Hindawi Wireless Communications and Mobile Computing Volume 2018, Article ID 7209475, 9 pages https://doi.org/10.1155/2018/7209475
Research Article A Fog Computing Security: 2-Adic Complexity of Balanced Sequences Wang Hui-Juan
and Jiang Yong
The Information Security Department, The First Research Institute of the Ministry of Public Security of P.R.C., Beijing 100084, China Correspondence should be addressed to Wang Hui-Juan;
[email protected] Received 9 January 2018; Accepted 5 March 2018; Published 9 September 2018 Academic Editor: Fuhong Lin Copyright © 2018 Wang Hui-Juan and Jiang Yong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the fog computing environment, the periodic sequence can provide sufficient authentication code and also reduce the power consumption in the verification. But the periodic sequence faces a known full-cycle attack threat in fog computing. This paper studies the 2-adic complexity attack ability of the periodic balance sequence in the fog computing environment. It uses the exponential function as a new approach to study the 2-adic properties of periodic balance sequence and presents that the 2-adic complexity of the periodic balanced sequence is not an attacking threat when used in fog computing.
1. Introduction Fog computing is a decentralized computing architecture compared to cloud computing and is currently used primarily for mobile and portable devices. Due to the current proliferation of IoT devices, the main advantage of fog computing is the ability to quickly provide scalable, decentralized solutions. Between data sources and cloud infrastructure, fog computing mainly processes and stores data. Fog computing can improve computational performance by reducing the amount of processing and storage that extra data consumes. Fog computing has real-time responsiveness and offers a cost-effective, flexible deployment of hardware and software in computing system deployments. Fog platform also faces a lot of network security issues. Such as code injection attacks (such as SQL injection), session and cookie hijacking (posing as legitimate users), illegal direct data access unsafe references, malicious redirect and driver attacks, web attacks, and other cyberattacks. Due to the relatively small computing resources (memory, processing, and storage) of the fog computing system, there is no security protection that can consume a large amount of secure authentication storage as cloud computing does. Fog computing should be defined for a broader range of ubiquitous connected devices, which requires the fog server to generate a large number of security
codes at one time and a relatively low computational load during verification. For secure communications and authentication, stream ciphers are recognized as fast certification, which require less computation and storage capacity. AESbased cipher type mentioned in [1] is an encryption algorithm which is suitable for fog platforms. But fog calculation of data encryption security needs to consider stream cipher antiattack performance. In the fog computing environment using stream ciphers, the security verification and data transmission should be considered between the length of the password and the verification algorithm. The safety of some fog calculations strongly depends on the security of the sequence itself by weakening the verification algorithm. In this case, the fog server will distribute a large amount of security codes, and it is easier for an attacker to collect large numbers of plain-texts and cipher-texts so that he may filter out full-period encrypted sequences. Currently, there are many attacks on the known periodic sequences in which a common one is the 2-adic complexity attack. For cryptographic applications, a good pseudorandom generator must be infeasible to find the corresponding initial state. Hence many modern stream ciphers are designed by combining the output sequences in various nonlinear ways. Goresky and Klapper first introduced feedback with carry shift registers (FCSRs) as shown in Figure 1 , which are a
2
Wireless Communications and Mobile Computing mn−1
an−1
an−2
q1
q2
······
an−r−1
an−r
qr−1
qr
+
Figure 1: Feedback with carry shift registers.
a binary balanced periodic sequence, we give a relationship with its 2-adic integer, the length of period, and 2-adic complexity and show that the 2-adic complexity is bigger than the half period of the sequence when its 2-adic number approaches half. Moreover, it is indicated that the 2-adic complexity of the binary balanced sequence is affected by the register bit values of the FCSR. In the following sections we only consider the binary strictly periodic sequences, and we denote them as periodic sequences for simplicity.
2. Preliminary Begin Input 𝑎 until the first nonzero 𝑎𝑘−1 is found 𝛼 = 𝑎𝑘−1 ⋅ 2𝑘−1 𝑓 = (0, 2) 𝑔 = (2𝑘−1 , 1) While there are more bits do In put a new bit 𝑎𝑘 𝛼 = 𝛼 + 𝑎𝑘 ⋅ 2𝑘 If 𝛼𝑔2 − 𝑔1 ≡ 0 (𝑚𝑜𝑑 2𝑘+1 ) then 𝑓 = 2𝑓 else if Φ(𝑔) < Φ(𝑓) then Let 𝑑 be odd and minimize Φ(𝑓 + 𝑑𝑔) ⟨𝑔, 𝑓⟩ = ⟨𝑓 + 𝑑𝑔, 2𝑔⟩ else Let 𝑑 be odd and minimize Φ(𝑔 + 𝑑𝑓) ⟨𝑔, 𝑓⟩ = ⟨𝑔 + 𝑑𝑓, 2𝑓⟩ fi fi 𝑘=𝑘+1 odd Return 𝑔 End Algorithm 1: Rational approximation algorithm.
