**5. Cryptanalysis of cross-layer scheme**

The cryptanalysis will be performed separately at the application and physical layers and later combined in the cross-layer scheme to demonstrate the high security level of the new scheme compared to separate implementations.

## **5.1 Cryptanalysis at the application layer**

The RSA public key cryptography is implemented at the application layer. The most successful method to break the RSA cryptosystem is the Number Field Sieve (NFS) method used for partial key exposure attacks. The NFS is based on a method known as "Fermat Factorization": one tries to find integers x, y, such that x2 � <sup>y</sup><sup>2</sup> mod n but x 6¼ � y mod n [12]. We assume that the two primes p and q should be close and approximately equal to the square root of n, where n = p.q. If one of the integers could be written as x = (p + q)/2 then number of steps, S1 required to determine the other integer, y could be computed as follows [23].

$$\mathbf{S}\_1 = \frac{\mathbf{p} + \mathbf{q}}{2} - \sqrt{\mathbf{n}} = \frac{\left(\sqrt{\mathbf{q}} - \sqrt{\mathbf{p}}\right)^2}{2} = \frac{\left(\sqrt{\mathbf{n}} - \mathbf{p}\right)^2}{2\mathbf{p}}\tag{10}$$

*<sup>S</sup>*<sup>2</sup> <sup>¼</sup> *<sup>p</sup>: <sup>q</sup>:*2*<sup>k</sup>*

< :

*DOI: http://dx.doi.org/10.5772/intechopen.82390*

10 � 8 � 1

*<sup>S</sup>* <sup>¼</sup> *<sup>S</sup>*1*: <sup>p</sup>: <sup>q</sup>:*2*<sup>k</sup>*

only attacks for different cascaded stages.

< :

**Table 6.**

**25**

� 2*:*28 10 � 8! 10 � 1 10 � 8 2 � �<sup>2</sup>

2-cascaded cryptosystem using ciphertext-only attack.

**5.3 Cryptanalysis of the new cross-layer security scheme**

security scheme should be a product of S1 and S2 given as [12].

� *<sup>q</sup>:*2*<sup>k</sup> p* � *k*! *p* � 1 *p* � *k* 2 � �<sup>2</sup>

**Operand key length Total number of steps** 8-bit 7.8 � <sup>10</sup><sup>34</sup> 16-bit 2.1 � <sup>10</sup><sup>57</sup> 32-bit 1.4 � 1057 64-bit 1.5 � <sup>10</sup><sup>62</sup> 128-bit 6.1 � 1071 256-bit 9.87 � <sup>10</sup><sup>87</sup> 512-bit 2.68 � 10112 1024-bit 1.42 � <sup>10</sup><sup>134</sup>

" # & ' *<sup>N</sup>* 8

**Table 7** gives a summary of the number of steps required to break the new cross-layer security scheme by using partial key exposure attack and ciphertext-

Comparing **Tables 5–7**, it can be seen that high security levels comparable to the traditional 1024-bit RSA implemented at the upper layer could be attained using short operand key lengths of 128 bits and 256 bits for cross-layer security

*p* � *k* � 1

*Number of steps required to break the (8, 8, 2) 2-cascaded cryptosystem.*

*<sup>S</sup>*<sup>2</sup> <sup>¼</sup> <sup>10</sup>*:* <sup>2</sup>*:*<sup>28</sup>

< : *p* � *k* � 1

*A New Cross-Layer FPGA-Based Security Scheme for Wireless Networks*

" # & ' <sup>2</sup> 8

� *<sup>q</sup>:*2*<sup>k</sup> p* � *k*! *p* � 1 *p* � *k* 2 � �<sup>2</sup>

For an (8, 8, 2) 2-cascaded cryptosystem, k = 8 and the least number of plaintext-ciphertext blocks required is p = 10 due to the number of rows and columns in the generator matrix. Assuming q = 2 states, S2 could be as

