Answer:
RSA with 1024 bits is eight times as slow as RSA with 512 bits
Explanation:
For simplicity it is assumed that [tex]k=k-1[/tex] in this problem.
Number of multiplications (and squarings) for one exponentiation with k{bit exponent:
[tex]N_m=1.5k[/tex]
complexity of one multiplication is given as
[tex]t=ck^2[/tex]
Complexity for the total exponentiation is given as
[tex]t_e=Nm*t\\t_e=1.5k*ck^2\\t_e=1.5ck^3[/tex]
Now assuming k1=512,k2=1024 and taking the ratios which are given as
[tex]\dfrac{t_e_{1024}}{t_e_{512}}=\dfrac{1.5ck_2^3}{1.5ck_1^3}\\k_2=2k_1 \\\dfrac{t_e_{1024}}{t_e_{512}}=\dfrac{1.5c(2k_1)^3}{1.5ck_1^3}\\\dfrac{t_e_{1024}}{t_e_{512}}=\dfrac{(2k_1)^3}{k_1^3}\\\dfrac{t_e_{1024}}{t_e_{512}}=\dfrac{8k_1^3}{k_1^3}\\\dfrac{t_e_{1024}}{t_e_{512}}=8[/tex]
So RSA with 1024 bits is eight times as slow as RSA with 512 bits
