A key is a string of hexadecimal characters from 0 to FFFF (decimal 65535).
When used in OpenVPN, the key determines what packets are allowed to pass and are allowed to encrypt.
The keydef file contains one or more keywords. OpenVPN can examine each key to see if it matches one of these keywords. If a key does not match, it is ignored. If a key matches multiple keywords, OpenVPN will ignore the first that it encounters.
The keydef file has one line per keyword. OpenVPN searches for each keyword until it either finds a match or ignores it.
Each line is a single keyword-value pair. The keyword is a case-insensitive string of alphanumeric characters, separated by spaces. The value is an ASCII string of hexadecimal numbers or a case-insensitive string of alpha-numeric characters.
The hexadecimal numbers in the value can be preceded by a + or – sign. In other words, -1F or +A would be valid values for the key in the keydef file. A plus or minus sign indicates that the hexadecimal string is optional.
encrypted Keywords set any packets sent through the VPN tunnel to be encrypted.
The following table lists the available keywords in the keydef file.
keytype A hexadecimal string in the range of 0-FFFF which specifies the encryption type to use for the given key. See the section on encryption types for more information on the types of encryption available.
If no keytype is specified, it is assumed to be an OpenVPN default. The default keytype is listed here:
Default Keytype Type Default Key Type Description
“128” OpenVPN Encryption Key “128” Authenticating Authenticating Key – Key you use for authentication.
“4096” RSA Encryption Key “4096” Advanced Encryption Advanced Encryption Key – Key used for strong encryption.
“5632” RSA Encryption Key “5632” Advanced Encryption Advanced Encryption Key – Key used for advanced encryption.
“65536” OpenVPN Encryption Key “65536” Authenticating Authenticating Key – Key you use for authentication.
“65536” RSA Encryption Key “65536” Advanced Encryption Advanced Encryption Key – Key used for strong encryption.
“8192” RSA Encryption Key “8192” Advanced 384a16bd22
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Show that if the prime numbers are infinite then the natural numbers are also infinite
I am asked to show that if the prime numbers are infinite, then the natural numbers are also infinite.
We know that $\phi$(N) is the number of the primes less than N and we know that $\phi$(N) = O(N/log(N)). We know that all primes $>2$ are of the form $6n+1$ or $6n+5$.
We also know that if a prime $p$ divides N, then $N = 6p+1$ or $N = 6p+5$.
If $p \gt 2$, then if $p | N$ then $6p+5 \lt N$. Therefore if $p \gt 2$ and $N \ge 7$, then $p | N$.
Therefore, if $p \gt 2$, $N \ge 7$, then $6p+1 | N$ (since $6p+5 | N$).
Then, we have $p|(6p+1)$, which implies that $p \mid 6$ since $6p+1$ is not a prime.
Hence, $N=6p+1$ or $N=6p+5$ and the primes $2,3,5,7,11,13,17,19,23$ are