An identity element leaves another element unchanged under a specified operation. For addition, zero satisfies a + 0 = a; for multiplication, one satisfies b × 1 = b. section 2.1
The identity element changes with the algebraic setting:
- Addition uses 0, because adding it preserves the original element. section 2.1
- Multiplication uses 1, because multiplying by it preserves the original element. section 2.1
- Elliptic-curve point addition uses the point at infinity, written 0 here; P + 0 = P and P − P = 0. section 3.1
- Taking the order of a point reaches the point-at-infinity identity, while taking an mth root of unity to the mth power reaches 1. section 15.1
Clock arithmetic and elliptic curves
A clock illustrates modular arithmetic: values wrap around after the modulus, so 15 is congruent to 3 modulo 12. On an elliptic curve, adding a point to its inverse produces the point at infinity rather than an ordinary plotted point. section 2.1 section 3.1
Figure 3.1 Elliptic curve \(y^2=x^3-5 x +5\)
Visualizes the elliptic curve whose point-at-infinity identity supports point addition. section 3.1