class of nonlinear sequence generators by [2], and used the arithmetic in the 2-adic number to analyze this stream generator. For the security of the stream, rational approximation algorithm given in [2] is an important adaptive synthesizing algorithm against FCSRs, as shown in Algorithm 1, by which if a key-stream can be generated by a short FCSR, then this FCSR can be efficiently determined from a small subsequence of the key-stream. Therefore, the rational approximation algorithm sets up a new measure of key-stream security and is referred to as 2-adic complexity. For the properties of FCSRs, it is well known that any strictly periodic sequence can be generated by an FCSR. Then any binary sequence with low 2-adic complexity is insecure for cryptographic applications. Although some properties of 2-adic complexity had been proven, such as the expected value and variance of 2-adic complexities of periodic binary sequences and the 2-adic complexity of 𝑚-sequence, the 2-adic complexity of binary sequences has not been quite clear. This paper studies one function of periodic balance sequence which can against the 2-adic complexity attack in the fog computing environment. This paper involves the exponential function and the structure principle of FCSR for the study of the 2-adic properties and 2-adic complexity of balanced binary sequences. For
In this section we briefly review some basic facts about feedback with carry shift register (FCSR) and 2-adic number. The FCSR is a feedback with 𝑟-stages shift register and its auxiliary memory contained nonnegative integer. Assume an odd integer 𝑞 has the binary representation as 𝑞 + 1 = 𝑞1 ⋅ 2 + 𝑞2 ⋅ 22 + ⋅ ⋅ ⋅ + 𝑞𝑟 ⋅ 2𝑟 . Then the 𝑟-stages connections of FCSR are given by the bits {𝑞1 , 𝑞2 , . . . , 𝑞𝑟 }. The FCSR with connection integer 𝑞 is described as follows: (1) Take an integer sum 𝜎𝑡 = ∑𝑟𝑘=1 𝑞𝑘 𝑎𝑡−𝑘 + 𝑚𝑡−1 . (2) Shift the contents one step to the right, outputting the right bit 𝑎𝑡−𝑟 . (3) Place 𝑎𝑡 = (𝜎𝑡 ) mod 2 into the left most cell of the shift register. (4) Replace the memory integer 𝑚𝑡−1 with 𝑚𝑡 = (𝜎𝑡 − 𝑎𝑡 )/2 = ⌊𝜎𝑡 /2⌋. The number of bits in the connection number coincides with the size of the basic register. For strictly periodic sequences, the extra memory is small and we can ignore it, but the eventually periodic sequence may require the amount of memory. In this paper, we just consider the strictly periodic sequences, and then we denote that the 2-adic complexity of sequences is to measure the number of bits in the basic FCSR. In the study of the output sequence of a given FCSR, we usually use the arithmetic in the 2-adic integer. 𝑡 A 2-adic integer is form power series 𝛼 = ∑∞ 𝑡=0 𝑎𝑡 ⋅ 2 , with 𝑎𝑡 ∈ {0, 1}, and a fact is that number −1 is represented by −1 = 1 + 2 + 22 + 23 + ⋅ ⋅ ⋅ . Then, the negative integer –𝑞 is associated with the product −𝑞 = (1 + 2 + 22 + 23 + ⋅ ⋅ ⋅) ⋅ (𝑞0 + 𝑞1 ⋅ 2 + 𝑞2 ⋅ 22 + ⋅ ⋅ ⋅ + 𝑞𝑟 ⋅ 2𝑟 ) .
(1)
Moreover, the multiplication of 2-adic integer also has unique inverse if the integer 𝑞 is an odd integer. Thus the 2adic integer contains every rational number 𝑝/𝑞, provided 𝑞 is odd. Proposition 1 (see [2]). There is a one-to-one correspondence between rational numbers 𝛼 = 𝑝/𝑞 (where 𝑞 is odd) and eventually periodic binary sequences 𝑎. We define the rational number 𝛼 as the 2-adic expansion of the binary sequences 𝑎. The sequence 𝑎 is strictly periodic if and only if 𝛼 ≤ 0 and |𝛼| < 1.
Wireless Communications and Mobile Computing
3
If a strictly sequence 𝑎 is generated by an FCSR with connection integer 𝑞, then the 2-adic integer 𝛼 = 𝑎0 + 𝑎1 ⋅ 2 + 𝑎2 ⋅ 22 + ⋅ ⋅ ⋅ of binary sequence 𝑎 has the following association.
even. When V𝑡 ∈ 𝑍/(𝑞) is an even integer, we assume V𝑡 = 2𝑘𝑡 over 𝑍/(𝑞), and we have 𝑞−1 (𝑞−1)/2
∑ ∑ 𝑒𝑞 (𝑏 ⋅ (
Proposition 2 (see [2]). Let a periodic sequence 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . be generated by an FCSR with connection integer 𝑞 and the 2-adic representation of sequence 𝑎 is −𝑝/𝑞. Then one has
𝑥=0
𝑏=1
𝑞−1 (𝑞−1)/2
= ∑ ∑ 𝑒𝑞 (𝑏 ⋅ ( 𝑏=0
𝑡 ∑𝑇−1 𝑝 𝑡=0 𝑎𝑡 2 = . 𝑞 2𝑇 − 1
V𝑡 − 𝑥)) 2
𝑥=0
𝑞−1 (𝑞−1)/2
(2)
=∑
∑ 𝑒𝑞 (𝑞 ⋅ (𝑘𝑡 − 𝑥)) + 𝑞 −
𝑏=1 𝑥=0,𝑥𝑡 =𝑘 ̸
From the above description about 2-adic integer and FCSR, the 2-adic complexity of periodic sequence 𝑎 can be regarded as 𝑞 𝜓(𝑎) = ⌊log2 ⌋. The binary sequences of 2-adic complexity can be got from rational approximation algorithm [2]. If the 2-adic complexity of a sequence is greater than half the period, then this sequence is resistant to 2-adic rational approximation attacks.
