[And four related questions/examples below:]
2) If a random 256-bit number is XOR'ed with its reverse (i.e. the big-endian version XOR'ed with the little-endian version), does this reduce the security properties in terms of bits, if the result is a palindrome, as there are only ${2^{128}}$ such palindromes in the range of ${2^{256}}$?
3) Can an adversary efficiently brute-force search just the palindromes in that range?
Example: an L
-bit random N
${\oplus}$ reverse(N
) = palindromic string L
-bits long.
Example using a random 256-bit binary string:
N=1000001110111000001110001111100111001000111111111111010110111001011010110110010110011001100010110110101101001000001110111111111111001110101001110001101101110101010101110011000111000010101011001010100000110110111000010101101000100111001010110110011011010110
Reverse(N)=0110101101100110110101001110010001011010100001110110110000010101001101010100001110001100111010101010111011011000111001010111001111111111110111000001001011010110110100011001100110100110110101101001110110101111111111110001001110011111000111000001110111000001
N
${\oplus}$ Reverse(N) = 1110100011011110111011000001110110010010011110001001100110101100010111100010011000010101011000011100010110010000110111101000110000110001011110110000100110100011100001101010100001100100011110100011010110011001000111100100100110111000001101110111101100010111
4) Are there any advantages in using a longer key with reduced security than a smaller key with the same security (i.e. could there be an advantage having a 256-bit key with 128-bits of security, compared to having a 128-bit key with a 128-bits of security?)
5) In addition, could there be benefits such as only having to retain half of the palindrome, in order to reconstruct the other half, in terms of compression/notation while avoiding information loss despite any reduced security?
Example using above palindrome where only the leading 128-bits are retained:
11101000110111101110110000011101100100100111100010011001101011000101111000100110000101010110000111000101100100001101111010001100
and where reconstruction of the original 256-bit palindrome is achieved by concatenating the reverse of the 128bits to the right-end of the retained 128 bits as follows: 11101000110111101110110000011101100100100111100010011001101011000101111000100110000101010110000111000101100100001101111010001100
||00110001011110110000100110100011100001101010100001100100011110100011010110011001000111100100100110111000001101110111101100010111