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That equality it's used in cryptography to speed up calculations in the ring $GF(2^k)[x]/(x^4+1)$. $GF(2^8)$, $GF(2^8)[x]$ and $GF(2^8)[x]/(x^4+1)$ are essential algebraic structures used in Advanced Encryption Standard (AES). The elements of $GF(2^8)$ are represented in computers as 8-bit words, whereas the elements of $GF(2^8)[x]/(x^4+1)$ as 32-bit words. ...


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Koblitz, Algebraic Aspects of Cryptography, by the co-inventor of Elliptic Curve Cryptography is a good choice.


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Braid-based cryptography uses braid groups to produce cryptographic schemes starting with the Anshel–Anshel–Goldfeld key exchange at around year 2000. Braid groups arise as the fundamental groups of topological spaces called the configuration spaces, and they also arise as the mapping class groups of punctured disks. However, braid-based cryptographic ...


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As mentioned in the comments the key leads to an identity mapping, so it's a weak key. The concern about $p,q$ being roughly the same size in bits is misplaced for a real world system. Let $N$ have size 4096 bits, as is recommended today. If $p,q$ are chosen randomly from the set of integers of size 2048 and tested for primality, they are chosen from a set ...


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You can calculate mod on any calculator with this algorithm: Example 27 / 6 = 4.5 27 / 6 = 4.5 4.5 - 4 = 0.5 Get the fractional part 6 * 0.5 = 3 Mul the denominator and fraction 27 mod 6 = 3 Works every time.



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