Rename Digital Sign Number with airSlate SignNow

Fax digital signature proof

[Music] today we are about to see another very practical application the elliptic curve digital signature algorithm once again we need our two friends Alice and Bob hi Alice hi hi Bob hi these guys can already exchange encryption keys so now they want to ensure the integrity and authenticity of their communication this requires them to use some digital signature scheme as we already know they agreed to use the elliptic curve design throughout this series and the point four five as their generator we also know that Alice has the private key represented by her secret number three and that she sent Bob the corresponding public key point thirteen twenty two so now it's time for her to sign the message she wants to send to Bob to do it she first calculates some digest values set that represents the message she is willing to sign such value must be of the same bit size as the order of the curve used as the order of the curve is 23 it means five bit value is required the digest value set is done now Alice rolls dice to choose a random number K between 1 and the order of the curve I wrote 19 now she multiplies the generator by this number K look at the point K using the double and add algorithm I calculate the point K as 19 multiply of the generator the resulting point K has coordinates 970 the x coordinate of the point K is the first half of our signature usually labeled R it serves as a reference value used for verifying the signature now Alice calculates the other half of the signature by using modular arithmetic with the order of the curve as the modulus first I calculate the multiplicative inverse of K modulo the order of the curve the multiplicative inverse of 19 modulo 23 is 17 that's because 17 times 19 modulo 23 is 1 now I sum the digest value set with the product of our and my private number a everything modulo 23 10 plus 9 times 3 modulo 23 is 14 the product of the multiplicative inverse of K and this value is s the second half of the signature 17 times 14 modulo 23 is 8 the signature of the message with digest value set equal to 10 is there for the pair of numbers 9 8 when sending the message with digest values that equal them Alice H is the signature pair 9 8 to it Bob receives the signature and before starting the verification he calculates the message digest set and comes to the same value as Alice used the message digest values earth is 10 the next step he has to perform is calculating the value W from the signature number s the value W equals to the multiplicative inverse of number s modulo the order of the curve the number W is 3 because 3 times 8 module 23 is 1 now he calculates two multipliers u1 and u2 the former is the product of the digest values M and the number W modulo the order of the curve and the latter is the product of the first half of the signature R and the number W once again module the order of the curve multiplying 10 times 3 modulo 23 gives me the value 7 which is the coefficient T 1 and multiplying 9 times 3 module 23 gives me 4 which is the coefficient u 2 to finally verify the signature Bob multiplies to generated by the coefficient T 1 then he multiplies alice's public point a by the coefficient drew and some of these two points give him a signature verification point ie both s by using the table net algorithm i compute seven times g which gives me the point seventeen eight then using the same algorithm i compute the point four times a which is ten twelve summing these two points i end up with the point s with coordinates nine seventeen although it was a bit longer than you might have expected it is finally possible to verify the signature the value r is nine and the x coordinate of the point s is nine as well therefore the signature is valid this was our most complex elliptic curve algorithm so far thank you for staying with us for so long and again Thank You Alice and thank you Bob you're welcome no problem thank you for watching and remember to subscribe to our youtube channel and to follow us on Twitter there is more to come [Music]