feat(tricks): ordered reduced residue system

Signed-off-by: Pouyan Heyratpour <[email protected]>

tricks/mathx/arithmetic.go

@@ -159,21 +159,41 @@ func PrimitiveRootsWithRrs(n int) map[int][]int {
 
 	prr := make(map[int][]int)
 	for _, cp := range cpp {
-		mm := make([]int, 0, len(cpp)+1)
-		m := 1
-		for i := 1; i < n; i++ {
-			m = (m * cp) % n
-			mm = append(mm, m)
-
-			if m == 1 {
-				break
-			}
-		}
-
+		mm := orderedReducedResidueSystem(n, cp, make([]int, 0, len(cpp)+1))
 		if len(mm) == len(cpp)+1 {
 			prr[cp] = mm[:len(mm):len(mm)]
 		}
 	}
 
 	return prr
 }
+
+// OrderedReducedResidueSystem returns reduced residue system modulo n
+// ordered by powers of g. If g is not a primitive root of n it returns nil
+func OrderedReducedResidueSystem(n, g int) []int {
+	if !IsPrimitiveRoot(n, g) {
+		return nil
+	}
+
+	return orderedReducedResidueSystem(n, g, make([]int, 0, EulerTotient(n)))
+}
+
+// orderedReducedResidueSystem returns reduced residue system modulo n
+// // ordered by powers of g which have to be a primitive root of n
+func orderedReducedResidueSystem(n, pr int, rrs []int) []int {
+	if rrs == nil {
+		rrs = make([]int, 0, 10)
+	}
+
+	m := 1
+	for i := 1; i < n; i++ {
+		m = (m * pr) % n
+		rrs = append(rrs, m)
+
+		if m == 1 {
+			break
+		}
+	}
+
+	return rrs[:len(rrs):len(rrs)]
+}

tricks/mathx/arithmetic_test.go

@@ -310,3 +310,14 @@ func TestPrimitiveRootsWithRrs(t *testing.T) {
 		}
 	}
 }
+
+func TestOrderedReducedResidueSystem(t *testing.T) {
+	assert.Equal(t, []int{2, 4, 3, 1}, mathx.OrderedReducedResidueSystem(5, 2))
+	assert.Equal(t, []int{3, 4, 2, 1}, mathx.OrderedReducedResidueSystem(5, 3))
+	assert.Equal(t, []int{
+		63, 10, 95, 100, 94, 37, 84, 49, 91, 62, 54, 85, 5, 101, 50, 47, 72, 42, 78, 99, 31, 27, 96, 56, 104, 25, 77, 36,
+		21, 39, 103, 69, 67, 48, 28, 52, 66, 92, 18, 64, 73, 105, 88, 87, 24, 14, 26, 33, 46, 9, 32, 90, 106, 44, 97, 12,
+		7, 13, 70, 23, 58, 16, 45, 53, 22, 102, 6, 57, 60, 35, 65, 29, 8, 76, 80, 11, 51, 3, 82, 30, 71, 86, 68, 4, 38, 40,
+		59, 79, 55, 41, 15, 89, 43, 34, 2, 19, 20, 83, 93, 81, 74, 61, 98, 75, 17, 1,
+	}, mathx.OrderedReducedResidueSystem(107, 63))
+}