Abstract
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. AES-based 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 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 -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.
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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 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
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 -stages shift register and its auxiliary memory contained nonnegative integer. Assume an odd integer has the binary representation as . Then the -stages connections of FCSR are given by the bits . The FCSR with connection integer is described as follows:(1)Take an integer sum .(2)Shift the contents one step to the right, outputting the right bit .(3)Place into the left most cell of the shift register.(4)Replace the memory integer with .
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.
A 2-adic integer is form power series , with , and a fact is that number is represented by . Then, the negative integer is associated with the product
Moreover, the multiplication of 2-adic integer also has unique inverse if the integer is an odd integer. Thus the 2-adic integer contains every rational number , provided is odd.
Proposition 1 (see [2]). There is a one-to-one correspondence between rational numbers (where is odd) and eventually periodic binary sequences . We define the rational number as the 2-adic expansion of the binary sequences . The sequence is strictly periodic if and only if and .
If a strictly sequence is generated by an FCSR with connection integer , then the 2-adic integer of binary sequence has the following association.
Proposition 2 (see [2]). Let a periodic sequence be generated by an FCSR with connection integer and the 2-adic representation of sequence is . Then one hasFrom the above description about 2-adic integer and FCSR, the 2-adic complexity of periodic sequence can be regarded as .
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.
3. Main Results
In this section we mainly prove Theorem 7, and some lemmas are given to support the main result proof.
Lemma 3 (see [2]). Suppose a periodic sequence is generated by an FCSR with connection integer . Let be the (multiplicative) inverse of in the ring of integer modulo . Then there exists such that, for all , one has .
In this paper, we just consider the balanced binary strictly periodic sequence. Then the sequence in a period of length satisfies the fact that the number of even integers equals the number of odd integers. We assume that another sequence over in a period of length is bilateral symmetry with . In the following analysis of this paper, we introduce the exponential function as the tool to prove the main theorems. It is easy to get and . Since , we have
Lemma 4. Let the sequence over be a periodic sequence as described above; one has known that the sequence in a period of length of satisfies the following equation:
Proof. In a period of length of the sequence , the number of odd integers equals the number of even integers and is an even. When is an even integer, we assume over , and we haveSince , then we have for as an even integer. When is an odd integer, we get another with , andAs the variable ,Since , we haveThusThen, from the above analysis, we get
The sequences have a little limit in Lemma 4, and the sequence in a period of length satisfieswhen , and in a period of length satisfieswhen .
Lemma 5. For any positive integer , one has
Proof. If , we haveThat is,If , we haveThat is,Since and ,these are satisfyingNext we consider the formulaWe first have the inequalityIt follows thatThenThus we get the conclusion
Lemma 6. The binary strictly periodic sequence corresponds to the 2-adic integer , where the integer is odd and primitive with . The complement sequence of has the 2-adic representation .
Proof. From the description about the 2-adic integer, we have known that and the sequence have the 2-adic representation . The complement sequences and satisfy . Then, we have ; that is, .
Theorem 7. Let be a binary balanced period sequence with period , is the exponential of the sequence , the elements in are symmetry with , and the 2-adic integer of sequence has the property
Proof. Let sequence with the period and corresponding sequence as the described sequence ; we haveThe sequences have the following inequality:ThenWe haveThat is, are defined as the even numbers in a period of , are defined as the odd numbers in a period of , and then the inequality can be expressed asWe haveSincethenthat is,SoAs Lemma 3 , we have .
If the connection integer of balanced sequence is large enough and satisfies , we can have the following corollary.
Corollary 8. Let be a period balanced sequence with period , is the exponential of the sequence , and the elements in are symmetry with , then the 2-adic complexity of sequence has
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.
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 be the balanced strictly periodic sequence of a binary sequence and correspond to the 2-adic integer . Assume that the sequence with . Then, there must exist another sequence over satisfying
Proof. When is an odd integer,When is an even integer,The sequence in a period of length has the same number of even integers and odd integers.
If and have a minimum value, then it can be arrived at the minimum value when are bilateral symmetry with . Sowhere the sequence in a period of length of is also bilateral symmetry with .
Lemma 10. Let the binary strictly periodic sequence be generated by an FCSR with connection integer and the correspondence 2-adic integer is . Then, the sequence satisfies
Proof. From the definition of , we haveThen, when is an even integer, , and when is an odd integer, .
We have known thatAs is large integer, we getThen we have
From Lemma 3, is the complement with the binary sequence . Since the one-to-one correspondence between the binary sequence and 2-adic integer, we have .
We have , , and . ThenSo we haveThus, we get the conclusion
Theorem 11. Let be a balanced strictly periodic sequence; if the connection integer satisfies , then its correspondence 2-adic integer, its period , and the 2-adic complexity satisfy
Proof. From Lemmas 3 and 9, we have known thatFrom Lemmas 5 and 10, we haveAs , then we get the conclusionTheorem 11 needs the connection integer to satisfy .
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 2-adic complexity (except the long sequences) is approximated with their half period but we have not got faithful and accurate proving to analyze this result.
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.