In Bitcoin cryptography, we need to deal with astronomical numbers, which are so huge that it can easily exceed the total number of atoms in our universe, maybe the 64-bit value is not enough to represent this number, and operations like addition, multiplication, power can cause overflow if we use 64-bit integers, so we may need to use golang's big package, we will change the code of FieldNumber by using 'to represent its value field, the code will look like this:
package elliptic_curve
import (
"fmt"
"math/big"
)
//using big package to deal with Astronomical figures
type FieldElement struct {
order * //field order
num * //value of the given element in the field
}
func NewFieldElement(order *, num *) *FieldElement {
/*
constructor for FieldElement, its the __init__ if you are from python
*/
if (num) == -1 {
err := ("Num not in the range from 0 to %v", order)
panic(err)
}
return &FieldElement{
order: order,
num: num,
}
}
func (f *FieldElement) String() string {
//format the object to printable string
//its __repr__ if you are from python
return ("FieldElement{order: %v, num: %v}", *, *)
}
func (f *FieldElement) EqualTo(other *FieldElement) bool {
/*
two field element is equal if their order and value are equal
*/
return () == 0 && () == 0
}
func (f *FieldElement) checkOrder(other *FieldElement) {
if () != 0 {
panic("add need to do on field element with the same order")
}
}
func (f *FieldElement) Add(other *FieldElement) *FieldElement {
(other)
//remember to do the modulur
var op
return NewFieldElement(, ((, ), ))
}
func (f *FieldElement) Negate() *FieldElement {
/*
for a field element a, its negate is another element b in field such that
(a + b) % order= 0(remember the modulur over order), because the value of element
in the field are smaller than its order, we can easily get the negate of a by
order - a,
*/
var op
return NewFieldElement(, (, ))
}
func (f *FieldElement) Subtract(other *FieldElement) *FieldElement {
//first find the negate of the other
//add this and the negate of the other
return (())
}
func (f *FieldElement) Multiply(other *FieldElement) *FieldElement {
(other)
//multiplie over modulur of order
var op
mul := (, )
return NewFieldElement(, (mul, ))
}
func (f *FieldElement) Power(power *) *FieldElement {
var op
powerRes := (, power, nil)
modRes := (powerRes, )
return NewFieldElement(, modRes)
}
func (f *FieldElement) ScalarMul(val *) *FieldElement {
var op
res := (, val)
res = (res, )
return NewFieldElement(, res)
}Now we need to make sure that these changes don't break our logic, let's run the test again, in which we have the following code:
package main
import (
ecc "elliptic_curve"
"fmt"
"math/big"
"math/rand"
)
func SolveField19MultiplieSet() {
//randomly select a num from (1, 18)
min := 1
max := 18
k := (max-min) + min
("randomly select k is : %d\n", k)
element := ((19), (int64(k)))
for i := 0; i < 19; i++ {
("element %d multiplie with %d is %v\n", k, i,
((int64(i))))
}
}
func main() {
f44 := ((57), (44))
f33 := ((57), (33))
// 44 + 33 equal to (44+33) % 57 is 20
res := (f33)
("field element 44 add to field element 33 is : %v\n", res)
//-44 is the negate of field element 44, which is 57 - 44 = 13
("negate of field element 44 is : %v\n", ())
("field element 44 - 33 is : %v\n", (f33))
("field element 33 - 44 is : %v\n", (f44))
//it is easy to check (11+33)%57 == 44
//check (46 + 44) % 57 == 33
("check 46 + 44 over modulur 57 is %d\n", (46+44)%57)
//check by field element
f46 := ((57), (46))
("field element 46 + 44 is %v\n", (f44))
SolveField19MultiplieSet()
}Running the above code will get the following results:
field element 44 add to field element 33 is : FieldElement{order: 57, num: 20}
negate of field element 44 is : FieldElement{order: 57, num: 13}
field element 44 - 33 is : FieldElement{order: 57, num: 11}
field element 33 - 44 is : FieldElement{order: 57, num: 46}
check 46 + 44 over modulur 57 is 33
field element 46 + 44 is FieldElement{order: 57, num: 33}
randomly select k is : 2
element 2 multiplie with 0 is FieldElement{order: 19, num: 0}
element 2 multiplie with 1 is FieldElement{order: 19, num: 2}
element 2 multiplie with 2 is FieldElement{order: 19, num: 4}
element 2 multiplie with 3 is FieldElement{order: 19, num: 6}
element 2 multiplie with 4 is FieldElement{order: 19, num: 8}
element 2 multiplie with 5 is FieldElement{order: 19, num: 10}
element 2 multiplie with 6 is FieldElement{order: 19, num: 12}
element 2 multiplie with 7 is FieldElement{order: 19, num: 14}
element 2 multiplie with 8 is FieldElement{order: 19, num: 16}
element 2 multiplie with 9 is FieldElement{order: 19, num: 18}
element 2 multiplie with 10 is FieldElement{order: 19, num: 1}
element 2 multiplie with 11 is FieldElement{order: 19, num: 3}
element 2 multiplie with 12 is FieldElement{order: 19, num: 5}
element 2 multiplie with 13 is FieldElement{order: 19, num: 7}
element 2 multiplie with 14 is FieldElement{order: 19, num: 9}
element 2 multiplie with 15 is FieldElement{order: 19, num: 11}
element 2 multiplie with 16 is FieldElement{order: 19, num: 13}
element 2 multiplie with 17 is FieldElement{order: 19, num: 15}
element 2 multiplie with 18 is FieldElement{order: 19, num: 17}
By checking the results, we can make sure that changes in FieldElement do not break our previous logic. Now let's consider the following:
p = 7, 11, 17, 19, 31, what would the following set be:
{1 ^(p-1), 2 ^ (p-1), … (p-1)^(p-1)}
Let's write code in :
func ComputeFieldOrderPower() {
orders := []int{7, 11, 17, 31}
for _, p := range orders {
("value of p is: %d\n", p)
for i := 1; i < p; i++ {
elm := ((int64(p)), (int64(i)))
("for element: %v, its power of p - 1 is: %v\n", elm,
((int64(p-1))))
}
("-------------------------------")
}
}
func main() {
ComputeFieldOrderPower()
}The results are as follows:
value of p is: 7
for element: FieldElement{order: 7, num: 1}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 2}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 3}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 4}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 5}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 6}, its power of p - 1 is: FieldElement{order: 7, num: 1}
-------------------------------
value of p is: 11
for element: FieldElement{order: 11, num: 1}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 2}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 3}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 4}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 5}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 6}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 7}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 8}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 9}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 10}, its power of p - 1 is: FieldElement{order: 11, num: 1}
-------------------------------
value of p is: 17
