The Diffie-Hellman key exchange algorithm allows two users to securely exchange cryptographic keys over a public channel without directly sending the keys. The process involves selecting a prime number and a primitive root, with each user calculating their public key based on their private key. Both parties then compute a shared secret key using their private keys and the other's public key, ensuring that the keys remain secure despite being transmitted over a potentially insecure channel.

1. DIFFIE – HELLMAN
KEY EXCHANGE
ALGORITHM
Diffie–Hellman key exchange is a method of securely exchanging
cryptographic keys over a public channel and was one of the first public-
key protocols as originally conceptualized by Ralph Merkle and named after
Whitfield Diffie and Martin Hellman
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2. Diffie Hellman
• Not an encryption algorithm
• Used to share secret keys between 2 users .
• We will use assymmteric encryption (public and private key
concept) to exchange the secret key.
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3. Why this algo?
• When we are sending a key to a receiver, it can be attacked in
between
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4. Algorithm
1. Consider a prime number q.
2. Select Alpha(which is primitive root) such that it must be
primitive root of q and (alpha<q)
‘a’ is a primitive root of q
a1 mod q, a2mod q,…………………….aq-1 mod q gives result
(1,2,3…………………….q-1)
Values should not be repeated & we should have all values in the
o/p set from 1 to q—1.
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5. What is primitive root?
• A is primitive root of q if
• a mod q, a2 mod q, a3 mod q………………………………….a(q-1)
mod q
• Example q=7,we want to check whthr 3 is primitive root or
not.
• We will check value until q value id less than one.(7-1=6)
• If q=7, then we must get all values less than 7(like 1,2,3,4,5
and 6).
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6. We get all values less than 7,so we can 3 is
the primitive root of 7
31 MOD 7 = 3
32 MOD 7 = 2
33 MOD 7 = 6
34 MOD 7 = 4
35 MOD 7 = 5
36 MOD 7 = 1
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7. Check 5 whether is primitive root
or not q=7.
51 MOD 7 = 5
52 MOD 7 = 4
53 MOD 7 = 6
54 MOD 7 = 3
55 MOD 7 = 2
56 MOD 7 = 1
We can 5 is primitive root of q. Primitive root must always be less
than q.
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8. Consideraprimenumberq.
Letsq=7
Sowewillcheckfrom1to7
Checkfor1.
Values are repeated so we will cancel 1 or discard immediately.
11 MOD 7 = 1
12 MOD 7 = 1
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9. Check for 2
• Values repeated ,will discard this one too
21 MOD 7 = 2
22 MOD 7 = 4
23 MOD 7 = 1
24 MOD 7 = 2
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10. We will take Alpha =5
• Primitive root can be 2 or 3 or 4 or more.
• We have to select one that must be less than q.
• Suppose we select alpha = 5.
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11. Global Elements
• Alpha and q are global elements that are known to all
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12. 3rd step of algorithm
Note: x->private key of users(will choose by user)
y-> public key of users.
3. Assume Xa (private key of a) and Xa < q.
Once we know ,private key then public will be calculated i.e(Ya)
Calculate Ya = alphaxamod q A is a public key of A
Example
Key generation of person 1
Assume private key Xa=3 (3<7 ,yes condition satisfy)
Calculating public key Ya =alphaxamod q
We already assume apha=5
So, 5pow 3 mod 7 =125 mod 7 equals to Ya=6
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13. 4rth step of algorithm
Calculate private key of second person
• Assume Xb (private key of b) Xb < q
• Calculate Yb = alphaxbmod q (public key of b)
Key generation of person 2
Let private key Xb = 4
Caluclating public key Yb = alphaxbmod q
Yb=5 pow 4 mod 7
Yb=2
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14. Q=7
Q=5
Global elements
Public Keys
Ya=6
Yb=2
Xa=3
Private Key
Private Key
Xb=4
Person 1 Person 2
Now person 1 and person 2 knows their
private key.
Ya,Yb public key will be known to all.
Person A will generate secret key
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15. Now we will calculate secret
key
• To calculate the secret key ,both the sender and receiver will use
public key.
• Value of Yb is 2
• Value of xa=3
Ka = (2)3 mod 7 we get k=1
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16. Calculate kb for person 2
• Kb = (Ya )xb mod q
• Ya = 6
• Xb=4
• Kb = (6 )4 mod 7
• Kb = 1
Both values of ka and kb is 1. So keys are succesfully exhanged.
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18. • Step 1: Alice and Bob get public numbers P = 23, G = 9
• Step 2: Alice selected a private key a = 4 and Bob
selected a private key b = 3
• Step 3: Alice and Bob compute public values Alice: x
=(9^4 mod 23) = (6561 mod 23) = 6 Bob: y = (9^3 mod
23) = (729 mod 23) = 16
• Step 4: Alice and Bob exchange public numbers
• Step 5: Alice receives public key y =16 and Bob
receives public key x = 6
• Step 6: Alice and Bob compute symmetric keys Alice:
ka = y^a mod p = 65536 mod 23 = 9 Bob: kb = x^b mod
p = 216 mod 23 = 9
• Step 7: 9 is the shared secret.
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20. References
1. https://www.geeksforgeeks.org/implementation-diffie-
hellman-algorithm/
2. https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_ke
y_exchange
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