in modern cryptography tie algorithmic complexity solve problem one problem discrete logarithm problem it formulate follow Lets fix finite field element b one need fun x ax b detect x it likely modern mankind solve problem discrete logarithm sufficiently large field size for example field residue modulo prime number prime 1024 2048 bit consider safe however calculation large number place significant load server perform cryptographic operation for reason instead simple module residue field complex field for field fast algorithm use field structure know small field operation properly optimize Developer Nikolai trust generally accept method want invent recently read strange field nimber think great fit purpose the field nimber define set integer 0 22k 1 positive integer k Bitwise exclusive oplus operation addition one way define multiplication operation odot follow property 0 odot odot 0 0 1 odot odot 1 odot b b odot odot b odot c odot b odot c odot b oplus c odot b oplus odot c if 22n integer n 0 b odot b cdot b if 22n integer n 0 odot frac32cdot for example 4 odot 4 6 8 odot 8 4 odot 2 odot 4 odot 2 4 odot 4 odot 2 odot 2 6 odot 3 4 oplus 2 odot 3 4 odot 3 oplus 2 odot 2 oplus 1 4 odot 3 oplus 2 odot 2 oplus 2 odot 1 12 oplus 3 oplus 2 13 32 odot 64 16 odot 2 odot 16 odot 4 16 odot 16 odot 2 odot 4 24 odot 8 16 oplus 8 odot 8 16 odot 8 oplus 8 odot 8 128 oplus 13 141 5 odot 6 4 oplus 1 odot 4 oplus 2 4odot 4 oplus 4 odot 2 oplus 4 odot 1 oplus 1 odot 2 6 oplus 8 oplus 4 oplus 2 8 Formally algorithm describe follow pseudocode multiplya b an 0 p1 bitsa number bit equal p2 bitsb number bit b equal ans an xor multiply_powers_of_21 p1 1 p2 return ansmultiply_powers_of_2a b 1 b 1 return b n maximal value 22n maxa b power 22n power b power return multiplypower 3 2 multiply_powers_of_2a power b power power return multiply_powers_of_2a power b power return multiply_powers_of_2a b power power it show operation form field moreover sense game theory operation that s related problem with help appropriate cache group operation possible calculate product quickly important improve speed cryptoalgorithm more formal definition additional property clarify wikipedia article link the author task hope property list statement solution powering muliplication define way formally aodot k underbracea odot odot cdot odot a_ktexttttimesyou need analyze propose scheme strength for pair number b need find x aodot x b determine do not exist