2

What do each of these structs represent? What is the difference between them?

typedef struct {
  fe X;
  fe Y;
  fe Z;
} ge_p2;

typedef struct {
  fe X;
  fe Y;
  fe Z;
  fe T;
} ge_p3;

typedef struct {
  fe X;
  fe Y;
  fe Z;
  fe T;
} ge_p1p1;

typedef struct {
  fe yplusx;
  fe yminusx;
  fe xy2d;
} ge_precomp;

typedef struct {
  fe YplusX;
  fe YminusX;
  fe Z;
  fe T2d;
} ge_cached;
2

They are all group elements (hence the ge_ prefix) and are different representations.

From some ed25519 source:

ge means group element.

Here the group is the set of pairs (x,y) of field elements (see fe.h)
satisfying -x^2 + y^2 = 1 + d x^2y^2
where d = -121665/121666.

Representations:
  ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
  ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
  ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
  ge_precomp (Duif): (y+x,y-x,2dxy)

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