J. Nig. Soc. Phys. Sci. 4 (2022) 874

Journal of the
Nigerian Society

of Physical
Sciences

Application of hourglass matrix in Goldreich-Goldwasser-Halevi
encryption scheme

Olayiwola Babarinsaa,∗, Olalekan Ihinkalub, Veronica Cyril-Okemea, Hailiza Kamarulhailic, Arif
Mandangand, Azfi Zaidi Mohammad Sofie, Akeem B. Disuf

aDepartment of Mathematics, Federal University Lokoja, Kogi State, Nigeria
bDepartment of Computer Sciences, Federal University Lokoja, Kogi State, Nigeria

cSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia
dFaculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Malaysia
eFaculty of Bioengineering & Technology, Universiti Malaysia Kelantan, 16100 Kota Bharu, Malaysia

fDepartment of Mathematics, National Open University of Nigeria, Abuja, Nigeria

Abstract

Goldreich-Goldwasser-Halevi (GGH) encryption scheme is lattice-based cryptography with its security based on the shortest vector problem
(SVP) and closest vector problem (CVP) with immunity to almost all attacks, including Shor’s quantum algorithm and Nguyen’s attack of higher
lattice dimension. To improve the efficiency and security of the GGH Scheme by reducing the size of the public basis to be transmitted, we use an
hourglass matrix obtained from quadrant interlocking factorization as a public key. The technique of quadrant interlocking factorization to yield
a nonsingular hourglass matrix compensates the encryption scheme with better efficiency and security.

DOI:10.46481/jnsps.2022.874

Keywords: Goldreich-Goldwasser-Halevi encryption scheme, Hourglass matrix, Quadrant interlocking factorization.

Article History :
Received: 18 June 2022
Received in revised form: 25 July 2022
Accepted for publication: 27 July 2022
Published: 08 October 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license
(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: S. Fadugba

1. Introduction

Cryptography aims to achieve information security in confi-
dentiality, data integrity, authentication, and non-repudiation
[1]. Integer factorization problem (IFP), elliptic curve discrete
logarithm problem (ECDLP), and discrete logarithm problem
(DLP) are number theoretic hard problems established in cryp-
tographic schemes based on the hardness of their security, which

∗Corresponding author tel. no: +2348060032554
Email address: olayiwola.babarinsa@fulokoja.edu.ng (Olayiwola

Babarinsa )

is mostly deployed in Rivest-Shamir-Adleman (RSA), El-Gamal
and elliptic curve cryptosystems [2-4]. Notwithstanding, the
security goals of the schemes can be attacked by a powerful
algorithm, for instance, Shor’s quantum algorithm, to compute
the problems in less amount of time, see [5-7]. The immu-
nity of some lattice problems such as the shortest vector prob-
lem (SVP) and closest vector problem (CVP) against Shor’s
quantum algorithm is exploited by the idea behind lattice-based
cryptography [8,9]. The security of GGH cryptosystem relies
on the smallest-basis problem (SBP) and CVP [10].

The earliest lattice-based encryption scheme which was the

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Babarinsa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 874 2

most considered practical scheme is Goldreich-Goldwasser-Halevi
encryption scheme or GGH scheme [11]. The security of this
scheme lies in the hardness of the underlying GGH-CVP in-
stance which is proven to be NP-hard [12]. However, the se-
curity of GGH Scheme has been compromised by Nguyen’s
attack [13]. There are significant efforts to improve the effi-
ciency of the GGH Scheme, such as the application of Hermite
Normal Form (HNF) or Jensen-Based cryptographic scheme as
the public key [14, 15]. The GGH Scheme survives against
Nguyen’s attack when being implemented in lattice dimensions
above 400 [16]. However, the implementation of the scheme
in lattice dimensions beyond 400 immediately makes the GGH
Scheme inefficient, impractical, and uncompetitive compared
to other existing encryption schemes. This is due to the large
key sizes involve in the scheme once it is implemented in a large
lattice dimensions. This is because the keys of this scheme are
bases of lattice which could be represented in matrices form.
That means the implementation of the scheme in a large lat-
tice dimension requires the submission of a large lattice ba-
sis as a public key from Alice to Bob. For instance, in a lat-
tice dimension of 400, the public basis can be represented as
a 400 × 400 matrix with 4002 = 160000 entries. The trans-
mission of a matrix with 160000 entries from Alice to Bob re-
quires a high computational cost. By transforming the underly-
ing GGH-CVP instance into its simpler form, Nguyen’s attack
successfully breaks the security of the GGH Scheme when be-
ing implemented in lattice dimensions smaller than 400. From
the simplified GGH-CVP instance, Nguyen’s attack derives an
easier SVP instance which could be solved by using lattice re-
duction methods such as LLL and BKZ algorithms. In low lat-
tice dimension, these algorithms efficiently work for solving the
derived SVP-instance which makes Nguyen’s attack succeeds.
As the lattice dimension increases, the efficiency of these algo-
rithms declines significantly. Consequently, the attack failed to
break the security of the GGH Scheme in a lattice dimension of
400 and above [17].

