Foundational cryptography concepts built on discrete math, such as modular exponentiation and public key basics.
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- What is modular arithmetic?
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Arithmetic where computations are performed with respect to a modulus, working only with remainders after division by a fixed number.- What does the notation a ≡ b (mod n) mean?
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a and b are congruent modulo n, meaning they have the same remainder when divided by n.- Define modular exponentiation.
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Computing the result of a^b mod n efficiently without first calculating a^b.- Why is modular exponentiation important for cryptography?
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It allows efficient computation of large powers while keeping intermediate results bounded, enabling practical implementation of cryptographic algorithms.- State Fermat's Little Theorem.
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If p is prime and gcd(a, p) = 1, then a^(p-1) ≡ 1 (mod p).- State Euler's Theorem.
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If gcd(a, n) = 1, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.- What is Euler's totient function φ(n)?
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The count of positive integers less than or equal to n that are coprime with n.- What is symmetric key cryptography?
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An encryption method using a single shared secret key for both encryption and decryption.- What is public key cryptography?
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An encryption method using a pair of mathematically related keys: a public key for encryption and a private key for decryption.- What two numbers form an RSA public key?
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The modulus n and the public exponent e.- What two numbers form an RSA private key?
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The modulus n and the private exponent d.- How is the RSA modulus n constructed?
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By multiplying two distinct large prime numbers: n = p * q.- What is the relationship between the RSA exponents e and d?
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e and d are multiplicative inverses modulo φ(n), meaning e * d ≡ 1 (mod φ(n)).- Define the discrete logarithm problem.
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Given g, h, and prime p, find the exponent x such that g^x ≡ h (mod p), believed to be computationally hard.- Why is the discrete logarithm problem important to cryptography?
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Many public key cryptosystems derive their security from the computational difficulty of solving the discrete logarithm problem.
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