6.1 Number Theory Background and RSA Cryptosystem

Number theory forms the backbone of modern cryptography. Modular arithmetic, prime numbers, and concepts like GCD enable secure calculations and key generation. These mathematical foundations are crucial for understanding cryptographic algorithms and their security.

RSA, a widely-used public-key cryptosystem, relies on the difficulty of factoring large numbers. Its security is threatened by quantum computing, spurring research into post-quantum alternatives. Understanding RSA's components and vulnerabilities is essential for grasping modern cryptographic challenges.

Number Theory Foundations

Fundamentals of number theory

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• Modular arithmetic performs operations with respect to a modulus enables efficient computation in cryptography
  • Congruence relation $a \equiv b \pmod{m}$ indicates a and b have the same remainder when divided by m
  • Properties include closure, associativity, commutativity, distributivity facilitate secure calculations
  • Applications encompass key generation, encryption, decryption (RSA, ElGamal)
• Prime numbers serve as building blocks for many cryptographic algorithms
  • Fundamental Theorem of Arithmetic states every integer has a unique prime factorization
  • Prime factorization decomposes a number into its prime factors (12 = 2 × 2 × 3)
  • Cryptographic importance stems from difficulty of factoring large primes
• Greatest Common Divisor (GCD) finds largest number dividing two integers
  • Euclidean algorithm efficiently computes GCD through repeated division
  • Extended Euclidean algorithm finds coefficients for Bézout's identity
  • Multiplicative inverses in modular arithmetic derived from extended Euclidean algorithm
• Euler's totient function $\phi(n)$ counts numbers coprime to n
  • For prime p, $\phi(p) = p - 1$
  • For coprime a and b, $\phi(ab) = \phi(a) \times \phi(b)$
• Euler's theorem states $a^{\phi(n)} \equiv 1 \pmod{n}$ for coprime a and n
  • Generalizes Fermat's Little Theorem to composite moduli
  • Crucial for RSA cryptosystem's mathematical foundation

RSA Cryptosystem and Security

RSA cryptosystem and factoring

• RSA algorithm components form basis of widely-used public-key cryptosystem
  • Key generation process:
    1. Select two large prime numbers p and q
    2. Compute modulus n = pq
    3. Calculate $\phi(n) = (p-1)(q-1)$
    4. Choose public exponent e coprime to $\phi(n)$
    5. Determine private exponent d such that $ed \equiv 1 \pmod{\phi(n)}$
  • Encryption transforms plaintext m to ciphertext c: $c \equiv m^e \pmod{n}$
  • Decryption recovers plaintext from ciphertext: $m \equiv c^d \pmod{n}$
• Mathematical basis relies on difficulty of factoring large composite numbers
  • Security stems from computational infeasibility of finding p and q given only n
  • Euler's theorem proves correctness: $m^{ed} \equiv m \pmod{n}$
• Key sizes directly impact security levels
  • Larger keys provide stronger security but increase computational cost
  • Current recommendations: 2048-bit minimum for general use, 3072-bit for long-term security
• Padding schemes enhance security and prevent certain attacks
  • OAEP (Optimal Asymmetric Encryption Padding) adds randomness to encryption
  • PKCS#1 v1.5 widely used but vulnerable to some attacks

Security analysis of RSA

• Classical attacks target various aspects of RSA implementation
  • Brute force key search attempts all possible private keys
  • Factoring algorithms (Quadratic Sieve, General Number Field Sieve) try to find p and q
  • Low private exponent attacks exploit small values of d
  • Common modulus attacks target systems using same modulus for multiple users
• Quantum threats pose significant risk to RSA security
  • Shor's algorithm enables efficient quantum factoring in polynomial time
  • Theoretical ability to break RSA encryption renders current implementations vulnerable
• Post-quantum cryptography develops quantum-resistant alternatives
  • Lattice-based cryptography relies on hardness of certain lattice problems
  • Code-based systems use error-correcting codes for security
  • Multivariate cryptography employs systems of multivariate polynomials
  • Hash-based signatures leverage properties of cryptographic hash functions
• Hybrid cryptosystems combine strengths of asymmetric and symmetric encryption
  • Key encapsulation mechanisms securely transmit symmetric keys using asymmetric encryption
• Ongoing research in RSA security addresses evolving threats
  • Advancements in factoring algorithms continually challenge RSA's security assumptions
  • Implementation security focuses on preventing side-channel attacks (timing, power analysis)