To systematically reduce the number of non-zero elements
in the public basis while maintaining all the required properties
of the public basis, especially the linear independency and or-
thogonality properties, we use a nonsingular hourglass matrix
H as a public basis and its corresponding factorization matrix R
as a private basis of the GGH Scheme. These matrices are re-
lated as R = U H where U is unimodular matrix. Section 2 gives
the detail of an hourglass matrix and its factorization algorithm,
while Section 3 entails the application of hourglass matrix and
its factorization technique in GGH encryption scheme.

2. Hourglass matrix

Babarinsa and Kamarulhaili [18] gave details on hourglass ma-
trix and its factorization algorithm by restricting the computed
entries of the factorization to be nonzero in comparison with
an hourglass device. They also suggested its applications in
mathematics, graph theory, statistics, and computer science, see
[19-24]. An hourglass matrix is defined as a nonsingular ma-
trix of order n (n ≥ 3) with nonzero entries from the ith to the
(n− i + 1) element of the ith and (n− i + 1) row of the matrix,

0’s otherwise for i = 1,2,...,bn+12 c [18]. Unlike Z-matrix with
nonzero restricted entries, hourglass matrix conforms with the
shape of an hourglass device, see Figure 1 which illustrates the
structural comparison between the hourglass device and hour-
glass matrix with nonzero elements denoted with black dots. To
buttress the shape of hourglass matrix, Figure 1

Figure 1. Structural comparison between hourglass device and hourglass ma-
trix.

For the factorization algorithm of hourglass matrix, we com-
pute w(k)i,k and w

(k)
i,n−k+1 from a dense square matrix R by solv-

ing 2×2 linear systems in equation (1) using Cramer’s rule to
generalize for every update of R to H and proceed similarly
for the inner square matrices of size (n −2k) and so on, for
k = 1,2,...,bn−12 c.{

h(k−1)k,k w
(k)
i,k −h

(k−1)
n−k+1,kw

(k)
i,n−k+1 = h

(k−1)
i,k

h(k−1)k,n−k+1w
(k)
i,k −h

(k−1)
n−k+1,n−k+1w

(k)
i,n−k+1 = h

(k−1)
i,n−k+1

(1)

Then we compute for kth steps of h(k)i, j as:

h(k)i, j = h
(k−1)
i, j + w

(k)
i,k h

(k−1)
k, j + w

(k)
i,n−k+1h

(k−1)
n−k+1, j (2)

where i, j = k + 1,...,n−k. From equation (2), if one of the
computed entries is zero, then apply possible row-interchange
in no more than (n−2k) times in H(k−1) and re-factorize, else
the factorization breakdown to produce H. From every suc-
cessful loop for each stage (with bn−12 c total stages in the fac-

torization), there are
bn2−1c

∑
k=1

(n−2k) of 2×2 linear systems to be

solved during the factorization using Cramer’s rule.
Hourglass matrix (H-matrix) is nonsingular and its W -matrix

is a unimodular matrix with det(W ) = (−1)Pn =±1, where Pn
is the number of permutation matrix in the factorization. Based
on the structure of hourglass matrix, the matrix could be poten-
tially used as the key (basis) in the GGH encryption scheme.
The usage of hourglass matrix is expected to be able to reduce
the size of bases, especially the public key. Almost half of the
entries of the hourglass matrix are zero entries, which means
the size of the public key can be reduced if the public key is
generated in the form of hourglass matrix. This reduction will
allow the GGH Scheme to be implemented in a higher lattice
dimensions while still being able to be efficient and practical.

