Cryptography Reference
In-Depth Information
c e (mod n ) .If m
c e (mod n ) , then accept the signature as valid, since
ver k ( m,c )=1 . Otherwise, reject the signature, since ver k ( m,c )=0 .
4.34. n = 17438441, φ ( n ) = 17430084, d = 15845531, m = 210314, and
c = 2673099.
4.35. n = 29778839, φ ( n ) = 29767920, d = 17234059, m = 186677, and
c = 17284872.
4.36. n = 42486991, φ ( n ) = 42473952, d = 16989581, m = 249917, and
c = 14191108.
4.37. n = 42486991, φ ( n ) = 42473952, d = 16989581, m = 249917, and
c = 14191109.
In Exercises 4.38-4.41, use the description of the DSA given on pages 183
and 184, applied to the given parameters in each case, to verify that Bob
should accept Alice's digital signature. For simplicity we use very small
parameters, as with the above applications of other algorithms, and in this
case we assume that h ( m )= m , to further simplify the calculations. Fur-
thermore, the primes p and q are not selected with the values suggested in
the description of DSS, rather are artificially small for pedagogical pur-
poses.
4.38. p = 1549, q = 43, α = 104, β = 252, m = 21, γ = 29, σ =7.
4.39. p = 2699, q = 71, α = 896, β = 1850, m = 21, γ = 11, σ = 33.
4.40. p = 3359, q = 73, α = 2451, β = 1185, m = 45, γ = 43, σ = 48.
4.39. p = 9439, q = 13, α = 4139, β = 2471, m =4, γ =5, σ =8.
In Exercises 4.42-4.45, employ the Elgamal cryptosystem described on pages
185 and 186 to recover the plaintext m from the ciphertext c via the pa-
rameters given by the prime p and Bob's private key a in each case.
4.42. p = 2099, a = 17, c =( α b ,mα ab ) = (1700 , 304).
4.43. p = 3313, a =7, c =( α b ,mα ab ) = (1697 , 770).
4.44. p = 4657, a = 19, c =( α b ,mα ab ) = (1640 , 4556).
4.43. p = 7177, a = 35, c =( α b ,mα ab ) = (1416 , 7104).
Exercises 4.44-4.47 pertain to the ElGamal signature scheme delineated on
pages 187 and 188. For the given parameters, determine if Bob should
accept the signature as valid.
4.44. for p = 463, α =3, y = 454, β = 243, and γ = 153, so Alice sends,
m = 96 and sig k ( m,r )=( β,γ ) = (243 , 153).
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