for element: FieldElement{order: 17, num: 1}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 2}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 3}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 4}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 5}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 6}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 7}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 8}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 9}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 10}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 11}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 12}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 13}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 14}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 15}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 16}, its power of p - 1 is: FieldElement{order: 17, num: 1}
-------------------------------
value of p is: 31
for element: FieldElement{order: 31, num: 1}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 2}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 3}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 4}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 5}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 6}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 7}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 8}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 9}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 10}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 11}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 12}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 13}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 14}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 15}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 16}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 17}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 18}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 19}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 20}, its power of p - 1 is: FieldElement{order: 31, num: 1}
my@MACdeMacBook-Air bitcoin % go run
value of p is: 7
for element: FieldElement{order: 7, num: 1}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 2}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 3}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 4}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 5}, its power of p - 1 is: FieldElement{order: 7, num: 1}
for element: FieldElement{order: 7, num: 6}, its power of p - 1 is: FieldElement{order: 7, num: 1}
-------------------------------
value of p is: 11
for element: FieldElement{order: 11, num: 1}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 2}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 3}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 4}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 5}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 6}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 7}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 8}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 9}, its power of p - 1 is: FieldElement{order: 11, num: 1}
for element: FieldElement{order: 11, num: 10}, its power of p - 1 is: FieldElement{order: 11, num: 1}
-------------------------------
value of p is: 17
for element: FieldElement{order: 17, num: 1}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 2}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 3}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 4}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 5}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 6}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 7}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 8}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 9}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 10}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 11}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 12}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 13}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 14}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 15}, its power of p - 1 is: FieldElement{order: 17, num: 1}
for element: FieldElement{order: 17, num: 16}, its power of p - 1 is: FieldElement{order: 17, num: 1}
-------------------------------
value of p is: 19
for element: FieldElement{order: 19, num: 1}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 2}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 3}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 4}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 5}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 6}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 7}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 8}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 9}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 10}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 11}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 12}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 13}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 14}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 15}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 16}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 17}, its power of p - 1 is: FieldElement{order: 19, num: 1}
for element: FieldElement{order: 19, num: 18}, its power of p - 1 is: FieldElement{order: 19, num: 1}
-------------------------------
value of p is: 31
for element: FieldElement{order: 31, num: 1}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 2}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 3}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 4}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 5}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 6}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 7}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 8}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 9}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 10}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 11}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 12}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 13}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 14}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 15}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 16}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 17}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 18}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 19}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 20}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 21}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 22}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 23}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 24}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 25}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 26}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 27}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 28}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 29}, its power of p - 1 is: FieldElement{order: 31, num: 1}
for element: FieldElement{order: 31, num: 30}, its power of p - 1 is: FieldElement{order: 31, num: 1}
-------------------------------
You can see that all elements in the set are 1, regardless of the order of the fields, which means that for any elements k and order p in any finite field, we will have:
k ^(p-1) % p == 1
This is an important conclusion, which we will use in subsequent videos to drive our encryption algorithm.