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Babarinsa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 874 3

Hourglass matrix has linearly independent columns forming the
basis of a lattice, which makes it suitable for GGH scheme. The
fixed zero entries in hourglass matrix will not only minimize
the memory cache used but also reduce computational time. In
addition, the generation of hourglass matrix from QIF can be
executed in polynomial time.

3. GGH scheme with hourglass matrix

Consider (n,σ) ∈ N to be the security parameter, where n is
a lattice dimension and σ is a threshold parameter. Denote
~m,~e ∈ Z n as the message vector and error vector respectively,
where the entries of ~e are ei ∈{−σ,+σ}. Let i = 1,2,...,n,
then denote H as hourglass matrix such that H = [~h1,~h2,...,~hn],
where ~hi ∈ Z n are the column vectors of H, denote R as a non-
singular matrix such that R = [~r1,~r2,...,~rn] where~ri ∈Zn are the
column vectors of R, and denote U as a unimodular matrix such
that U = [~u1,~u2,...,~un], where ~ui ∈ Zn are the column vectors
of U and det(U) =±1.

Proposition 3.1. [22] Any two bases for a lattice L are re-
lated by a unimodular matrix U that has integer coefficients
and det(U) =±1.

Definition 3.1. [23] Let ~b1,~b2,...,~bn be n linearly independent
vectors of Rm with n ≤ m. The set of all integer linear combi-
nations of the vectors ~b1,~b2,...,~bn is called lattice and can be
denoted in the form

L (~b1,,...,~bn) =

{
n

∑
i=1

mi~bi|mi ∈Z

}
(3)

The linearly independent vectors ~b1,~b2,...,~bn form the columns
of the basis for the lattice L (B). Suppose that B,H ∈ Rn×n be
nonsingular square matrices with linearly independent vectors
~b1,~b2,...,~bn and ~h1,~h2,...,~hn as their columns respectively. The
lattice L(B) ⊂ Rn that is spanned by the basis B is defined as
follows [22].

L(B) =

{
n

∑
i, j=1

µi, j~bi | ~bi ∈ B and µi, j ∈Z, ∀i, j = 1,2,...,n

}
(4)

and the lattice L(H) ⊂ Rn that spanned by the basis and
the lattice L(B) ⊂ Rn that spanned by the basis B is defined as
follows

L(H) =

{
n

∑
i, j=1

τi, j~hi | ~hi ∈ H and τi, j ∈Z, ∀i, j = 1,2,...,n

}
(5)

To ensure that the bases B and H are spanning the same
lattice (i.e L(B) = L(H)), the matrix W is required to be a uni-
modular matrix with det(U) =±1.

The desired properties for the public and private basis are
as the following:

1. Both H and R
a) Two different bases span the same lattice L, i.e., L (R) =
L = L (H).
b) To be a lattice basis, both matrices H and R must sat-
isfy the following conditions:

i. The vectors {~h1,~h2,...,~hn} and {~r1,~r2,...,~rn} are
linearly independent, where the only solution for the fol-
lowing equations

α1~h1 + α2~h2 +...+ αn~hn =~0 (6)

and

β1~r1 + β2~r2 +...+ βn~rn =~0 (7)

are the trivial solutions, i.e., αi = 0 and βi = 0 ∀,i =
1,2,...,n.

ii. The vectors {~h1,~h2,...,~hn} and {~r1,~r2,...,~rn} span
the whole space Z, i.e.,

span{~h1,~h2,...,~hn}= Z (8)

and

span{~r1,~r2,...,~rn}= Z (9)

c) Since both H and R are spanning the same lattice L,
these bases have the same determinant, i.e., det(H) =
det(R).
d) Both H and R are mathematically related by unimod-
ular matrix U , as R = U H.

