Elliptic Curves in Cryptography

U ovom radu smo definirali eliptičke krivulje te smo dali geometrijsku i algebarsku interpretaciju operacije uz koju skup točaka na eliptičkoj krivulji E(K) postaje Abelova grupa. Odredili smo problem diskretnog logaritma za eliptičke krivulje (ECDLP) i komentirali vremensku složenost problema. Naveli smo algoritam ”dupliciraj i zbrajaj”, poznat pod nazivom binarne ljestve, za računanje vrijednosti višekratnika neke točke na eliptičkoj krivulji. Opisali smo upotrebu eliptičkih krivulja u kriptografiji kroz eliptički Diffie-Hellman protokol (ECDH) te eliptički ElGamalov kriptosustav zasnovan na problemu računanja diskretnog logaritma u konačnim grupama gdje prilikom šifriranja poruka postaje otprilike četiri puta dulja. Na kraju smo naveli varijantu ElGamalovog kriptosustava pod nazivom MenezesVanstoneov kriptosustav kod kojeg je šifrirana poruka dva puta dulja.In this work, we defined the elliptic curves and gave the geometric and algebraic interpretation of the operation in which a set of points on an elliptic curve E(K) becomes the Abel group. In addition, we determined the elliptic curve discrete logarithm problem (ECDLP) and commented time complexity of algorithm. We have listed a “double-and-add” algorithm, known as a binary ladder, to compute the multiple value of a point on an elliptic curve. Also, we have described the use of elliptic curves in cryptography through the elliptical Diffie-Hellman protocol (ECDH) and the elliptical ElGamal cryptosystem based on the problem of computing a discrete logarithm in finite groups where during encrypting messages become approximately four times longer. Finally, we have listed a variant of ElGamal’s cryptosystem called the Menezes-Vanstone cryptosystem in which the encrypted message is twice as long