The Curve

Doubling a Point

Doubling a point uses a different slope formula. The table recomputes the tangent-style row.

highlighted = computed this step

Why doubling is a special add

Doubling uses the tangent formula because P and Q are the same point. The table marks that this row is a double.

P+PP+P
Doubling a pointPoint-addition rows recomputed from the two input points.Doubling a point - R=(6,3)quantityvalueP(5,1)Q(5,1)doubletrueslope s13x3=s^2-x1-x26y3=s(x1-x3)-y13R=P+Q(6,3)

Use P twice

The point is P=(5,1).

2P2P
Doubling a pointPoint-addition rows recomputed from the two input points.Doubling a point - R=(6,3)quantityvalueP(5,1)Q(5,1)doubletrueslope s13x3=s^2-x1-x26y3=s(x1-x3)-y13R=P+Q(6,3)

Read the doubling slope

The doubling slope is s=13.

s=13s=13
Doubling a pointPoint-addition rows recomputed from the two input points.Doubling a point - R=(6,3)quantityvalueP(5,1)Q(5,1)doubletrueslope s13x3=s^2-x1-x26y3=s(x1-x3)-y13R=P+Q(6,3)

Read the doubled point

The doubled point is (6,3).

2P=(6,3)2P=(6,3)
Doubling a pointPoint-addition rows recomputed from the two input points.Doubling a point - R=(6,3)quantityvalueP(5,1)Q(5,1)doubletrueslope s13x3=s^2-x1-x26y3=s(x1-x3)-y13R=P+Q(6,3)

Summary

Doubling G gives (6,3). NOTE: toy-modulus; no-side-channel; no-production; never-roll-your-own. Exact point arithmetic over a deliberately tiny prime field; real ECC uses about 256-bit curves such as Curve25519 in reviewed protocols.

doubling recomputed\text{doubling recomputed}
Doubling a pointPoint-addition rows recomputed from the two input points.Doubling a point - R=(6,3)quantityvalueP(5,1)Q(5,1)doubletrueslope s13x3=s^2-x1-x26y3=s(x1-x3)-y13R=P+Q(6,3)