![](bg8.png)
LOW BOUND Pr[X = τ]/Pr[Y = τ] WHEN τ ∈ T
good
.
When τ ∈ T
good
, all pairs in τ are distinct. In X,
τ mainly records the randomly chosen secret key as
well as the total (q + σ + r) fresh calls to P (or P
±
) ,
that is
Pr[X = τ] =
1
2
k
·
1
2
n
q+σ+r
(2
n
−q −σ −r)!
, (9)
while in Y , τ records the dummy key, q distinct
randomly-chosen L
i
s, σ blocks of random bits and r
calls to P
±
, that is
Pr[Y = τ] =
1
2
k
·
1
2
n
q
·
1
2
nσ
·
1
2
n
r
(2
n
−r)!
. (10)
Obviously, according to (9) (10), when τ ∈ T
good
,
Pr[X = τ] > Pr [Y = τ].
According to Lemma 1, (6), (8),
Adv
Φ
⊕
−ind−bw−cpa
PCBC
(q,r) (11)
≤
qr
2
k
+
σ(σ −1) + qr + 2σ(q + r)
2
n
.
By so far, Theorem 1 is proved.
5 CONCLUSION
In this paper, we study how to provide confidential
protection with a single cryptographic permutation,
and propose 4 practical encryption schemes - PCBC,
POFB, PCFB and PCTR, by adding two simple kinds
of operations - XOR and the finite multiplication.
And we prove that, when using a non-repeated nonce,
these 4 permutation-based encryption schemes are in-
distinguishable from the random function against the
blockwise adaptive chosen plaintext attack in the Φ
⊕
-
related-key setting. Meanwhile they are more effi-
cient than the Sponge construction.
ACKNOWLEDGEMENTS
The work of this paper is supported by the Funda-
mental Theory and Cutting Edge Technology Rese-
arch Program of Institute of Information Engineering,
CAS (Grant No. Y7Z0251103), and the National Na-
tural Science Foundation of China (Grants 61472415,
61732021, 61772519).
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