𝑞−1
(𝑞−1)/2
𝑞−1 (𝑞−1)/2
∑ ∑ (𝑒𝑞 (𝑏 ⋅ ( 𝑥=0
𝑏=0
V𝑡 V − 𝑥)) + 𝑒𝑞 (𝑏 ⋅ ( 𝑠 − 𝑥))) 2 2
𝑞−1 (𝑞−1)/2
Lemma 3 (see [2]). Suppose a periodic sequence 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . is generated by an FCSR with connection integer 𝑞. Let 𝑟 = 2−1 ∈ 𝑍/(𝑞) be the (multiplicative) inverse of 2 in the ring 𝑍/(𝑞) of integer modulo 𝑞. Then there exists 𝐴 ∈ 𝑍/(𝑞) such that, for all 𝑡 = 0, 1, 2, . . ., one has 𝑎𝑡 = (𝐴 ⋅ 𝑟𝑡 mod 𝑞) mod 2. In this paper, we just consider the balanced binary strictly periodic sequence. Then the sequence 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 in a period of length 𝑇 satisfies the fact that the number of even integers equals the number of odd integers. We assume that another sequence V = {V𝑡 }∞ 𝑡=0 over 𝑍/(𝑞) in a period of length 𝑇 is bilateral symmetry with (𝑞 − 1)/2. In the following analysis of this paper, we introduce the exponential function 𝑒𝑞 (∗) = 𝑒(2𝜋𝑖⋅∗)/𝑞 as the tool to prove the main theorems. It is easy to get 𝑒𝑞 (𝑞) = 𝑒(2𝜋𝑖⋅𝑞)/𝑞 = 1 and 𝑒𝑞 (𝑞/2) = 𝑒(𝜋𝑖⋅𝑞)/𝑞 = −1. Since 𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞, we have 𝑒𝑞 (𝑢𝑡 ) = 𝑒(2𝜋𝑖⋅𝑢𝑡 )/𝑞 = 𝑒(2𝜋𝑖⋅𝑝⋅2
)/𝑞
= 𝑒𝑞 (𝑝 ⋅ 2−𝑡 ) .
Lemma 4. Let the sequence V over 𝑍/(𝑞) be a periodic sequence as described above; one has known that the sequence V in a period of length of 𝑇 satisfies the following equation:
Proof. In a period of length 𝑇of the sequence V, the number of odd integers equals the number of even integers and 𝑇 is an
(6)
𝑞 − 1 V𝑡 − − 𝑥))) . 2 2 As the variable (𝑞 − 1)/2 − 𝑥 ∈ [0, (𝑞 − 1)/2], 𝑞−1 (𝑞−1)/2
∑ ∑ (𝑒𝑞 (𝑏 ⋅ (
𝑏=0
𝑥=0
V𝑡 V − 𝑥)) + 𝑒𝑞 (𝑏 ⋅ ( 𝑠 − 𝑥))) 2 2
𝑞−1 (𝑞−1)/2
V𝑡 V − 𝑥)) + 𝑒𝑞 (𝑏 ⋅ (𝑥 − 𝑡 ))) 2 2
= ∑ ∑ (𝑒𝑞 (𝑏 ⋅ ( 𝑥=0
𝑏=0
(𝑞−1)/2
= ∑ +
(7)
𝑒𝑞 (𝑞 ⋅ (V𝑡 /2 − 𝑥)) − 1 𝑒𝑞 (V𝑡 /2 − 𝑥) − 1
𝑥=0
𝑒𝑞 (𝑞 ⋅ (𝑥 − V𝑡 /2)) − 1 𝑒𝑞 (𝑥 − V𝑡 /2) − 1
.
Since 𝑒𝑞 (𝑞 ⋅ (V𝑡 /2 − 𝑥)) − 1 = 𝑒𝑞 (𝑞 ⋅ (𝑥 − V𝑡 /2)) − 1 = −2, we have (𝑞−1)/2
𝑒𝑞 (𝑞 ⋅ (V𝑡 /2 − 𝑥)) − 1 𝑒𝑞 (V𝑡 /2 − 𝑥) − 1
𝑥=0
+
𝑒𝑞 (𝑞 ⋅ (𝑥 − V𝑡 /2)) − 1 𝑒𝑞 (𝑥 − V𝑡 /2) − 1
(8)
𝑞+1 = 𝑞 + 1. 2
=2⋅ Thus 𝑞−1 (𝑞−1)/2 𝑏=1
(4)
V𝑡 − 𝑥)) 2
+ 𝑒𝑞 (𝑏 ⋅ (
∑ ∑ (𝑒𝑞 (𝑏 ⋅ (
𝑇−1 𝑞−1 (𝑞−1)/2
𝑇 (𝑞 − 1) V . ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ ( 𝑡 − 𝑥)) = 2 4 𝑡=0 𝑏=1 𝑥=0
𝑥=0
𝑏=0
∑ (3)
𝑞+1 . 2
∑𝑏=1 ∑𝑥=0 𝑒𝑞 (𝑏 ⋅ (V𝑡 /2 − 𝑥)) = (𝑞 − 1)/2 for V𝑡 as an even integer. When V𝑡 = 2𝑘𝑡 + 1 is an odd integer, we get another V𝑠 with V𝑠 = 𝑞 − 1 − V𝑡 , and
= ∑ ∑ (𝑒𝑞 (𝑏 ⋅ (
In this section we mainly prove Theorem 7, and some lemmas are given to support the main result proof.
(5)
Since 𝑒𝑞 (𝑞 ⋅ (𝑘𝑡 − 𝑥)) = 𝑒(𝑞⋅(𝑘𝑡 −𝑥)⋅2𝜋𝑖)/𝑞 = 1, then we have
3. Main Results
−𝑡
𝑞+1 V𝑡 − 𝑥)) − 2 2
𝑥=0
V𝑡 V − 𝑥)) + 𝑒𝑞 (𝑏 ⋅ ( 𝑠 − 𝑥))) 2 2
𝑞−1 (𝑞−1)/2
= ∑ ∑ (𝑒𝑞 (𝑏 ⋅ ( 𝑏=0
𝑥=0
(9) V𝑡 V − 𝑥)) + 𝑒𝑞 (𝑏 ⋅ ( 𝑠 − 𝑥))) 2 2
− 𝑞 − 1 = 𝑞 + 1 − 𝑞 − 1 = 0.
