# Need expert to take my “cryptography”

Problem 1 [10 points]
1. 푓1(푛) = 푓 (푛) · 푛

, for some constant 푐 > 0 and some negligible function 푓 (푛)
2. 푓2(푛) = 푛
−log log 푛
Problem 2 [15 points]
Recall that two ensembles {푋푛 }, {푌푛 } where 푋푛 , 푌푛 ∈ {0, 1}
푝표푙 푦(푛) are computationally indistinguishable if for any efficient adversary 풜, there exists a negligible function 휖(푛) such that
|Pr[풜(1

, 푥) = 1; 푥 ← 푋푛] − Pr[풜(1

, 푦) = 1; 푦 ← 푌푛]| ≤ 휖(푛)
Let 푓 : {0, 1}
∗ → {0, 1}
∗ be a polynomial-time computable function. Show that if {푋푛 }, {푌푛 } are
computationally indistinguishable, then { 푓 (푋푛)}, { 푓 (푌푛)} are also computationally indistinguishable.
[Hint: Use reductions.]
Problem 3 [15 points]
If {푋푛 }, {푌푛 } are computationally indistinguishable, and {푌푛 }, {푍푛 } are computationally indistinguishable, show that {푋푛 }, {푍푛 } are computationally indistinguishable.
Problem 4 [15 points]
Give examples of non-negligible functions 푓1(푛) and 푓2(푛) such that 푓1(푛) · 푓2(푛) is actually a
negligible function.
Problem 5 [25 points]
Let 퐺 : {0, 1}
푛 → {0, 1}
3푛 be a secure pseudorandom generator, prove that the following pseudorandom generators are secure.
푠, 푟 ∈ {0, 1}
푛 below are perfectly random strings. You can think of them as the seeds of the
PRG.
1. 퐺1(푠) : {0, 1}
푛 → {0, 1}
2푛
, 퐺1(푠) = 퐺(푠)[2푛]
. That is, 퐺1(푠) is obtained by dropping the last
푛 bits of 퐺(푠).
2. 퐺2(푟, 푠) : {0, 1}
2푛 → {0, 1}
6푛
, 퐺2(푟, 푠) = (퐺(푟), 퐺(푠)).
Problem 6 [20 points]
1. Is the following function 푓 : {0, 1}
푛 → {0, 1}
푛 a one-way function? ∀푥 ∈ {0, 1}

,
푓 (푥) = 푥 + 10 (mod 2푛
)1.
1Here 푥 ∈ {0, 1}

is interpreted as a number between 0 and 2
푛 − 1. For instance, 10 ≡ (0 . . . 001010)2
2
2. Suppose 푓 : {0, 1}
푛 → {0, 1}

is a one-way function. Using 푓 , can you construct another
function 푓
0
: {0, 1}
푛 → {0, 1}
푛+1
such that 푓
0
is also a one-way function but not a secure
pseudorandom generator.

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