SharedConcatenation_DIFG_IS.sagewsOpen in CoCalc
Deep-iterates-Interchange-structure-based attack on the Concatenation combiner of two hash functions
n = var('n')
l = var('l')
g1 = var('g1')
g2 = var('g2')
r = var('r')

restriction_g1 = (g1 >= n/2, g1 >= n - l)
restriction_r = ( 2*r <=l)
restrictions = (g1 >= n/2, g1 >= n - l, 2*r <= l)

Phase1 = l
Phase2 = 2 * n - g1 - l
Phase3_1 = g2
Phase3_2 = (3 * g1 - n - 2 * r) + (r + n - g2)
Phase3_3 = n/2 + 2 * r

show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Phase 1: ', Phase1)
show('Phase 2: ', Phase2)
show('Phase 3 term 1: ', Phase3_1.simplify_full())
show('Phase 3 term 2: ', Phase3_2.simplify_full())
show('Phase 3 term 3: ', Phase3_3.simplify_full())
show('--------------------------------------------------------------------')

g2s = solve([Phase3_1 == Phase3_2], g2)[0]
show('--------------------------------------------------------------------')
show('Balance the first two terms in Phase 3 by setting: ')
show(g2s)
g2s = g2s.rhs()

Phase3_1  = Phase3_1(g2 = g2s)
Phase3_2 = Phase3_2(g2 = g2s)

show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Phase 1: ', Phase1)
show('Phase 2: ', Phase2)
show('Phase 3 term 1: ', Phase3_1.simplify_full())
show('Phase 3 term 2: ', Phase3_2.simplify_full())
show('Phase 3 term 3: ', Phase3_3.simplify_full())

g1s, rs = solve([Phase2 == Phase3_1 , Phase3_1 == Phase3_3], g1, r)[0]
show('--------------------------------------------------------------------')
show('Balance Phase 2, Phase 3 term 1 and Phase 3 term 2 by setting:')
show(g1s)
show(rs)
g1s = g1s.rhs()
rs = rs.rhs()
show('This balance only valid under the restrictions: ')
show(restrictions)
show('Which implies: ')
bound_l = solve([g1s >= n - l, g1s >= n/2, rs <= l/2, l < n], l)[-1]
bound_l_ = (bound_l[0], bound_l[1])
show(bound_l_)

Phase1_case1 = Phase1(g1 = g1s)
Phase2_case1 = Phase2(g1 = g1s)
Phase3_1_case1 = Phase3_1(g1 = g1s, r = rs)
Phase3_2_case1 = Phase3_2(g1 = g1s, r = rs)
Phase3_3_case1 = Phase3_3(g1 = g1s, r = rs)

show('--------------------------------------------------------------------')
show('For the case')
show(bound_l_)
show('Complexity of Phases are (log2): ')
show('Phase 1: ', Phase1_case1)
show('Phase 2: ', Phase2_case1)
show('Phase 3 term 1: ', Phase3_1_case1.simplify_full())
show('Phase 3 term 2: ', Phase3_2_case1.simplify_full())
show('Phase 3 term 3: ', Phase3_3_case1.simplify_full())
show('--------------------------------------------------------------------')

ls = solve([Phase1_case1 == Phase2_case1], l)[0]
show('The optimal complexity is (log2):')
show('Phase1: ', Phase1_case1(l = ls.rhs()).simplify_full())
show('Phase2: ', Phase2_case1(l = ls.rhs()).simplify_full())
show('Phase 3, term 1:', Phase3_1_case1(l = ls.rhs()).simplify_full())
show('Phase 3, term 2:', Phase3_2_case1(l = ls.rhs()).simplify_full())
show('Phase 3, term 3:', Phase3_3_case1(l = ls.rhs()).simplify_full())
show('Obtained for ')
show(ls)
show('--------------------------------------------------------------------')

bound_l_low = (bound_l[0].rhs() <= bound_l[0].lhs())
rs = (r == l/2)
Phase1_case2 = Phase1(r = l / 2)
Phase2_case2 = Phase2(r = l / 2)
Phase3_1_case2 = Phase3_1(r = l / 2)
Phase3_2_case2 = Phase3_2(r = l / 2)
Phase3_3_case2 = Phase3_3(r = l / 2)

show('--------------------------------------------------------------------')
show('For the case')
show(bound_l_low)
show('Set: ', rs)
show('Complexity of Phases are (log2): ')
show('Phase 1: ', Phase1_case2)
show('Phase 2: ', Phase2_case2)
show('Phase 3 term 1: ', Phase3_1_case2.simplify_full())
show('Phase 3 term 2: ', Phase3_2_case2.simplify_full())
show('Phase 3 term 3: ', Phase3_3_case2.simplify_full())

g1s = solve([Phase2_case2 == Phase3_1_case2], g1)[0]
show('--------------------------------------------------------------------')
show('Balance Phase 2 and Phase 3 term 1 by setting:')
show(g1s)
g1s = g1s.rhs()
show('This balance only valid under the restriction: ')
show(restriction_g1)
show('Which implies: ')
bound_l = solve([g1s >= n - l, g1s >= n/2, l < n], l)[-1]
show(bound_l[0])