4
Wireless Communications and Mobile Computing If (𝑞 − 1)/2 ≡ (𝑇/2 + 1) mod 2, we have
Then, from the above analysis, we get 𝑇−1 𝑞−1 (𝑞−1)/2
∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (
𝑡=0 𝑏=1
=
𝑥=0
V𝑡 − 𝑥)) 2
𝑞−1 (𝑞−1)/2 𝑇 (𝑞 − 1) 𝑘 𝑇 . ∑ ∑ 𝑒 (𝑏 ⋅ ( 𝑡 − 𝑥)) = 2 𝑏=1 𝑥=0 𝑞 2 4
𝑇−1
{V𝑡 }𝑡=0 = { (10)
𝑇−1
{V𝑡 }𝑡=0
(11) 𝑞−1 𝑇 𝑞−1 𝑇 𝑞−1 𝑇 − + 1, − + 2, . . . , + }, 2 2 2 2 2 2
when (𝑞 − 1)/2 ≡ (𝑇/2) mod 2, and V𝑡 in a period of length 𝑇 satisfies 𝑇−1
{V𝑡 }𝑡=0
𝑞−1 𝑇 𝑞−1 ={ − ,..., − 1} 2 2 2 ∪{
𝑞−1 𝑇 𝑞−1 + 1, . . . , + }, 2 2 2
(12)
(17)
(𝑞−1)/2
(𝑞−1)/2 𝑏=1
(𝑞−1)/2 𝑞−1 𝑞 ) + ) ∑ 𝑒𝑞 (𝑏 ⋅ (−2𝑥)) . 2 2 𝑥=0
Since |𝑒𝑞 (𝑏 ⋅ ((𝑞 − 1)/2 − 𝑇/2 − 1))| ≤ 1 and |𝑒𝑞 (𝑏 ⋅ ((𝑞 − 1)/2) + 𝑞/2)| ≤ 1, 𝑇−1
{V𝑡 }𝑡=0 = { (13)
1 1 < 𝑞 ( lncot(𝜋/𝑞) + ) . 𝜋 6
𝑞−1 𝑇 𝑞−1 − ,..., − 1} 2 2 2
𝑞−1 𝑇 𝑞−1 + 1, . . . , + }; ∪{ 2 2 2
(18)
these are satisfying
Proof. If (𝑞 − 1)/2 ≡ (𝑇/2) mod 2, we have 𝑇−1 {V𝑡 }𝑡=0
(14) 𝑞−1 𝑇 𝑞−1 𝑇 𝑞−1 𝑇 − + 1, − + 2, . . . , + }. 2 2 2 2 2 2
That is, (𝑞−1)/2 (𝑞−1)/2 𝑁−1 ∑ ∑ ∑ 𝑒 (2𝑏 ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 = ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2+𝑇/2 𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) = ∑ ∑ ∑ 𝑏=1 𝑥=0 𝑡=(𝑞−1)/2−𝑇/2+1 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 ≤ ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) . 𝑏=1 𝑥=0 𝑡=0
(16)
𝑇+2 𝑞−1 𝑇 − − 1)) ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) 2 2 𝑥=0 𝑡=1
+ ∑ 𝑒𝑞 (𝑏 ⋅ (
Lemma 5. For any positive integer 𝑁, one has
={
𝑞−1 𝑇 𝑞−1 + 1, . . . , + }. 2 2 2
(𝑞−1)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 (𝑏 ⋅ (V − 2𝑥)) 𝑞 𝑡 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2+𝑇/2 𝑒 (𝑏 ⋅ (𝑡 − 2𝑥)) = ∑ ∑ ∑ 𝑏=1 𝑥=0 𝑡=(𝑞−1)/2−𝑇/2+1 𝑞 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 𝑞−1 − ∑ ∑ 𝑒𝑞 (𝑏 ⋅ ( − 2𝑥)) = ∑ 𝑒𝑞 2 𝑏=1 𝑥=0 𝑏=1 ⋅ (𝑏 ⋅ (
when (𝑞 − 1)/2 ≡ (𝑇/2 + 1) mod 2.
