Try the following examples from H. Cohen and G. Frey,

The following are examples of Galois Field operations. The base is hexadecimal. The reduction polynomial is 805:

- Reduction: 2D4 = 11902B (mod 805)

- Multiplication: A6A6 = 36 × 6F9

- Inversion: 702 = 1
**/**173

Coordinates |
P |
Q |
P + Q |
2 × P |
2FB × P |

Affine |
(420, 5B3) |
(4B8, 167) |
(724, 7B3) |
(14D, 4CB) |
(84, 475) |

Projective |
(64F, 5BA, 1C9) |
(4DD, 1F0, 3FA) |
(675, 6D5, 4D5) |
(4D5, 21E, 705) |
(582, 14, 543) |

Jacobian |
(4DA, 1F7, 701) |
(383, 5BA, 1E1) |
(12, 46B, 5F) |
(5B1, 417, 7D) |
(2F7, 572, 3E2) |

Lopez-Dahab |
(6BE, 15F, 7B3) |
(757, 3EF, 219) |
(7C5, 1D2, 3D2) |
(444, 4A0, 193) |
(2F, 265, 220) |

- Text box – gives diagnostic messages
- Undo button – undo the last operation
- Redo button – undo the undo button

- Text box – the place to key in numbers in the current base
- Push button – push the value in the entry text box onto the stack
- Pop button – pop the last value from the stack into the entry text box

- Drop – delete the last item from the bottom of the stack
- Dup – duplicate the last item and push it onto the bottom of the stack
- Swap – swap the last two items on the stack, reversing their order
- Over – copy the second last item and pushes it onto the bottom of the stack
- Rot – rotate the last three items, moving the third last item to the bottom of the stack
- Pick – copy the currently selected stack item and pushes it onto the bottom of the stack
- Roll – move the currently selected stack item to the bottom of the stack

- Binary – base 2
- Quaternary – base 4
- Octal – base 8
- Hexadecimal – base 16

- Reduce – reduce the last value on the stack by the reduction polynomial
- Add – add the last two values on the stack
- Sqr – square the last value on the stack
- Mul – multiply the last two values on the stack
- → Point – assemble the scalars on the bottom of the stack into a point in the current coordinate system
- Inv – get the multiplicative inverse of the last value, producing a reduced result
- Div – divide the last two values, by inverting the last number and multiplying by the second last
- Sqrt – take the square root of the last value, producing a reduced result
- Trace – compute trace of the last value, producing a reduced result
- Half-Trace – compute half-trace of the last value, producing a reduced result

- Text box – the encoded value of the reduction polynomial
- Push button – push the value from the reduction polynomial text box onto the stack
- Pop button – pop the last value from the stack into the reduction polynomial text box

- Affine – (A
_{x}, A_{y}): normal two-dimensional coordinate system - Projective – (P
_{x}, P_{y}, P_{z}) = (*z*·A_{x},*z*·A_{y},*z*), z ≠ 0

- Jacobian – (J
_{x}, J_{y}, J_{z}) = (*z*²·A_{x},*z*³·A_{y},*z*), z ≠ 0 - Lopez-Dahab – (LD
_{x}, LD_{y}, LD_{z}) = (*z*·A_{x},*z*²·A_{y},*z*), z ≠ 0

- ∞ – push the point at infinity onto the bottom of the stack
- Neg – get the additive inverse of the last value on the stack
- Dbl – double the last value on the stack
- Add – add the last two values on the stack
- Mul – perform scalar multiplication where the second last value on the stack is the scalar and the last value is the point
- → Scalar – separate the point on the bottom of the stack into its component scalars in the current coordinate system

- Text box – the a
_{2}' value used in elliptic curve operations - Push button – push the value in the a
_{2}' text box onto the stack - Pop button – pop the last value from the stack into the a
_{2}' text box

- Text box – the a
_{6}' value used in elliptic curve operations - Push button – push the value in the a
_{6}' text box onto the stack - Pop button – pop the last value from the stack into the a
_{6}' text box

- Del – delete the currently selected item
- Move ↑ – move the currently selected item up one line
- Move ↓ – move the currently selected item down one line
- Push – push the currently selected storage item onto the stack
- Pop – pop the last value from the stack to the currently selected storage slot