**3.1 Illustrative example**

Consider an arbitrary array of integers for plaintext as follows {398, 453, 876, 200, 356, 165, 265, 897}.

• Source

has exactly one solution modulo the product m1, m2, …, mN. The solution to the

(Eq. (8)) establishes the uniqueness of the solution. In this research, the integers

by solving the linear congruence in (Eq. (7)). In other words, if Mj = M/mj, then the multiplicative inverses, Tj are computed by solving the congruence TjMj = 1

• Security services achieved using RSA cryptography are authentication and non-repudiation based on digital signatures and confidentiality based on encryption. Implementation of these services are summarized in **Table 3**

The new scheme is implemented at the application and physical layers. The detail operations of the application and physical layers at the source and destination

a. Step 1: Residue number system (RNS) converts the message points into

residues based on the moduli set;

∣ often referred to as the multiplicative inverses of m^ <sup>j</sup> are computed a priori

� � � � mj *:* xj � � � � � M

Choose plaintext X. Use Bob's public key (m, e) to

compute <sup>C</sup> � <sup>X</sup><sup>e</sup> (mod m). Send ciphertext, C to Bob

(8)

simultaneous linear congruence is formally given as [15, 17, 18].

**Authentication and non-repudiation Confidentiality**

where <sup>m</sup>^ <sup>j</sup> <sup>¼</sup> <sup>M</sup>

Choose plaintext X. Compute Xs � <sup>X</sup><sup>d</sup> (mod m)

*Computer and Network Security*

message, X

**Table 3.**

).

• Key creation

are as follows:

• Source:

**18**

◦ Application layer:

◦ Physical layer:

Tj <sup>¼</sup> <sup>∣</sup> xj m^ <sup>j</sup>

(mod mj

mj

**2.3 RSA public-key cryptography**

◦ Choose encryption exponent, e

◦ Public key: (m, e) and Private key: d

a. Traditional RSA encryption

and M <sup>¼</sup> <sup>Q</sup>

Send (X, Xs) to Alice. Xs is the RSA digital signature of

*Summary of security services of RSA public-key cryptography.*

N j¼1 mj.

◦ Choose secrete primes p and q and compute m = p.q

◦ Compute d satisfying e.d � 1 mod ((p – 1). (q – 1)).

**3. Protocol of implementation of cross-layer security scheme**

j j x <sup>M</sup> ¼ ∑

L j¼1 m^ j xj m^ j � � � �

� � � � �

◦ Application layer: Traditional RSA encryption

a. Primes, p = 13; q = 37 ) n = p. q = 481

b.Encryption key, e = 5

c. Decryption key: e. d � 1 mod (432) ) d = 173

Array due to RSA encryption is given as {151, 293, 252, 135, 304, 315, 265, 182}

	- a. Step 1: Moduli set of {107, 109, 113} is used to convert the RSA ciphertext into 8-bit data point arrays. The residue set, r1 for m1 = 107 is as follows:
	- r1 = {44, 79, 38, 28, 90, 101, 51, 75}
		- b.Step 2: Subband coding is performed to split residues obtained using moduli set into three levels of decomposition since m = 8 = 2<sup>3</sup> data points are used. Subband coding is basically down sampling by 2 using (Eq. (3)). The corresponding arrays for the first level of decomposition are as follows:

r11 = {�9, �48, �79, �27}; � r12 = {�9, �42, �75, �21}; � r13 = {�9, �30, �67, �9}

Note that r11 refers to first level of decomposition array for modulus, m1 = 107 and the first element is obtained using (Eq. (3)) with integer lifting filter coefficients set, h = {2, 0, 0} as follows: r1(1) – 2 � r1(0) = 79– 2 � 44 = �9.

The same procedure is performed for the second and third levels of decomposition.

c. Step 3: **Table 4** summarizes the manual computation of the encryption and decryption process of the convolutional cryptosystem for the data r11(0) = �9 from the subband encoding stage based on the entries of the product cipher and combinational logic of the non-linear (8,8,2) 2-cascaded convolutional transducer in **Figure 4**.

	- Physical layer:

The entire process is reversed starting with convolutional decoding at the physical layer which was illustrated in **Table 4**.

> a. Subband decoding: It involves upsampling and the use of quadrature mirror filter (QMF) bank as depicted in **Figure 5** [12, 13]

r31 from subband encoding namely a0 = 2 and d0 = 44 is used. From **Figure 5**, upsampling performed on the approximation, a0 and detail, d0 data points gives the sets y\_1 = {2, 0} and z\_1 = {44, 0} respectively. Using 4-tap Daubechies integer

> w\_1(0) = z\_1(0) – h(2)\*y\_1(0) – h(3)\*y\_1(1) = 44–0 – 0 = 44 w\_1(1) = z\_1(1) – h(2)\*y\_1(0) – h(3)\*y\_1(1) = 0–0 – 0=0

Hence the first level approximation data points, a1 are obtained as follows.