(𝑞−1)/2 (𝑞−1)/2 𝑁−1 ∑ ∑ ∑ 𝑒 (2𝑏 ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑏=1 𝑥=0 𝑡=0
∪{ That is,
The sequences V𝑡 have a little limit in Lemma 4, and the sequence V𝑡 in a period of length 𝑇 satisfies
={
𝑞−1 𝑇 𝑞−1 − ,..., − 1} 2 2 2
(𝑞−1)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 (𝑏 ⋅ (V − 2𝑥)) 𝑞 𝑡 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇+2 ≤ ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) . 𝑏=1 𝑥=0 𝑡=0
(19)
Next we consider the formula ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) . 𝑏=1 𝑥=0 𝑡=0
(𝑞−1)/2 (𝑞−1)/2 𝑁
(15)
(20)
We first have the inequality (𝑞−1)/2 𝑁 ∑ ∑𝑒 (𝑏 ⋅ (𝑡 − 2𝑥)) = 𝑒𝑞 (𝑏 ⋅ 𝑁 + 1) − 1 𝑒𝑞 (𝑏) − 𝑒𝑞 (−𝑏) 𝑥=0 𝑡=0 𝑞 1 ≤ . sin (2𝜋𝑏/𝑞)
(21)
Wireless Communications and Mobile Computing
5 Proof. Let sequence 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . with the period 𝑇 and corresponding sequence 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 as the described sequence V; we have
It follows that (𝑞−1)/2 (𝑞−1)/2 𝑁 ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2
1 1 ≤ + ∑ sin (2𝜋𝑏/𝑞) sin (2𝜋/𝑞) 𝑏=2 𝑑((𝑞−1)/4)𝑡
1 < + 2∫ sin (2𝜋/𝑞) 2
(22)
csc (2𝜋𝑥/𝑞) 𝑑𝑥
𝑞 cot(2𝜋/𝑞) 1 < . + 𝜋 ln sin (2𝜋/𝑞) Then ∑ ∑𝑒𝑞 (𝑏 ⋅ (𝑡 − 2𝑥)) 𝑥=0 𝑡=0
(𝑞−1)/2 (𝑞−1)/2 𝑁
∑
𝑏=1
(23)
1 1 < 𝑞 ( lncot(𝜋/𝑞) + ) . 𝜋 6 Thus we get the conclusion (𝑞−1)/2 (𝑞−1)/2 𝑁−1 ∑ ∑ ∑ 𝑒 (2𝑏 ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑏=1 𝑥=0 𝑡=0
(24)
(𝑞−1)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 (2𝑏 ⋅ ( 𝑢𝑡 − 𝑥)) 2 𝑏=1 𝑥=0 𝑡=0 𝑞 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 V = ∑ ∑ ∑ 𝑒𝑞 (2𝑏 ⋅ ( 𝑡 − 𝑥)) . 2 𝑏=1 𝑥=0 𝑡=0 The sequences V have the following inequality: 𝑇−1 𝑞−1 (𝑞−1)/2 ∑ ∑ ∑ 𝑒 (𝑏 ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑡=0 𝑏=1 𝑥=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 V ≤ ∑ ∑ ∑ 𝑒𝑞 (2𝑏 ⋅ ( 𝑡 − 𝑥)) 2 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 V + ∑ ∑ ∑ 𝑒𝑞 ((2𝑏 − 1) ⋅ ( 𝑡 − 𝑥)) . 2 𝑏=1 𝑥=0 𝑡=0
𝑒𝑞 ((2𝑏 − 1) (
V𝑡 − 𝑥)) . 2
(28)
We have (𝑞−1)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 ((2𝑏 − 1) ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 = ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡=0 V𝑡 ⋅ 𝑒𝑞 (−1 ⋅ ( − 𝑥)) . 2
Theorem 7. Let 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . be a binary balanced period sequence with period 𝑇, 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 is the exponential of the sequence 𝑎, the elements in 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 are symmetry with (𝑞 − 1)/2, and the 2-adic integer of sequence 𝑎 has the property (25)
V𝑡 − 𝑥)) 2
= 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) ⋅ 𝑒𝑞 (−1 ⋅ (
Proof. From the description about the 2-adic integer, we have known that −1 = 1 + 2 + 22 + 23 + ⋅ ⋅ ⋅ and the sequence 1 = 1, 1, 1, . . . have the 2-adic representation −1. The complement sequences 𝑎 and 𝑎 satisfy 𝛼 + 𝛼 = −1. Then, we have −𝑝/𝑞 + 𝛼 = −1; that is, 𝛼 = −(𝑞 − 𝑝)/𝑞.
𝜋 𝑞 ⋅ 𝑇 < log2 . 12
(27)
Then
1 1 < 𝑞 ( lncot(𝜋/𝑞) + ) . 𝜋 6
Lemma 6. The binary strictly periodic sequence 𝑎 corresponds to the 2-adic integer 𝛼 = −𝑝/𝑞, where the integer 𝑞 is odd and primitive with 𝑝. The complement sequence 𝑎 of 𝑎 has the 2adic representation 𝛼 = −(𝑞 − 𝑝)/𝑞.
(26)
That is,
(𝑞−1)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 (𝑏 ⋅ (V − 2𝑥)) 𝑒 (−1 ⋅ ( V𝑡 − 𝑥)) 𝑞 𝑡 𝑞 2 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 V𝑒 = ∑ ∑ ∑ (𝑒𝑞 (𝑏 ⋅ (V𝑡𝑒 − 2𝑥)) ⋅ (𝑒𝑞 (− ( 𝑡 − 𝑥)) − 1) + 𝑒𝑞 (𝑏 ⋅ (V𝑡𝑒 − 2𝑥))) 2 𝑏=1 𝑥=0 𝑡
(29)
6
Wireless Communications and Mobile Computing (𝑞−1)/2 (𝑞−1)/2
+ ∑ 𝑏=1
∑ ∑ (𝑒𝑞 (𝑏 ⋅
𝑥=0
𝑡
(V𝑡𝑜
V𝑡𝑜 𝑜 − 2𝑥)) (𝑒𝑞 (− ( − 𝑥)) + 1) − 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥))) 2
(𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 𝑒𝑞 (((𝑞 + 1) /2) (V𝑡𝑒 − 2𝑥) − 𝑒𝑞 (V𝑡𝑒 − 2𝑥)) 𝑒 (V − 2𝑥) + 1) + = ∑ ∑ (𝑒 ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡𝑒 − 2𝑥)) 𝑞 𝑡 𝑒𝑞 (V𝑡𝑒 − 2𝑥) − 1 𝑥=0 𝑡=0 𝑡 𝑥=0 𝑏=1 (𝑞−1)/2 𝑇−1 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 𝑜 − ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) = ∑ ∑ (−1 + 1) + ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡𝑒 − 2𝑥)) − ∑ ∑ ∑𝑒𝑞 (𝑏 𝑥=0 𝑡=0 𝑡 𝑡 𝑡 𝑥=0 𝑥=0 𝑥=0 𝑏=1 𝑏=1 𝑏=1 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 (𝑞−1)/2 𝑜 𝑒 𝑜 ⋅ (V𝑡 − 2𝑥)) = ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) − ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) , 𝑏=1 𝑥=0 𝑡 𝑡 𝑥=0 𝑏=1 (30)
V𝑡𝑒 are defined as the even numbers in a period of V, V𝑡𝑜 are defined as the odd numbers in a period of V, and then the inequality can be expressed as (𝑞−1)/2 (𝑞−1)/2 𝑇−1 V ∑ ∑ ∑ 𝑒𝑞 ((2𝑏 − 1) ⋅ ( 𝑡 − 𝑥)) 2 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 (31) = ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 − 2 ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡𝑜 − 2𝑥)) . 𝑡 𝑥=0 𝑏=1 We have 𝑇−1 𝑞−1 (𝑞−1)/2 ∑ ∑ ∑ 𝑒 (𝑏 ⋅ ( V𝑡 − 𝑥)) 𝑞 2 𝑡=0 𝑏=1 𝑥=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 (32) ≤ 2 ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑜 + 2 ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) . 𝑏=1 𝑥=0 𝑡 Since (𝑞−1)/2 (𝑞−1)/2 𝑜 ∑ ∑ ∑𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) 𝑏=1 𝑥=0 𝑡 (33) (𝑞−1)/2 (𝑞−1)/2 𝑇/4 ≤ ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ ((2𝑡 + 1) 2𝑥)) , 𝑏=1 𝑥=0 𝑡=0 then (𝑞−1)/2 (𝑞−1)/2 ∑ ∑ ∑𝑒 (𝑏 ⋅ (V𝑜 − 2𝑥)) 𝑞 𝑡 𝑏=1 𝑥=0 𝑡 (34) (𝑞−1)/2 sin ((𝑇) 𝑏𝜋/2𝑞) ; ≤ ∑ sin (2𝑏𝜋/𝑞) sin (𝑏𝜋/𝑞) 𝑏=1
that is, sin ((𝑇) 𝑏𝜋/2𝑞) ≤ (cos𝑇/4 ( 𝑏𝜋 ) 𝑞 sin (2𝑏𝜋/𝑞) sin (𝑏𝜋/𝑞) + cos
𝑇/4−1
𝑏𝜋 𝑏𝜋 1 ( ) + ⋅ ⋅ ⋅ + cos ( )) ⋅ . 𝑞 𝑞 sin (𝑏𝜋/𝑞)
(35)
So (𝑞−1)/2 (𝑞−1)/2 ∑ ∑ ∑𝑒 (𝑏 ⋅ (V𝑜 − 2𝑥)) 𝑞 𝑡 𝑏=1 𝑥=0 𝑡 (𝑞−1)/2
≤ ∑ (cos𝑇/4 ( 𝑏=1
+ cos ( ≤
𝑏𝜋 𝑏𝜋 ) + cos𝑇/4−1 ( ) + ⋅ ⋅ ⋅ 𝑞 𝑞
(36)
𝑇/4
𝑞 1 1 𝑏𝜋 ≤ ⋅ ( ∑ ) lncot(𝜋/𝑞) )) ⋅ 𝑞 sin (𝑏𝜋/𝑞) 𝜋 𝑡=1 𝑡
𝑞 𝑞 ⋅ ln𝑇/4 log2 . 𝜋 𝑞
𝑞
As Lemma 3 (𝑇/4)(𝑞 − 1) < (𝑞/𝜋) ⋅ log2 + (2𝑞/𝜋)log2 , we have 𝑞 (𝜋/12) ⋅ 𝑇 < log2 . If the connection integer 𝑞 of balanced sequence 𝑎 is large 𝑞 enough and satisfies lncot(𝜋/𝑞) < ⌊log2 ⌋, we can have the following corollary. Corollary 8. Let 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . be a period balanced sequence with period 𝑇, 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 is the exponential of the sequence 𝑎, and the elements in 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 are symmetry with (𝑞 − 1)/2, then the 2-adic complexity of sequence 𝑎 has 𝜋 ⋅ 𝑇 < 𝜓 (𝑎) . 12
(37)
The balanced binary sequence described in Theorem 7 is resistant to 2-adic attack, but the higher sequence requirements are difficult to achieve. In general, when considering the complexity, it cannot get its exponential representation.
Wireless Communications and Mobile Computing
7
How to get a relatively broad condition to reflect the relationship between 2-adic complexity and periodicity of binary balanced sequences is the problem we need to consider. Lemma 9. Let 𝑎 be the balanced strictly periodic sequence of a binary sequence and correspond to the 2-adic integer 𝑝/𝑞. Assume that the sequence 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 with 𝑎𝑡 = 𝑢𝑡 mod 2. Then, there must exist another sequence V over 𝑍/(𝑞) satisfying (𝑞−3)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 ((𝑏 + 1 ) ⋅ (V − 2𝑥)) 𝑞 𝑡 2 𝑏=0 𝑥=0 𝑡=0 (𝑞−1)/2 𝑇−1 (𝑞−3)/2 1 < ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (𝑢𝑡 − 2𝑥)) . 2 𝑥=0 𝑡=0 𝑏=0
(38)
(39)
(41)
Lemma 10. Let the binary strictly periodic sequence 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . be generated by an FCSR with connection integer 𝑞 and the correspondence 2-adic integer is −𝑝/𝑞. Then, the sequence 𝑢 = {𝑢𝑡 = 𝑝 ⋅ 2−𝑡 mod 𝑞}∞ 𝑡=0 satisfies (𝑞−1)/2 𝑇−1 (𝑞−3)/2
When 𝑢𝑡 = 2𝑘𝑡 is an even integer, (𝑞−3)/2 ∑ 𝑒 ((𝑏 + 1 ) ⋅ (𝑢 − 2𝑥)) 𝑞 𝑡 2 𝑏=0 1 . = 2 cos (((2𝑘𝑡 − 2𝑥) /𝑞) 𝜋)
(𝑞−3)/2 (𝑞−1)/2 𝑇−1 ∑ ∑ ∑ 𝑒 ((𝑏 + 1 ) ⋅ (V − 2𝑥)) 𝑞 𝑡 2 𝑏=0 𝑥=0 𝑡=0 (𝑞−1)/2 𝑇−1 (𝑞−3)/2 1 < ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (𝑢𝑡 − 2𝑥)) , 2 𝑥=0 𝑡=0 𝑏=0
where the sequence V in a period of length of 𝑇 is also bilateral symmetry with (𝑞 − 1)/2.