" # & ' *<sup>N</sup>* 8

**Table 6** gives a summary of the number of steps required to break the (8, 8, 2)

At the upper layer, huge key lengths such as 1024 bits and 2048 bits are used to implement the RSA. Such implementations will greatly compromise throughput at the physical layer due to modular exponentiation. Hence, the main objective of the new cross-layer security scheme is to increase security level at the physical layer despite the small valued data points transmitted derived from the RNS-based RSA in order to enhance throughput. Cryptanalysis is performed on the small residue RSA encrypted values. The analysis will be based on partial key exposure and ciphertext-only attacks at the physical layer for eavesdropper who could wiretap the transmitted data. The number of steps, S required to break the new cross-layer

� <sup>2</sup><sup>2</sup>

9 =

� 22

9 = ;

(13)

� 22

9 = ;

; <sup>¼</sup> <sup>7</sup>*:*<sup>8</sup> � <sup>10</sup><sup>34</sup> (12)

(11)

It is partial key exposure attack since the number of steps, S1 required for the attack depends on one of the primes.

**Table 5** gives a summary of the number of steps required to break the traditional RSA cryptography implemented at the application layer using Fermat Factorization.

#### **5.2 Cryptanalysis at the physical layer**

Security at the physical layer is ensured by the multi-level convolutional cryptosystem which encrypts already encrypted data emanating from the RNS-based RSA. The cryptanalysis of the multi-level convolutional cryptosystem will be based on the ciphertext-only attack whereby, it is assumed that the attacker knows ciphertext of several messages encrypted with the same key and/or several keys. The keys used in the encryption are those mentioned in Section 2.1.2 for the nonlinear (8, 8, 2) 2-cascaded convolutional cryptosystem.

It is shown in [12] that, for an (n, k, L) convolutional code, each generator matrix reveals at most p – k – 1 values of a private parameter, using Gaussian elimination for p blocks of input data. Hence, if q is the number of states, then to completely break the (k, k, L) N-cascaded cryptosystem, the minimum number of plaintext-ciphertext pairs (u, v) required is [12].


**Table 5.**

*Number of steps required to break the traditional RSA.*

*A New Cross-Layer FPGA-Based Security Scheme for Wireless Networks DOI: http://dx.doi.org/10.5772/intechopen.82390*

$$\mathcal{S}\_2 = \left\{ \left[ p. \left[ \frac{q.2^k}{p - k - 1} \times \frac{q.2^k}{p} \times \frac{k!}{p} \times \frac{1}{p} \times \left( \frac{k}{2} \right)^2 \times 2^2 \right] \right]^N \right\} \tag{11}$$

For an (8, 8, 2) 2-cascaded cryptosystem, k = 8 and the least number of plaintext-ciphertext blocks required is p = 10 due to the number of rows and columns in the generator matrix. Assuming q = 2 states, S2 could be as

$$S\_2 = \left\{ \left[ 10. \left[ \frac{2.2^8}{10 - 8 - 1} \times \frac{2.2^8}{10} \times \frac{8!}{10} \times \frac{1}{10} \times \left( \frac{8}{2} \right)^2 \times 2^2 \right] \right]^2 \right\} = 7.8 \times 10^{34} \tag{12}$$

**Table 6** gives a summary of the number of steps required to break the (8, 8, 2) 2-cascaded cryptosystem using ciphertext-only attack.

#### **5.3 Cryptanalysis of the new cross-layer security scheme**

At the upper layer, huge key lengths such as 1024 bits and 2048 bits are used to implement the RSA. Such implementations will greatly compromise throughput at the physical layer due to modular exponentiation. Hence, the main objective of the new cross-layer security scheme is to increase security level at the physical layer despite the small valued data points transmitted derived from the RNS-based RSA in order to enhance throughput. Cryptanalysis is performed on the small residue RSA encrypted values. The analysis will be based on partial key exposure and ciphertext-only attacks at the physical layer for eavesdropper who could wiretap the transmitted data. The number of steps, S required to break the new cross-layer security scheme should be a product of S1 and S2 given as [12].

$$\mathcal{S} = \mathcal{S}\_1 \cdot \left\{ \left[ p. \left[ \frac{q. 2^k}{p - k - 1} \times \frac{q. 2^k}{p} \times \frac{k!}{p} \times \frac{1}{p} \times \left( \frac{k}{2} \right)^2 \times 2^2 \right] \right]^N \right\} \tag{13}$$

**Table 7** gives a summary of the number of steps required to break the new cross-layer security scheme by using partial key exposure attack and ciphertextonly attacks for different cascaded stages.