2. The public basis H
The column vectors {~h1,~h2,...,~hn} are long vectors from
the origin and highly nonorthogonal vectors where the
Hadamard ratio of the matrix H is far from 1 and closer
to 0.

3. The private basis R
a) The column vectors {~r1,~r2,...,~rn} are short vectors
from the origin and reasonably orthogonal vectors where
the Hadamard ratio of the matrix R is far from 0 and
closer to 1.
b) The rounding vector bR~ee=~0 to ensure the decryption
process succeeds.

Proposition 3.2. [19] Let B,H ∈ Rn×n be nonsingular square
matrices such that B = HW , where B is a basis for lattice L(B),
H is a basis for lattice L(H). If W is a unimodular matrix, then
L(B) = L(H) where W ∈Zn×n.

3.1. GGH Scheme algorithm using hourglass matrix

The algorithm of the GGH Scheme by using hourglass matrix
is as follows:

1. Key Generation by Alice (recipient)

• Sets the security parameters n,σ ∈ N where n is an
even number.

• Generates a non-singular n×n -matrix R with the
following properties:

a) Represent the matrix R as

R = [~r1 ~r2 ... ~rn] (10)

where the vectors~ri ∈Z are the columns of R. These
vectors are required to be linearly independent.

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Babarinsa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 874 4

b) Compute the Hadamard ratio of the matrix
R as

H (R) =


|det(R)|n

∏
i=1
‖~ri‖




1
n

(11)

where ‖~ri‖ is the Euclidean norm of the vectors ~ri.
The Hadamard ratio is required to measure the or-
thogonality of the vectors~ri. The closure the Hadamard
ratio to 1, the more orthogonal the vectors ~ri are.
We accept R to be the private basis if

H (R)∈ [0.7,1) (12)

to make sure that the private basis R is a reasonably
orthogonal basis.

• Computes the factorization of the matrix R as fol-
lows

R = U H

where U ∈Zn×n is a unimodular matrix with det(U) =
±1 and H ∈ Zn×n is an Hourglass matrix. For the
Hourglass matrix H,

a) Represent the matrix H as

H = [~h1 ~h2 ... ~hn] (13)

where the vectors ~hi ∈ Zn are the columns of H.
These vectors are required to be linearly indepen-
dent.

b) Since H and R are related by a unimodular
matrix U as R = U H, then

det(U H) = det(R) =±det(H) (14)

c) Compute the Hadamard ratio of the matrix
H as

H (H) =


|det(H)|n

∏
i=1
‖~hi‖




1
n

(15)

where ‖~hi‖ is the Euclidean norm of the vectors ~hi.
The closure of the Hadamard ratio to 0, the more
highly non-orthogonal the vectors ~hi are. We accept
H to be the public basis if

H (H)∈ (0,0.3] (16)

to make sure that the public basis H is a highly non-
orthogonal basis.

• Keeps the matrix R as her private basis and U as her
private matrix.

• Sends the Hourglass matrix H as her public basis to
Bob together with her security parameters n,σ ∈ N.

2. Encryption by Bob (sender)

• Sets the message as ~m =




m1
m2
...

mn


∈Zn

• Generates the error vector as ~e =




e1
e2
...

en


 where

ei ∈{±σ}
• Encrypts the message ~m as follows

~c = H~m +~e (17)

where ~c ∈Zn is the ciphertext vector.
• Sends the ciphertext ~c to Alice.

3. Decryption by Alice (recipient)

• Computes a vector T ∈Rn as follows

T = R−1~c (18)

• Rounds each entry of the matrix T to the nearest in-
teger, i.e., btie∈Z where ti ∈ T for all i = 1,2,...,n.