Proof. When 𝑢𝑡 = 2𝑘𝑡 + 1 is an odd integer, (𝑞−3)/2 ∑ 𝑒 ((𝑏 + 1 ) ⋅ (𝑢 − 2𝑥)) 𝑞 𝑡 2 𝑏=0 1 . = 2 sin (((2𝑘𝑡 + 1 − 2𝑥) /𝑞) 𝜋)
The sequence 𝑢 in a period of length 𝑇 has the same number of even integers and odd integers. If |2 sin(((2𝑘𝑡 + 1 − 2𝑥)/𝑞)𝜋)| = |2 cos(((2𝑘𝑡 − 2𝑥)/𝑞)𝜋)| and |1/2 sin(((2𝑘𝑡 + 1 − 2𝑥)/𝑞)𝜋)| + |1/2 cos(((2𝑘𝑡 − 2𝑥)/𝑞)𝜋)| have a minimum value, then it can be arrived at the minimum value when 𝑢𝑡 are bilateral symmetry with (𝑞 − 1)/2. So
(40)
1 ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (𝑢𝑡 − 2𝑥)) 2 𝑥=0 𝑡=0 𝑏=0 𝑞 𝑝/(𝑞−𝑝) . < 𝑇 log2 𝜋
(42)
Proof. From the definition of 𝑒𝑞 (𝑎), we have
(𝑞−1)/2 𝑇−1 (𝑞−3)/2
(𝑞−1)/2 𝑇−1 𝑒𝑞 ((𝑞 − 3) /2 + 1 + 1/2) (𝑢𝑡 − 2𝑥) − 𝑒𝑞 ((1/2) (𝑢𝑡 − 2𝑥)) 1 ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (𝑢𝑡 − 2𝑥)) = ∑ ∑ 2 𝑒 (𝑢 − 2𝑥) − 1 𝑞 𝑡 𝑥=0 𝑡=0 𝑏=0 𝑥=0 𝑡=0 (𝑞−1)/2 𝑇−1 1 . = ∑ ∑ 1 ± 𝑒 (𝑢 /2 − 𝑥) 𝑞 𝑡 𝑥=0 𝑡=0
Then, when 𝑢𝑡 is an even integer, |1 + 𝑒𝑞 (𝑢𝑡 /2 − 𝑥)| = 2| cos((𝑢𝑡 − 2𝑥)𝜋/2𝑞)|, and when 𝑢𝑡 is an odd integer, |1 − 𝑒𝑞 (𝑢𝑡 /2 − 𝑥)| = 2| sin((𝑢𝑡 − 2𝑥)𝜋/2𝑞)|. We have known that 1 ∑ sin ((𝑢 − 2𝑥) 𝜋/2𝑞) 𝑡 𝑥=0 𝑞 < lntan(𝑢𝑡 −𝑞)/2𝑞 − lntan(𝑢𝑡 )/2𝑞 . 𝜋
(𝑞−1)/2
1 𝑞−𝑢 𝑞 𝑢𝑡 ∑ < log2 − log2 𝑡 . 𝜋 𝑥=0 1 ± 𝑒𝑞 (𝑢𝑡 /2 − 𝑥)
Then we have 𝑞 𝑇−1 𝑢𝑡 /(𝑞−𝑢𝑡 ) 1 < ∑ log . ∑ ∑ 𝜋 2 𝑡=0 𝑥=0 1 ± 𝑒𝑞 (𝑢𝑡 /2 − 𝑥) 𝑡=0
𝑇−1 (𝑞−1)/2
(46)
(44)
As 𝑞 is large integer, we get (𝑞−1)/2
(43)
(45)
From Lemma 3, 𝑎𝑡 = 𝑢𝑡 mod 2 is the complement with the binary sequence {𝑎𝑡 = (𝑞 − 𝑢𝑡 ) mod 2}∞ 𝑡=0 . Since the oneto-one correspondence between the binary sequence and 2adic integer, we have 𝑎𝑡 = (𝑝 − 𝑢𝑡 ) mod 2 = ((𝑞 − 𝑝) ⋅ 2−𝑡 mod 𝑞) mod 2. We have 𝑢𝑡 = 𝑝 ⋅ 2−𝑡 − 𝑛𝑡 𝑞, 𝑞 − 𝑢𝑡 = (𝑞 − 𝑝) ⋅ 2−𝑡 − 𝑛𝑡 𝑞, and 𝑛𝑡 ≈ ((𝑞 − 𝑝)/𝑝)𝑛𝑡 . Then
8
Wireless Communications and Mobile Computing Table 1: The security of binary sequences.
