To compare the entropy of a base64 string and a hex string generated by `openssl rand`, we need to understand how each encoding scheme affects the information content of the data.
Entropy Basics
Entropy, in the context of information theory, measures the amount of uncertainty or randomness in a system. It is typically calculated using the formula $$ \log_2(n) $$, where $$ n $$ is the number of possible states or outcomes. For character-based encoding schemes, the entropy per character can be calculated by considering the number of possible characters in the alphabet used by the scheme.
Base64 Encoding
Base64 encoding uses a 64-character alphabet, which includes uppercase and lowercase letters, digits, and two special characters (usually `+` and `/`). However, in some implementations, padding characters (`=`) are added to ensure the encoded string length is a multiple of four characters. Each character in the base64 alphabet can represent one of 64 possible values.
To calculate the entropy per character in base64, we use the formula:
$$ \text{Entropy per character} = \log_2(64) \approx 6 \text{ bits} $$
However, since base64 encodes three bytes into four characters, the effective entropy per byte is slightly less than the maximum theoretical value due to the encoding efficiency.
Hex Encoding
Hex encoding uses a 16-character alphabet (0-9 and A-F), where each byte is represented by two characters. Therefore, each character in the hex alphabet can represent one of 16 possible values.
To calculate the entropy per character in hex, we use the formula:
$$ \text{Entropy per character} = \log_2(16) = 4 \text{ bits} $$
Since each byte is encoded into two characters, the entropy per byte is effectively $$ 2 \times 4 = 8 $$ bits, which matches the original byte's entropy.
Comparison
When comparing the entropy of base64 and hex strings generated by `openssl rand`, both encoding schemes preserve the original entropy of the random bytes. However, base64 is more compact, encoding three bytes into four characters, while hex encodes one byte into two characters. This means base64 can represent more data with fewer characters, but both schemes maintain the same overall entropy per byte.
In terms of entropy per character, base64 has a higher value (approximately 6 bits per character) compared to hex (4 bits per character). However, when considering the encoding efficiency and the fact that both schemes preserve the original entropy of the data, the choice between them often depends on specific requirements such as compactness, readability, or compatibility with certain protocols.
OpenSSL Rand
`openssl rand` generates cryptographically secure pseudo-random bytes using a CSPRNG. The entropy of these bytes is determined by the quality of the seeding process, which typically involves operating system entropy sources. Both base64 and hex encoding schemes preserve this entropy when converting the bytes to strings.
In summary, while base64 and hex encoding schemes have different entropy values per character, they both preserve the original entropy of the data generated by `openssl rand`. The choice between them should be based on specific application needs rather than entropy considerations alone.
Citations:
[1] https://therootcompany.com/blog/how-many-bits-of-entropy-per-character/
[2] https://jameshfisher.com/2017/03/10/openssl-rand/
[3] https://stackoverflow.com/questions/9020409/is-it-ok-to-remove-the-equal-signs-from-a-base64-string
[4] https://security.stackexchange.com/questions/104861/is-there-an-entropy-difference-between-hashing-an-encoded-string
[5] https://docs.openssl.org/3.0/man1/openssl-rand/
[6] https://jupyterhub.readthedocs.io/_/downloads/en/0.9.2/pdf/
[7] https://stackoverflow.com/questions/3183841/base64-vs-hex-for-sending-binary-content-over-the-internet-in-xml-doc
[8] https://dev.to/mochafreddo/a-deep-dive-into-cryptographic-random-number-generation-from-openssl-to-entropy-16e6
[9] https://manpages.ubuntu.com/manpages/focal/en/man1/rand.1ssl.html
[10] https://docs.openssl.org/1.1.1/man1/rand/