Problem 63 We define a digital signature scheme DS kl K S V which has two

# Problem 63 we define a digital signature scheme ds kl

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Problem 6.3. We define a digital signature scheme DS k,l = ( K , S , V ) which has two parameters: an integer k (the RSA modulus size) and an integer l . The message space for the scheme is { 0 , 1 } l . For any l -bit message M let M j denote its j -th bit, for j = 1 , . . . , l . Then the key generation, signing and verifying algorithms are as follows: Algorithm K ( N, e ) , ( N, d ) \$ ← K rsa For b = 0 , 1 do For j = 1 , . . . , l do Y [ b, j ] \$ Z * N EndFor EndFor Return ( N, e, Y ) , ( N, d, Y ) Algorithm S ( N,d,Y ) ( M ) For j = 1 , . . . , l do X [ j ] ( Y [ M j , j ]) d mod N EndFor Return X Algorithm V ( N,e,Y ) ( M, X ) flag 1 For j = 1 , . . . , l do If ( X [ j ]) e mod N 6 = Y [ M j , j ] Then flag 0 EndIf EndFor Return flag Here K rsa is the standard RSA key generation algorithm. The public scheme of our scheme consists of the RSA modulus N , the RSA public exponent e , and a 2 by l array Y each of whose entries is a random point in Z * N . The secret key is the RSA modulus N , the RSA secret exponent d , and the same array Y . The size of N is k bits.
The signature of M is a one-dimensional array X of size l consisting of pre-images under RSA N,e of certain points in the two-dimensional array Y , one per column of Y , the choice of which row being made according to the corresponding bit in the message.