This document discusses the multiplicative group Zn* modulo n. Zn* is the set of integers from 1 to n that are relatively prime to n, meaning they do not share any factors. It provides examples of Zn* for different values of n, such as Z9* = {1, 2, 4, 5, 7, 8} and Z97* containing all integers from 1 to 97 that are relatively prime to 97. The document also discusses cyclic groups and using a generator g to generate the elements of a cyclic group modulo n.

1. ℤₙ* The Multiplicative
group for ℤₙ modulo n
Prof Bill Buchanan OBE, The Cyber Academy
http://asecuritysite.com

2. A puzzle
• For the numbers up to 10, which numbers do not share any factors with
10?

3. Multiplicative group for ℤₙ modulo n
• For the numbers up to 10, which numbers do not share any factors with
10?
• And so we get 1, 3, 7 and 9 [here]. This has the name of the multiplicative
group for ℤₙ modulo n.

4. Multiplicative group for ℤₙ modulo n
• In number theory, ℤₙ is the set of non-negative integers less than n
({0,1,2,3…n-1}).
• ℤₙ* is then a subnet of this which is the multiplicative group for ℤₙ
modulo n.
• The set ℤₙ* is the set of integers between 1 and n that are relatively
prime to n (ie they do not share any factors).
• If n is prime, then ℤₙ* is the values up to (n − 1). ℤ11* =
{1,2,3,4,5,6,7,8,9,10}
• Note that is is also shown as (ℤ/nℤ)ˣ.

5. Examples
If we want g to be generated from multiplicative group for ℤₙ modulo n,
we would define: g ∈ ℤₙ*
For Z9* we get: {1, 2, 4, 5, 7, 8}
For Z91* we get (13x7):
{ 1 2 3 4 5 6 8 9 10 11 12 15 16 17 18 19 20 22 23 24 25 27 29 30 31 32 33
34 36 37 38 40 41 43 44 45 46 47 48 50 51 53 54 55 57 58 59 60 61 62 64
66 67 68 69 71 72 73 74 75 76 79 80 81 82 83 85 86 87 88 89 90}
For Z97* we get [here]:
{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 }
number=1763*1763
import syscount=0
def gcd(a,b):
while b > 0:
a, b = b, a % b
return aprint
"Value is:t",number
print "Multiplicative group for Zn up to 100 is:”
for i in range(1,100):
if (gcd(number,i)==1):
print i,
count=count+1
print "nThe number of values is: ",count

6. Cyclic
Y = Gˣ (mod N)
If N is prime we get a cyclic result. For N=7, we can select 2,3,4,5 or 6, so let’s select
6:
>>> print 6**1 % 7
6
>>> print 6**2 % 7
1
>>> print 6**3 % 7
6
>>> print 6**4 % 7
1
Output: {6,1,6,1,…}

7. Cyclic
Y = Gˣ (mod N)
Sometimes, though, something special happens, and we get a sequence from
0 to N-1. So let’s select N=11, we pick a g value of 2.
Y =2¹ (mod 11) = 2 Y = 2² (mod 11) = 4
Y = 2³ (mod 11) = 8 Y =2⁴ (mod 11) = 5
Y =2⁵ (mod 11) = 10 Y =2⁶ (mod 11) = 9
Y =2⁷ (mod 11) = 7 Y =2⁸ (mod 11) = 3
Y =2⁹ (mod 11) = 6 Y = 2¹⁰ (mod 11) = 1
Y =2¹¹ (mod 11) = 2 (repeat) Y =2¹² (mod 11) = 4 (repeat)
{2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8 …}
Cylic group G of order n with a generator g. (ℤ/nℤ)*. [Link]

8. ℤₙ* The Multiplicative
group for ℤₙ modulo n
Prof Bill Buchanan OBE, The Cyber Academy
http://asecuritysite.com