A cryptographically secure pseudorandom number generator (CSPRNG) is a specialized form of pseudorandom number generator (PRNG) engineered to satisfy rigorous security criteria, ensuring its suitability for cryptographic applications. These applications include key generation, initialization vectors, nonces, and salts, where the quality of randomness is paramount.

Requirements

CSPRNGs must fulfill two primary requirements:

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   Statistical Randomness: The output should pass all statistical tests for randomness, including the next-bit test. This test asserts that, given the first k bits of a sequence, there is no polynomial-time algorithm that can predict the (k+1)th bit with a probability significantly greater than 50%. Andrew Yao demonstrated in 1982 that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness. (en.wikipedia.org)

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   Resilience to State Compromise: If any part of the generator's internal state is exposed, it should be infeasible to reconstruct previous outputs (backtracking resistance) or predict future outputs (forward secrecy). This ensures that past and future random numbers remain secure even if the current state is compromised. (en.wikipedia.org)

Design Approaches

CSPRNGs are typically designed using one of the following approaches:

Based on Cryptographic Primitives

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  Block Ciphers in Counter Mode: Secure block ciphers, such as the Advanced Encryption Standard (AES), can be operated in counter mode (CTR) to produce pseudorandom output. In this mode, the cipher encrypts successive values of a counter, generating a stream of pseudorandom bits. (en.wikipedia.org)

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  Cryptographic Hash Functions: Hash functions like SHA-256 can be utilized to generate pseudorandom numbers by repeatedly hashing a seed value and its subsequent outputs. (en.wikipedia.org)

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  HMAC-based Generators: Hash-based Message Authentication Code (HMAC) constructions can serve as the foundation for CSPRNGs, combining a secret key with a hash function to produce secure random numbers. (en.wikipedia.org)

Based on Hard Mathematical Problems

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  Blum Blum Shub Algorithm: This generator relies on the difficulty of the quadratic residuosity problem, which is as hard as integer factorization. While theoretically secure, it is computationally intensive and thus less practical for general use. (en.wikipedia.org)

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  Blum-Micali Algorithm: Based on the hardness of the discrete logarithm problem, this generator offers a high level of security but suffers from inefficiency in practical applications. (en.wikipedia.org)

Practical Implementations

Several practical CSPRNG implementations are widely used:

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  /dev/random and /dev/urandom: Unix-like operating systems provide these special files as interfaces to the kernel's random number generator, which collects environmental noise to produce random numbers. (en.wikipedia.org)

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  Yarrow and Fortuna: These generators, developed by Bruce Schneier and colleagues, are designed to be secure and efficient. Yarrow was used in macOS until it was replaced by Fortuna, which offers improvements in security and performance. (en.wikipedia.org)

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  CryptGenRandom: Part of Microsoft's CryptoAPI, this function provides applications with cryptographically secure random numbers. (en.wikipedia.org)

Security Considerations

The security of a CSPRNG is contingent upon the secrecy of its internal state and the quality of its entropy sources. Notable security incidents include:

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  Dual_EC_DRBG Controversy: The Dual Elliptic Curve Deterministic Random Bit Generator (Dual_EC_DRBG), standardized by NIST, was found to contain a potential backdoor inserted by the NSA, allowing for the prediction of its output under certain conditions. (en.wikipedia.org)

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  DUHK Attack: This attack exploited devices using hardcoded seeds in the ANSI X9.31 RNG algorithm, enabling attackers to recover encryption keys and decrypt communications. (en.wikipedia.org)

Testing and Validation

To ensure the reliability of CSPRNGs, various statistical test suites are employed:

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  NIST SP 800-22: A suite of tests developed by the National Institute of Standards and Technology to assess the randomness of binary sequences generated by cryptographic random number generators. (en.wikipedia.org)

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  Diehard Tests: A collection of statistical tests for measuring the quality of random number generators. (en.wikipedia.org)

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  TestU01: A software library offering a comprehensive suite of tests for random number generators. (en.wikipedia.org)

Applications

CSPRNGs are integral to various cryptographic operations, including:

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  Key Generation: Producing secure cryptographic keys for encryption and decryption processes.

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  Nonces and Initialization Vectors: Generating unique values to ensure that encrypted messages are distinct, even when the same plaintext is encrypted multiple times.

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  Salts: Creating random values added to passwords before hashing to prevent precomputed attacks such as rainbow table attacks.

The robustness and security of CSPRNGs are critical to maintaining the confidentiality and integrity of cryptographic systems.