Comparing **Tables 5–7**, it can be seen that high security levels comparable to the traditional 1024-bit RSA implemented at the upper layer could be attained using short operand key lengths of 128 bits and 256 bits for cross-layer security


#### **Table 6.**

*Number of steps required to break the (8, 8, 2) 2-cascaded cryptosystem.*

**5. Cryptanalysis of cross-layer scheme**

*Computer and Network Security*

**5.1 Cryptanalysis at the application layer**

*<sup>S</sup>*<sup>1</sup> <sup>¼</sup> <sup>p</sup> <sup>þ</sup> <sup>q</sup>

linear (8, 8, 2) 2-cascaded convolutional cryptosystem.

plaintext-ciphertext pairs (u, v) required is [12].

*Number of steps required to break the traditional RSA.*

**Table 5.**

**24**

attack depends on one of the primes.

**5.2 Cryptanalysis at the physical layer**

The cryptanalysis will be performed separately at the application and physical layers and later combined in the cross-layer scheme to demonstrate the high secu-

The RSA public key cryptography is implemented at the application layer. The most successful method to break the RSA cryptosystem is the Number Field Sieve (NFS) method used for partial key exposure attacks. The NFS is based on a method known as "Fermat Factorization": one tries to find integers x, y, such that x2 � <sup>y</sup><sup>2</sup> mod n but x 6¼ � y mod n [12]. We assume that the two primes p and q should be close and approximately equal to the square root of n, where n = p.q. If one of the integers could be written as x = (p + q)/2 then number of steps, S1 required to

> ffiffiffi <sup>q</sup> <sup>p</sup> � ffiffiffi p � � p <sup>2</sup>

It is partial key exposure attack since the number of steps, S1 required for the

**Table 5** gives a summary of the number of steps required to break the traditional RSA cryptography implemented at the application layer using Fermat Factorization.

Security at the physical layer is ensured by the multi-level convolutional cryptosystem which encrypts already encrypted data emanating from the RNS-based RSA. The cryptanalysis of the multi-level convolutional cryptosystem will be based on the ciphertext-only attack whereby, it is assumed that the attacker knows ciphertext of several messages encrypted with the same key and/or several keys. The keys used in the encryption are those mentioned in Section 2.1.2 for the non-

It is shown in [12] that, for an (n, k, L) convolutional code, each generator matrix reveals at most p – k – 1 values of a private parameter, using Gaussian elimination for p blocks of input data. Hence, if q is the number of states, then to completely break the (k, k, L) N-cascaded cryptosystem, the minimum number of

**Operand key length Total number of steps**

16-bit 1 32-bit 1 64-bit 1.8 � <sup>10</sup><sup>8</sup> 128-bit 8.0 � <sup>10</sup><sup>17</sup> 256-bit 1.26 � 1025 512-bit 2.53 � <sup>10</sup><sup>63</sup> 1024-bit 3.3 � <sup>10</sup><sup>140</sup>

2 ¼

ffiffiffi n p ð Þ � p

2

2 p (10)

rity level of the new scheme compared to separate implementations.

determine the other integer, y could be computed as follows [23].

<sup>2</sup> � ffiffiffi <sup>n</sup> <sup>p</sup> <sup>¼</sup>


**Table 7.**

*Number of steps required to break the cross-layer security scheme.*

implemented at the physical layer. It is worth noting that, the security level could be much higher compared to the values displayed in **Table 7** if the S-boxes were implemented using 4-bit and 8-bit shuffling instead of the aforementioned 2-bit shuffling.