• Decrypts the ciphertext as follows
~M = UbTe (19)

where bTe is a matrix T with rounded entries. De-
cryption succeeds if

~M = ~m (20)

Now, consider the following scenario. Suppose that Bob
wants to send a secret message to Alice. As the recipient, Alice
generates the private basis R as a good basis. Then, she derived
the public basis H as R = U H. The public basis H is accepted if
the basis H is a highly non-orthogonal basis. Otherwise, the key
generation process will be re-initiated. Once the proper bases
R and H are completely generated, Alice sends the public basis
H, the security parameter {n,α} to Bob and keeps the other
basis and parameters secret. Upon receiving the information
from Alice, Bob encoded his secret message in a vector ~m ∈
Zn. Then, he generates the error vector ~e ∈{−α,α}n and then
proceeds to the encryption process which is done as~c = H~m+~e.
The ciphertext ~c ∈ Zn is then sent to Alice. Upon receiving ~c
from Bob, Alice proceed with the decryption process which can
be done by solving the underlying CVP instance using Babai’s
round-off method. She computes ~m = UbR−1~ce.

Proposition 3.3. Suitability of hourglass matrix in GGH scheme
is attainable if the encrypted message M is successfully de-
crypted to message m, such that

~M = ~m.

Proof. Since R = U H, then U = H−1R.
Thus,

~M = UbTe
= UbR−1~ce

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Babarinsa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 874 5

= UbR−1(H~m +~e)e
= UbR−1H~m + R−1~ee
=bU R−1H~m +U R−1~ee
=bH−1RR−1H~m +U R−1~ee
=b~m +U R−1~ee
=b~me+bU R−1~ee.

Since ~m ∈ Zn, bme = m. For the decryption to succeed, the
rounding vector is required to be bR−1~ee=~0. Therefore,

~M = ~m +UbR−1~ee

If all the entries of the rounding vector bR−1~ee are smaller than
1
2 , then we have bR

−1~ee=~0. Therefore, we have

~M = ~m +U~0

~M = ~m

3.2. A numerical example of hourglass matrix in GGH scheme
We consider a numerical example of 4×4 hourglass matrix in
GGH scheme. Let R, U, H be the hourglass matrix, unimodular
matrix, nonsingular matrix, error, and encrypted message to be
sent from Alice to Bob, and e, m are vectors.

H =




143 123 −211 103
0 −14 33 0
0 −14 −124 0
−211 −14 −122 213


, ~m =




5
6
7
8


, ~e =



−3
3
−3
−3


.

Then det(H) = 114,718,016 and the Hadamard ratio of H is
0.4495. Then U is given as

U =



−1 2 1 −2
2 −1 1 2
1 −1 1 2
1 1 1 −1




where det(U) = 1. Thus, R is computed as

R = H ·U =



−5 477 158 −565
5 −19 19 38
−152 138 −138 −276
274 −73 −134 −63




Vividly, det(R) = 114,718,016. Thus, the Hadamard ration
of R is 0.2606. Now compute H−1 as

H−1 =




0.00408 0.033828 0.004000 −0.001973
0 −0.056415 −0.015014 0
0 0.006370 −0.006370 0

0.00404 0.033451 −0.000673 0.002740




and

H−1 ·e =



−0.107163
0.214286

0
−0.077988




The round-off bH−1 ·ee gives a zero vector, ~0 =




0
0
0
0


. The

message m sent by Alice to Bob be

m =




5
6
7
8




R ·m =



−577
348
−3106
−510




c = R·m + e =



−574
345
−3109
−507




T = H−1c =



−2.107163
27.214286
22.000000
9.922012




The round-off bTe to nearest integers gives

bTe=



−2
27
22
10




HbTe=



−577
348
−3106
−510




The decryption M is

M = R−1HT =




4.9999999994
5.9999999780
6.9999999926
8.0000000102




The round-off decrypted message bMe by Bob is

bMe=




5
6
7
8




Hence, ~M−~m =~0.

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Babarinsa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 874 6

4. Conclusion

The advantage of the hourglass matrix as a public key has been
explored in the encryption scheme to enhance the efficiency of
the GGH. The orthogonal columns of hourglass matrix keep the
GGH encryption scheme efficient and practical by reducing the
number of non-zero elements on the public basis while main-
taining all the required properties of the public basis, especially
the linear independency and orthogonality properties. With bet-
ter security and efficiency, the scheme is expected to be highly
competitive in the post-quantum era. Due to the simplicity and
practicality that can be offered by the scheme, it may be widely
adopted for providing security in devices with small computing
capacity.

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