The hardest operation on finite domain elements is division, we have multiplication operations, for elements 3 and 7 in the field (in order 19), their product is (3 * 7) % 19 = 2. Now given two field elements 2 and 7, how do we get 7? We define a division operation, which is the inverse operation of multiplication, i.e. 2/7 = 3, which is quite intuitive. Here we need to make sure that the denominator is not 0.
Remember in a finite definition, if a is in the field, then there is still a b in the field, such that a * b = 1. For 3 7 = 2 (note the multiplication that represents the modulus order), if we can find b so that b * 7 = 1, then we will have 3 * 7 * b = 2 * b => 3 * (7 * b) = 2 * b => 3 = 2 * b, which means 2 / 7 is the result of 2 times b, b. That is, if we want to do division a / b, we can find the multiplication inverse of b, called it c, and multiply c with the modulus order.
Now the question is, how do we find the multiplication inverse element of b? Remember our above questions? b ^ (p - 1) % p = 1 => b * b ^(p-2) % p = 1 => The multiplication inverse of b is b ^ (p-2).
If you can't determine why for a given element b is in a field and b^(p-1) % p = 1, we have a small code snippet to get the result, we need to make it mathematically solid, and then we have its proof, the conclusion b^(p-1) % p = 1 is called Fermat's small theorem:
For any field element k (k!=0) and order p, we have {1, 2, 3 …, p-1} <=> {k 1 % p, …, k (p-1) %p} =>
[1 2 3… (p-1)] % p == (k1) (k2) … (k* (p-1)) % p = k^(p-1) * [1 2 … p-1] % p, both sides eliminate [12…p-1] We get 1 % p == k ^(p-1) % p => 1 == k^(p-1)%p
Now let's see how to implement division operations using code:
func (f *FieldElement) Multiply(other *FieldElement) *FieldElement {
(other)
// Multiply by modulo order var op
mul := (, )
return NewFieldElement(, (mul, ))
}Because b ^ (p - 1) % p = 1, when we calculate the T power of field element k, we can optimize to first get t = T % (p-1), and then calculate k^ (t) % p, here is the code:
func (f *FieldElement) Power(power *) *FieldElement {
/*
k^ (p-1) % p = 1, we can calculate t = power % (p-1)
Then k^ power %p == k^t %p
*/
var op
t := (power, (, (int64(1))))
powerRes := (, t, nil)
modRes := (powerRes, )
return NewFieldElement(, modRes)
}Now we can check our code in:
package main
import (
ecc "elliptic_curve"
"fmt"
"math/big"
"math/rand"
)
func main() {
f2 := ((int64(19)), (int64(2)))
f7 := ((int64(19)), (int64(7)))
("field element 2 / 7 with order 19 is %v\n", (f7))
f46 := ((57), (46))
("field element 46 * 46 with order 57: %v\n", (f46))
("field element 46 ^ (58) is %v\n", ((int64(58))))
}Run the above code and we get the following results:
field element 2 / 7 with order 19 is FieldElement{order: 19, num: 3}
field element 46 * 46 with order 57: FieldElement{order: 57, num: 7}
field element 46 ^ (58) is FieldElement{order: 57, num: 7}
This is exactly what we expect, and this is the implementation of the field element.
This is the article about golang implementing the Bitcoin kernel: processing astronomical numbers in the elliptic curve. For more related content of golang Bitcoin kernel, please search for my previous articles or continue browsing the related articles below. I hope everyone will support me in the future!