openssl-prime

Generate a 2048bit prime number and display it in hexadecimal

$ openssl prime -generate -bits 2048 -hex

$ openssl prime [number]

openssl prime [-help] [-hex] [-safe] [-rand file...] [-in file] [-out file] [-generate] [-bits n] [number...]

-help
    Prints a usage message for the command.

-hex
    Specifies that the input number(s) provided are in hexadecimal format.

-safe
    When generating primes, ensures that the prime p is a "safe" prime. A safe prime p is one where (p-1)/2 is also prime.

-rand file...
    Specifies one or more files containing random data used to seed the random number generator, crucial for generating cryptographically secure primes.

-in file
    Specifies an input file from which to read numbers (one per line) for primality checking. If not provided, numbers are taken from the command line arguments.

-out file
    Specifies an output file to write generated prime numbers to. If this option is omitted, primes are printed to standard output (stdout).

-generate
    Instructs the command to generate prime numbers instead of checking existing ones.

-bits n
    Specifies the desired bit length n for the prime numbers to be generated. This option is used in conjunction with -generate.

number...
    One or more numbers to be checked for primality. These can be decimal or hexadecimal (when -hex is used).

The openssl prime command is a versatile utility integrated into the OpenSSL cryptographic toolkit, serving dual purposes: primality testing and prime number generation. When used for testing, it efficiently determines whether one or more input numbers are prime, leveraging a sophisticated probabilistic algorithm like Miller-Rabin. This makes it a practical choice for verifying large numbers encountered in cryptographic contexts.

For generation, the command can produce new prime numbers of a specified bit length, which are crucial components for algorithms such as RSA and Diffie-Hellman key exchange. A particularly useful feature is the ability to generate "safe" primes (where p and (p-1)/2 are both prime), which offer enhanced security properties in certain cryptographic constructions. Its directness and integration with OpenSSL make it an indispensable tool for developers and security professionals working with foundational number theory aspects of cryptography.

The primality test used by openssl prime (typically Miller-Rabin) is probabilistic. While highly reliable for cryptographic purposes, it offers a very low probability of declaring a composite number as prime. Generating very large primes, especially safe primes, can be computationally intensive and time-consuming. For security-sensitive applications, always ensure the random number generator is adequately seeded using the -rand option.

When performing primality checks, if the -in file option is not specified, the numbers to be checked are read directly from the command line arguments. Similarly, when generating prime numbers, if the -out file option is not used, the generated prime(s) will be printed to standard output.

The openssl prime command primarily employs the Miller-Rabin primality test. This is a probabilistic algorithm, meaning it does not offer absolute certainty but rather a very high probability of correctness. For the practical purposes of cryptography, the likelihood of a false positive (a composite number being identified as prime) is astronomically low, making it a reliable choice for key generation and validation.

The prime command is a foundational utility within the OpenSSL project, providing essential functionality for prime number operations. These operations are fundamental to many asymmetric cryptographic algorithms like RSA and Diffie-Hellman. Its development has progressed alongside the general evolution of the OpenSSL toolkit, continuously refined for efficiency and security over the years, reflecting the ongoing needs of cryptographic applications.

openssl(1), openssl-genrsa(1), openssl-gendsa(1), openssl-rand(1)