Phase2_case2 = Phase2_case2(g1 = g1s )
Phase3_1_case2 = Phase3_1_case2(g1 = g1s )
Phase3_2_case2 = Phase3_2_case2(g1 = g1s )
Phase3_3_case2 = Phase3_3_case2(g1 = g1s )

show('--------------------------------------------------------------------')
show('Complexity of Phases are (log2): ')
show('Phase 1: ', Phase1_case2)
show('Phase 2: ', Phase2_case2)
show('Phase 3 term 1: ', Phase3_1_case2.simplify_full())
show('Phase 3 term 2: ', Phase3_2_case2.simplify_full())
show('Phase 3 term 3: ', Phase3_3_case2.simplify_full())
show('--------------------------------------------------------------------')

--------------------------------------------------------------------
Complexity of Phases are (log2):
Phase 1: $\displaystyle l$
Phase 2: $\displaystyle -g_{1} - l + 2 \, n$
Phase 3 term 1: $\displaystyle g_{2}$
Phase 3 term 2: $\displaystyle 3 \, g_{1} - g_{2} - r$
Phase 3 term 3: $\displaystyle \frac{1}{2} \, n + 2 \, r$
--------------------------------------------------------------------
--------------------------------------------------------------------
Balance the first two terms in Phase 3 by setting:
$\displaystyle g_{2} = \frac{3}{2} \, g_{1} - \frac{1}{2} \, r$
--------------------------------------------------------------------
Complexity of Phases are (log2):
Phase 1: $\displaystyle l$
Phase 2: $\displaystyle -g_{1} - l + 2 \, n$
Phase 3 term 1: $\displaystyle \frac{3}{2} \, g_{1} - \frac{1}{2} \, r$
Phase 3 term 2: $\displaystyle \frac{3}{2} \, g_{1} - \frac{1}{2} \, r$
Phase 3 term 3: $\displaystyle \frac{1}{2} \, n + 2 \, r$
--------------------------------------------------------------------
Balance Phase 2, Phase 3 term 1 and Phase 3 term 2 by setting:
$\displaystyle g_{1} = -\frac{5}{11} \, l + \frac{19}{22} \, n$
$\displaystyle r = -\frac{3}{11} \, l + \frac{7}{22} \, n$
This balance only valid under the restrictions:
($\displaystyle g_{1} \geq \frac{1}{2} \, n$, $\displaystyle g_{1} \geq -l + n$, $\displaystyle 2 \, r \leq l$)
Which implies:
($\displaystyle \frac{7}{17} \, n < l$, $\displaystyle l < \frac{4}{5} \, n$)
--------------------------------------------------------------------
For the case
($\displaystyle \frac{7}{17} \, n < l$, $\displaystyle l < \frac{4}{5} \, n$)
Complexity of Phases are (log2):
Phase 1: $\displaystyle l$
Phase 2: $\displaystyle -\frac{6}{11} \, l + \frac{25}{22} \, n$
Phase 3 term 1: $\displaystyle -\frac{6}{11} \, l + \frac{25}{22} \, n$
Phase 3 term 2: $\displaystyle -\frac{6}{11} \, l + \frac{25}{22} \, n$
Phase 3 term 3: $\displaystyle -\frac{6}{11} \, l + \frac{25}{22} \, n$
--------------------------------------------------------------------
The optimal complexity is (log2):
Phase1: $\displaystyle \frac{25}{34} \, n$
Phase2: $\displaystyle \frac{25}{34} \, n$
Phase 3, term 1: $\displaystyle \frac{25}{34} \, n$
Phase 3, term 2: $\displaystyle \frac{25}{34} \, n$
Phase 3, term 3: $\displaystyle \frac{25}{34} \, n$
Obtained for
$\displaystyle l = \frac{25}{34} \, n$
--------------------------------------------------------------------
--------------------------------------------------------------------
For the case
$\displaystyle l \leq \frac{7}{17} \, n$
Set: $\displaystyle r = \frac{1}{2} \, l$
Complexity of Phases are (log2):
Phase 1: $\displaystyle l$
Phase 2: $\displaystyle -g_{1} - l + 2 \, n$
Phase 3 term 1: $\displaystyle \frac{3}{2} \, g_{1} - \frac{1}{4} \, l$
Phase 3 term 2: $\displaystyle \frac{3}{2} \, g_{1} - \frac{1}{4} \, l$
Phase 3 term 3: $\displaystyle l + \frac{1}{2} \, n$
--------------------------------------------------------------------
Balance Phase 2 and Phase 3 term 1 by setting:
$\displaystyle g_{1} = -\frac{3}{10} \, l + \frac{4}{5} \, n$
This balance only valid under the restriction:
($\displaystyle g_{1} \geq \frac{1}{2} \, n$, $\displaystyle g_{1} \geq -l + n$)
Which implies:
$\displaystyle \frac{2}{7} \, n < l$
--------------------------------------------------------------------
Complexity of Phases are (log2):
Phase 1: $\displaystyle l$
Phase 2: $\displaystyle -\frac{7}{10} \, l + \frac{6}{5} \, n$
Phase 3 term 1: $\displaystyle -\frac{7}{10} \, l + \frac{6}{5} \, n$
Phase 3 term 2: $\displaystyle -\frac{7}{10} \, l + \frac{6}{5} \, n$
Phase 3 term 3: $\displaystyle l + \frac{1}{2} \, n$
--------------------------------------------------------------------