The process is repeated to obtain the second and third levels approximation data points. The third level approximation data points, a3 should be equal to residue set, r1 obtained from the RNS-based RSA ciphertext using the modulus,

b.RNS-based Chinese Remainder Theorem (CRT): It is applied to the residue sets

(Eq. (8))will be used to retrieve the RSA ciphertext from the residue sets. Using (Eq. (8)) and moduli set, m = {107, 109, 113} to compute the first data point of the

� ð Þ ð44 � 12*;* 317 � 9Þ þ ð42 � 12*;* 091 � 68Þ þ ð Þ 38 � 11*;* 663 � 33 ð Þ mod 1*;* 317*;* 919

The same process is repeated to obtain all the other data points of the RSA ciphertext set. The RSA ciphertext set is fed to the RSA decryption block at the

Decryption of the first data point is given as M = 151d mod n = 151<sup>173</sup> mod 481 = 398 which represents the original data which was sent at the source. The same process is repeated to obtain all the other data points of the

a\_1(0) = y\_1(0) – h(1)\*w\_1(0) = 2–2\*44 = �86 a\_1(1) = y\_1(1) – h(1)\*w\_1(1) = 0–2\*0 = 0

a1(0) = w\_1(0) = 44 and a1(1) = w\_1(1) + z\_1(0) = 0–86 = �86

X � ð Þ r11 � *m*^ <sup>1</sup> � T1 ð Þ r21 � *m*^ <sup>2</sup> � T2 ð Þ r31 � *m*^ <sup>3</sup> � T3 ð Þ mod M

◦ Application layer: RSA decryption

*The first two stages of the wavelet inverse transform.*

lifting filter coefficients set, h = {2, 0, 0} we have.

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

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

) a1 = {44, �86}.

r1, r2 and r3.

ciphertext set we have.

) X ¼ 151

application layer.

plaintext set.

**Figure 5.**

**21**

m1 = 107.

The (↑2) symbol represents upsampling by 2, which means that zeros are inserted between samples. The QMF bank are the coefficients derived from the 4-tap Daubechies filter bank [12, 13, 19]. Hence, using upsampling and the QMF bank coefficients the residue sets r1, r2 and r3 are retrieved.

**Figure 5** will be used to perform a numerical illustration of subband decoding for the first level of decomposition of the array of modulus m1 = 107 to obtain approximation coefficients a1.

The moduli sets obtained from subband encoding for the three levels of decomposition for m1 = 107 are as follows:


For subband decoding, the entire process is reversed with level 3 of encoding becoming level 1 for decoding.

r31 from subband encoding namely a0 = 2 and d0 = 44 is used. From **Figure 5**, upsampling performed on the approximation, a0 and detail, d0 data points gives the sets y\_1 = {2, 0} and z\_1 = {44, 0} respectively. Using 4-tap Daubechies integer lifting filter coefficients set, h = {2, 0, 0} we have.

> w\_1(0) = z\_1(0) – h(2)\*y\_1(0) – h(3)\*y\_1(1) = 44–0 – 0 = 44 w\_1(1) = z\_1(1) – h(2)\*y\_1(0) – h(3)\*y\_1(1) = 0–0 – 0=0 a\_1(0) = y\_1(0) – h(1)\*w\_1(0) = 2–2\*44 = �86 a\_1(1) = y\_1(1) – h(1)\*w\_1(1) = 0–2\*0 = 0

Hence the first level approximation data points, a1 are obtained as follows. a1(0) = w\_1(0) = 44 and a1(1) = w\_1(1) + z\_1(0) = 0–86 = �86 ) a1 = {44, �86}.

The process is repeated to obtain the second and third levels approximation data points. The third level approximation data points, a3 should be equal to residue set, r1 obtained from the RNS-based RSA ciphertext using the modulus, m1 = 107.

b.RNS-based Chinese Remainder Theorem (CRT): It is applied to the residue sets r1, r2 and r3.

(Eq. (8))will be used to retrieve the RSA ciphertext from the residue sets. Using (Eq. (8)) and moduli set, m = {107, 109, 113} to compute the first data point of the ciphertext set we have.

$$\begin{aligned} \mathbf{X} &\equiv (\mathbf{r}\_{11} \times \hat{m}\_1 \times \mathbf{T}\_1) \left( \mathbf{r}\_{21} \times \hat{m}\_2 \times \mathbf{T}\_2 \right) (\mathbf{r}\_{31} \times \hat{m}\_3 \times \mathbf{T}\_3) (\text{mod M}) \\ &\equiv ((44 \times 12, 317 \times 9) \quad + \ (42 \times 12, 091 \times 68) \quad + \ (38 \times 11, 663 \times 33)) (\text{mod } 1, 317, 919) \\ &\Rightarrow \mathbf{X} = \mathbf{151} \end{aligned}$$

The same process is repeated to obtain all the other data points of the RSA ciphertext set. The RSA ciphertext set is fed to the RSA decryption block at the application layer.

◦ Application layer: RSA decryption

Decryption of the first data point is given as M = 151d mod n = 151<sup>173</sup> mod 481 = 398 which represents the original data which was sent at the source.

The same process is repeated to obtain all the other data points of the plaintext set.

**Figure 5.** *The first two stages of the wavelet inverse transform.*

• Destination

◦ Physical layer:

*Computer and Network Security*

approximation coefficients a1.

becoming level 1 for decoding.

79, 27}.

**Table 4.**

**20**

position for m1 = 107 are as follows:

layer which was illustrated in **Table 4**.

The entire process is reversed starting with convolutional decoding at the physical

mirror filter (QMF) bank as depicted in **Figure 5** [12, 13] The (↑2) symbol represents upsampling by 2, which means that zeros are inserted between samples. The QMF bank are the coefficients derived from the 4-tap Daubechies filter bank [12, 13, 19]. Hence, using upsampling and the QMF

**Figure 5** will be used to perform a numerical illustration of subband decoding for the first level of decomposition of the array of modulus m1 = 107 to obtain

The moduli sets obtained from subband encoding for the three levels of decom-


For subband decoding, the entire process is reversed with level 3 of encoding

*Manual computation for the first sample of first level of decomposition for modulus 107.*

bank coefficients the residue sets r1, r2 and r3 are retrieved.

a. Subband decoding: It involves upsampling and the use of quadrature
