n = var('n')
l = var('l')
lp = var('lp')
k = var('k')
kappa = var('kappa')
s = var('s')
t = var('t')
d = var('d')
r = var('r')
Step1 = n / 2
Step2 = t
Step3 = n / 2
Step4 = lp
Step5 = n - l
Step6 = r + n - t
show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Step 1: ', Step1.simplify_full())
show('Step 2: ', Step2.simplify_full())
show('Step 3: ', Step3.simplify_full())
show('Step 4: ', Step4.simplify_full())
show('Step 5: ', Step5.simplify_full())
show('Step 6: ', Step6.simplify_full())
show('--------------------------------------------------------------------')
pairs1 = 2 * r
pairs2 = k * n - (k + 1) * d
pairs_eq = (pairs1 == pairs2)
show('--------------------------------------------------------------------')
show('When the message length is limited to be no more than (log2): ', n / 2)
show('the success of the attack requires: ', pairs_eq)
rs = solve([pairs_eq], r)[0]
show('which implies: ', rs)
rs = rs.rhs()
Step6 = Step6(r = rs)
show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Step 1: ', Step1.simplify_full())
show('Step 2: ', Step2.simplify_full())
show('Step 3: ', Step3.simplify_full())
show('Step 4: ', Step4.simplify_full())
show('Step 5: ', Step5.simplify_full())
show('Step 6: ', Step6.simplify_full())
ts = solve([Step2 == Step6], t)[0]
show('--------------------------------------------------------------------')
show('Balance the Step 2 and 6 by setting: ')
show(ts)
ts = ts.rhs()
Step2 = Step2(t = ts)
Step6 = Step6(t = ts)
show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Step 1: ', Step1.simplify_full())
show('Step 2: ', Step2.simplify_full())
show('Step 3: ', Step3.simplify_full())
show('Step 4: ', Step4.simplify_full())
show('Step 5: ', Step5.simplify_full())
show('Step 6: ', Step6.simplify_full())
show('--------------------------------------------------------------------')
show('Optimize the complexity by setting: ')
show(d == n/2)
show(lp == n/2)
Step2 = Step2(d = n/2, lp = n/2)
Step4 = Step4(lp = n/2)
Step6 = Step6(d = n/2, lp = n/2)
show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Step 1: ', Step1.simplify_full())
show('Step 2: ', Step2.simplify_full())
show('Step 3: ', Step3.simplify_full())
show('Step 4: ', Step4.simplify_full())
show('Step 5: ', Step5.simplify_full())
show('Step 6: ', Step6.simplify_full())
show('--------------------------------------------------------------------')
for i in range(2, 6):
show(' ------------------------------ ', k == i, ' ------------------------------')
Step2cplx = Step2(k = i)
Step2cplxs = Step2cplx(kappa=log(i, 2)).simplify_full()
show('Complexity is: ', Step2cplxs)
--------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: t Step 3: 21n Step 4: lp Step 5: −l+n Step 6: n+r−t --------------------------------------------------------------------
--------------------------------------------------------------------
When the message length is limited to be no more than (log2): 21n the success of the attack requires: 2r=−d(k+1)+kn which implies: r=−21dk+21kn−21d --------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: t Step 3: 21n Step 4: lp Step 5: −l+n Step 6: −21dk+21(k+2)n−21d−t --------------------------------------------------------------------
Balance the Step 2 and 6 by setting:
t=−41dk+41(k+2)n−41d --------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: −41dk+41(k+2)n−41d Step 3: 21n Step 4: lp Step 5: −l+n Step 6: −41dk+41(k+2)n−41d --------------------------------------------------------------------
Optimize the complexity by setting:
d=21n lp=21n --------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: 81(k+3)n Step 3: 21n Step 4: 21n Step 5: −l+n Step 6: 81(k+3)n --------------------------------------------------------------------
------------------------------ k=2 ------------------------------ Complexity is: 85n ------------------------------ k=3 ------------------------------ Complexity is: 43n ------------------------------ k=4 ------------------------------ Complexity is: 87n ------------------------------ k=5 ------------------------------ Complexity is: n ------------------------------ k=3 ------------------------------ Complexity is: 43n ------------------------------ k=4 ------------------------------ Complexity is: 87n ------------------------------ k=5 ------------------------------ Complexity is: n