Period Occurrence 𝐸(𝐶(𝜏)) 𝑉(𝐶(𝜏)) 2-adic complexity
𝐿-sequence 2𝑒−1 bit 𝐴0 = 𝐴1 0 𝑒+1 𝑂(2𝑒−1 /ln2 ) 2𝑒−2 ≻ 𝜑
𝑚-sequence 2𝑒 − 1 bit 𝐴0 = 𝐴1 − 1 0 𝑒 𝑂(2𝑒 − 1/ln4(2 −1) ) 𝜑 ≻ 2𝑒−1 − 1 (Test)
AES balance 𝑇 𝐴0 = 𝐴1 — — 𝜑 ≻ 𝑇/2 − 1 (Test)
Table 2: Hardware performance comparison.
Trivium AES FFCSR-2 FCSR-SS
Critical path (ns) 4.623 5.232 5.452 5.950
Frequency (MHz) 216.31 199.43 183.42 168.07
Throughout (Mbps) 216.31 216.31 1467.36 1344.56
Total logic elements 650 632 622 771
𝑇−1
𝑢 /(𝑞−𝑢𝑡 ) ∑ log2 𝑡 (47)
𝑝/(𝑞−𝑝) . ≈ 𝑇 log2 So we have
𝑞 𝑝/(𝑞−𝑝) 1 < 𝑇 log . ∑ ∑ 𝜋 2 𝑥=0 𝑡=0 1 ± 𝑒𝑞 (𝑢𝑡 /2 − 𝑥) Thus, we get the conclusion (𝑞−1)/2 𝑇−1 (𝑞−3)/2 1 ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (𝑢𝑡 − 2𝑥)) 2 𝑥=0 𝑡=0 𝑏=0 𝑞 𝑝/(𝑞−𝑝) . < 𝑇 log2 𝜋 (𝑞−1)/2 𝑇−1
𝜋 𝛼/(1−𝛼) 1 ) < 𝜓 (𝑎) + . − log 4 2 6 Proof. From Lemmas 3 and 9, we have known that 𝑇 (𝑞 − 1) 𝑇−1 𝑞−1 (𝑞−1)/2 V𝑡 = ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ ( − 𝑥)) 4 2 𝑡=0 𝑏=1 𝑥=0 (𝑞−3)/2 (𝑞−1)/2 𝑇−1 1 ≥ ∑ ∑ ∑ 𝑒𝑞 ((𝑏 + ) ⋅ (V𝑡 − 2𝑥)) 2 𝑏=0 𝑥=0 𝑡=0 (𝑞−1)/2 (𝑞−1)/2 𝑇−1 + ∑ ∑ ∑ 𝑒𝑞 (𝑏 ⋅ (V𝑡 − 2𝑥)) . 𝑏=1 𝑥=0 𝑡=0
The 2-adic complexity is the half period
(48) Period T
Figure 2: 2-adic complexity of binary sequences.
(49)
Theorem 11. Let 𝑎 = 𝑎0 , 𝑎1 , 𝑎2 , . . . be a balanced strictly periodic sequence; if the connection integer 𝛼 satisfies 1/3 ≤ |𝛼| ≤ 3/5, then its correspondence 2-adic integer, its period 𝑇, and the 2-adic complexity 𝜓(𝑎) satisfy 𝑇(
Connection integer Q
∏𝑇−1 ((𝑝(𝑞−𝑝)2−𝑡 −(𝑞−𝑝)𝑞𝑛 )/(𝑝(𝑞−𝑝)2−𝑡 −𝑝𝑞𝑛 )) 𝑡 𝑡 + log2 𝑡=0
Initialization cycles 1152 1146 182 182
2-adic complexity of the balances sequences (satisfies Theorem 11)
𝑡=0
𝑝/(𝑞−𝑝) = 𝑇 log2
Total registers 321 344 420 506
(50)
(51)
From Lemmas 5 and 10, we have 𝑇 (𝑞 − 1) 𝑞 𝑝/(𝑞−𝑝) 1 𝑞 1 . < 𝑞 ( log2 + ) + 𝑇 log2 4 𝜋 6 𝜋 As 𝛼 = −𝑝/𝑞, then we get the conclusion
(52)
1 𝜋 𝛼/(1−𝛼) ) < 𝜓 (𝑎) + . (53) − log 4 2 6 Theorem 11 needs the connection integer 𝛼 to satisfy 1/3 ≤ |𝛼| ≤ 3/5. 𝑇(
Note that the sequences of which the second half of one period is the bitwise complement of the first half are also the balanced sequences but their 2-adic complexities do not correspond to this result. The 2-adic complexity of these sequences with the analysis in this article inconformity is due to the bit proportion and distribution in a period of sequences, and it is well known that their 2-adic complexity is smaller than their half period because of the bitwise complement. However, through the experiment (Figure 2), their 2adic complexity (except the long sequences) is approximated with their half period but we have not got faithful and accurate proving to analyze this result.
Wireless Communications and Mobile Computing
4. Conclusion The vigorous development of fog calculation is increasing the security requirements on it. Stream ciphers are undoubtedly the most suitable (see Table 1) among the nodes in the situation of lightweight security encryption, and the security of the stream cipher directly affects the communication security of the fog computing nodes (see Table 2). In this correspondence, lower bounds of the 2-adic complexity of binary periodic sequences are presented, and they are influenced by the length of encryption sequences in fog computing. However, the tighter lower bounds are not determined, so better results are desirable.
Conflicts of Interest The authors declare that they have no conflicts of interest.
References [1] P. Mahajan and A. Sachdeva, “A study of encryption algorithms, de sand for security,” Global Journal of Computer Science and Technology, vol. 13, no. 15, pp. 15–22, 2013. [2] M. Goresky and A. Klapper, “Arithmetic crosscorrelations of feedback with carry shift register sequences,” Institute of Electrical and Electronics Engineers Transactions on Information Theory, vol. 43, no. 4, pp. 1342–1345, 1997.
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