

func (curve *SIEC255Params) ScalarBaseMult(k []byte) (x, y *big.Int)

ScalarBaseMult returns k*G, where G is the base point of the group and k is an integer in big-endian form.

func (*SIEC255Params) ScalarMult

func (curve *SIEC255Params) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int)

ScalarMult returns k*(Bx,By) where k is a number in big-endian form.