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) * n / 2 + n / 2 - lp
pairs_eq = (pairs1 == pairs2)
show('--------------------------------------------------------------------')
show('When the message length is not limited: ')
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())
show('--------------------------------------------------------------------')
show('Optimize the complexity by setting: ')
ts, lps = solve([Step2 == Step6, Step2 == Step4], t, lp)[0]
show(ts)
show(lps)
ts = ts.rhs()
lps = lps.rhs()
Step2 = Step2(t = ts, lp = lps)
Step4 = Step4(t = ts, lp = lps)
Step6 = Step6(t = ts, lp = lps)
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, 7):
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 not limited:
the success of the attack requires: 2r=−21(k+1)n+kn−lp+21n which implies: r=41kn−21lp --------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: t Step 3: 21n Step 4: lp Step 5: −l+n Step 6: 41(k+4)n−21lp−t --------------------------------------------------------------------
Optimize the complexity by setting:
t=101(k+4)n lp=101(k+4)n --------------------------------------------------------------------
Complexity of Phases are (log2):
Step 1: 21n Step 2: 101(k+4)n Step 3: 21n Step 4: 101(k+4)n Step 5: −l+n Step 6: 101(k+4)n --------------------------------------------------------------------
------------------------------ k=2 ------------------------------ Complexity is: 53n ------------------------------ k=3 ------------------------------ Complexity is: 107n ------------------------------ k=4 ------------------------------ Complexity is: 54n ------------------------------ k=5 ------------------------------ Complexity is: 109n ------------------------------ k=6 ------------------------------ Complexity is: n