ASymmetric key Encryption

What is asymmetric key encrption

Types of Asymmetric key Encryption

1. DH(Diffe-Helman)

Once both parties derive the shared secret key(ssk), DH Pvt, public keys are dropped(to prevent derivation of ssk by hacker if DH keys are compromised in future)

Understand Deeply //DH Keys sizes Group | No of Bits | Combinations | Strong -------|------------|----------------|---------- 1 | 768 | 2^768 | 2 | 1024 | | 5 | 1536 | | Strongest Prime number=p, Generator=g are sent publicly p(Always used in mod), g(everything is raised to power of g) Modulus Operation: 19mod7 = 19%7 = 5 //Algorithm Host-A Host B ----------Prime=p=7, Generator=g=5-------> Choose Private-key Choose private-key a=4 b=6 We can generate pvt key using: ssh-keygen -t ed25519 -C "your_email@example.com" Dervive Public-Key: gPow(a)modp gPow(b)modp 5pow(4)mod7 5pow(6)mod7 625mod7 = 2 15625mod7=1 -------My Public Key = 2-----------------> <------My Public Key = 1------------------ Derive Shared Secret Key(ssk) ssk=(Public Key)pow(a) mod(p) ssk=(Public Key)pow(b) mod(p) 1pow(4)mod7 = 1 2pow(6)mod(7) = 64mod7=1 A,B have ssk=1, which is stored secretly to derive AES symmetric key ssk is Never sent on network This completes DH Key exchange, now DH Pvt, public keys are dropped (to prevent derivation of ssk by hacker if DH keys are compromised in future)

Key size in DH

key size = Size of Prime number. Small number is used for easy calculation, in practice large sizes are choosen
1024, 2048(standard minumum), 3072(strong), 4096(max) bits

2. RSA (Ronald Rivest, Adi Shamir, Len Adleman)

A. Pre-calculate Public, pvt key
  1. Choose 2 numbers p(1024 bit),q(1024 bit). {p=3,q=11}
  2. Find n = p`*`q, z =(p-1)(q-1) {n=33, z=20}
  3. Choose a number d relatively prime to z. {d=7} //7 and 20 has no common factor
  4. Find e. So that e × d = 1 mod(z)
    e x 7 = 1 mod(20)
    e = mod(20)/7 = 3 (approx)

Public Key = (e,n). Private Key = (d,n)

B. Divide Plain-text into blocks input=10101111. {block1=1010 block2=1111}
C. Encrypt: cipher text© = Block-of-plain-texte (mod n)
  C = P^3 mod(33)

D. Decrypt: Plain text(P) = Cd (mod n)
          P = C^7 mod(33)

Public Key (n,  e)        Private Key (n, d) or 5-value
                Host-A                                                  Host-B
                        -----Prime-1=53, Prime-2=59------>    //In real calculations P & Q are large numbers (64 bytes)
                                                                   Modulus(n)=P*Q=64x64=128 bytes=1024 bit
                                                                   Phy(n)=(P-1)(Q-1)=3016
                                                                   Exponent(e)=coprime of Phy
            Public-key calculated                    Public-key= n&e
                                                                   Pvt key=2 (Phy(n) + 1)/e

                                                                    encryption of data: data pow(e)mod(n)
                        <---cipher-text------             89 pow(3)mod(3127)    //if data=89
        Decryption of data
        (cipher Text)pow(Pvt Key) mod(n)
           (1394) pow(2011) mod(3127) = 89
      

3. Elliptic Curve

Public, Pvt keys are calculated using elliptic curve(y² = x³ + ax + b) over Galois field.
Galois field: Field containing finite number of elements rather than real numbers.
Advantage of ECC? Smaller keys in ECC provides equivalent security to larger non-ECC based algos.
Applications of ECC?
Calculating keys for following: Key agreement, Digital Signature, Pseudo-random generators.
ECCs can be used after combining with Symmetric encryption schemes.

DH vs ECC keys: ECC provides the same cryptographic security as DH with significantly smaller key sizes. ECC achieves identical protection with DH on

3.1 (ECDH) Ecliptic Curve based Diffie Hellman

Allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel.
This shared secret maybe used directly as key or derive another key.
The key, or the derived key, can then be used to encrypt subsequent communications using a symmetric-key cipher.

3.2 secp256k1 ECC Algorithm (Used in Bitcoin/Ethereum/Blockchain)

Private Key: This is just a random number between (1 & (2²⁵⁶ = 10⁷⁷)). But should be generated using good source of randomness(called entropy).
K(Public Key) = k(pvt Key) * G(constant called Generator point). //G is same for all users