RSA Modulus Generation: n = p × q

Step 1: Generate Two Large Prime Numbers

512-bit

1024-bit

2048-bit

4096-bit

Prime p

32,317

1024-bit prime

5 decimal digits

Hex: 0x7e2d

✓ Primality Test Passed

Prime q

40,009

1024-bit prime

5 decimal digits

Hex: 0x9c49

✓ Primality Test Passed

Step 2: Compute RSA Modulus n = p × q

Prime p

32,317

1024 bits

Prime q

40,009

1024 bits

RSA Modulus n

1,292,886,653

PUBLIC COMPONENT

Modulus Properties and Security Considerations

Current Security Level Assessment

Weak Moderate Strong Very Strong

Mathematical Properties

n is a semi-prime (product of exactly two primes)
n is publicly shared in RSA cryptosystem
Bit length of n ≈ bit length of p + bit length of q
For 2048-bit RSA: p and q are each ~1024 bits
Factoring n = p × q must remain computationally infeasible

n = p × q (semi-prime)
|n| ≈ |p| + |q| (bit length)
φ(n) = (p-1)(q-1)
                    

Security Requirements

p and q must be kept absolutely secret
|p - q| should be large (not too close in value)
Both primes should pass rigorous primality tests
Generated using cryptographically secure randomness
Security relies on Integer Factorization Problem

Critical: Knowledge of p,q ≡ complete key compromise

RSA KEY GENERATION REQUIREMENTS

Prime Selection: Must use cryptographically secure PRNG

Generate random odd number of desired bit length
Test for primality using Miller-Rabin test
Repeat until prime is found
Ensure |p - q| is sufficiently large
                

Modulus Computation: Simple multiplication with security implications

Factorization Resistance: Current best algorithms are sub-exponential

Time Complexity: O(exp((64/9 * ln n)^(1/3) * (ln ln n)^(2/3)))
Key Sizes: 1024-bit deprecated, 2048-bit current standard, 4096-bit future-proof
                