The document discusses several cryptosystems used for public-key encryption, including RSA, Diffie-Hellman key exchange, and elliptic curve cryptography. It provides background on necessary mathematical concepts like modular arithmetic, primes, and discrete logarithms. The author analyzes the security, efficiency, and ability to withstand large adversaries of each cryptosystem to determine the most effective for public-key encryption.

1. TEMPLETON SCHOLARSHIP PROJECT
What is the most effective
cryptosystem for public-key
encryption?
Author
Alex Wang
Institution
Wilson’s
School
July 2022

2. Contents
1 Introduction to cryptography 3
1.1 Symmetric Encryption . . . . . . . . . . . . . . . . . . . . . 3
1.2 Asymmetric Encryption . . . . . . . . . . . . . . . . . . . . 3
1.3 Trapdoor Functions: P vs NP . . . . . . . . . . . . . . . . . 4
2 Divisibility and Primes 5
2.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . 6
2.3 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Modular Arithmetic 8
3.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . 9
3.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 RSA 11
4.1 A history of RSA . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 Decryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.4 Prime Factorisation . . . . . . . . . . . . . . . . . . . . . . . 13
4.5 Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 The Diffie-Hellman Key Exchange 14
5.1 A history of the Diffie-Hellman Key Exchange . . . . . . . . 14
1

3. 5.2 The Key Exchange Protocol . . . . . . . . . . . . . . . . . . 14
5.3 The Discrete Logarithm Problem . . . . . . . . . . . . . . . 15
5.4 Attacks - The Pohlig-Hellman Algorithm . . . . . . . . . . . 15
6 Elliptic Curve Cryptography 16
6.1 What is an Elliptic Curve? . . . . . . . . . . . . . . . . . . . 16
6.2 The Key Exchange Protocol . . . . . . . . . . . . . . . . . . 19
6.3 The Elliptic Curve Discrete Logarithm Problem . . . . . . . 19
7 Comparing Cryptosystems 20
7.1 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7.3 Effect of Large Adversaries . . . . . . . . . . . . . . . . . . . 22
8 Conclusion 22
2

4. 1 Introduction to cryptography
Cryptography is the study of the methods used to allow secure communica-
tion between 2 parties on a network. Typically we refer to these 2 parties as
Alice and Bob. The original data is called the plaintext and the encrypted
data is called the ciphertext.
1.1 Symmetric Encryption
Symmetric encryption is a data encryption method where the same key is
used to encrypt and decrypt messages. Before the 1970s, it was the only form
of encryption that existed and was widely used. Due to the same key being
used for encryption and decryption, everyone involved in the data transfer
process must know (or agree upon) the key. For this reason, symmetric
encryption is generally used for so data at rest. This is when data is stored
and encrypted locally, with no need to be sent across a network. Examples
of this include confidential business data stored on a local server [22].
There are 2 main types of symmetric encryption: block ciphers and stream
ciphers [1]. Block ciphers encrypts data in fixed length blocks, such as 64
bit or 128 bit. the most widely used block cipher is the AES (Advanced
Encryption Standard). Stream ciphers use an additive cipher where each
bit is replaced by another bit from a secret keystream. As a result, the
ciphertext would be the same length as the plaintext. Stream ciphers are
implemented in the RC4 ND HC-256 protocols.
1.2 Asymmetric Encryption
Modern symmetric encryption is very secure: data encrypted with cryp-
tosystems such as the AES have a 256-bit key. However, there remains a
problem for sending data over the internet: how can 2 parties (who have
never met before) agree on a key?
Consider 2 parties called Alice and Bob. Alice wants to send a message
to Bob. Internet transmission is public channel, so Alice cannot send Bob
her key over the internet without others in the middle intercepting internet
traffic. The only way for Alice and Bob to agree on a symmetric key is to
use a private channel of communication – to meet up in person to exchange
the key. If Alice and Bob can only communicate over a public channel, they
3

5. never have an opportunity to agree on a secret key without others listening
in. If the key is publicly visible, then the symmetric encryption used would
be useless [22].
Therefore, we need another way to send encrypted data over the internet.
Before the 1960s, asymmetric encryption was thought to be impossible. All
cryptographic transmission was by symmetric encryption. Any two parties
who wanted to communicate secretly must have exchanged keys privately,
perhaps through a courier. However, in the 1970s, several cryptosystems
were developed independent of each other to deal with this problem.
To develop an asymmetric cryptosystem, we need two keys: a public key and
a private key. In general, the key used to encrypt the message is public, and
the key used to decrypt the message is private. The encryption algorithms
use one-way functions (also known as trapdoor functions) which only remain
one-way if the decryption key is kept private. The encryption function can
be computationally easily applied to cipher-text, but its inverse function
(used for decryption) cannot be easily computed, without the decryption
key.
1.3 Trapdoor Functions: P vs NP
A trapdoor function is a function that is difficult to compute in the opposite
direction (by finding the inverse function), without specific information such
as the decryption key.
The analogy to a ”trapdoor” comes from the fact that it is easy to fall
through a trapdoor, but it is difficult to climb back out, unless you have a
ladder (which represents the decryption key).
A trapdoor function a problem is in the class NP (nondeterministic, poly-
nomial time). These problems are characterised by the fact that it is easy
to check the solution to an NP problem, but it is difficult to compute the
problem. For an asymmetric encryption algorithm, it is easy to decrypt
the algorithm if you have the decryption key (checking the answer to an
NP problem), but it is difficult to solve the inverse function by brute force
search [13].
As the size n of an NP problem increases, the computational resources re-
quired to check the solution increases in proportion to a polynomial function
of n, such as n2
. However, the computational resources required to compute
the answer increases in proportion to an exponential function of n: typically
4

6. 2n
. Exponential functions increase much more rapidly than polynomial func-
tions and so NP problems can become unsolvable when n is sufficiently large.
Figure 1: 2n
function (red) vs n2
function (black)
2 Divisibility and Primes
2.1 Divisibility
We can say that an integer b is divisible by an integer a if there is an integer
n such that b = an.
We can write ”a divides b” as a|b.
Properties of divisibility: [6]
1. a|b → a|bc c ∈ Z
2. a|b & b|c → a|c
3. a|b & a|c → a|(bx + cy) x, y ∈ Z
4. a|b & b|a → a = ±b
5. a|b, a > 0, b > 0 → a ≤ b
6. m ̸= 0, a|b ↔ ma|mb
5

7. The Division Algorithm:
The division algorithm divides one integer b by another integer a which
produces a quotient q and remainder r
Let a, b ∈ N
There exists q, r ∈ N such that b = qa + r, 0 ≤ r < a.
Greatest Common Divisor:
A common divisor of a and b is an integer n such that n|a and n|b. The
greatest of the common divisors is called the greatest common divisor of
a and b. This is often written as gcd(a, b).
2.2 The Euclidean Algorithm
The Euclidean Algorithm is an algorithm used to efficiently compute the
greatest common divisor of 2 integers, without factoring them. It is used in
many applications such as breaking the RSA cryptosystem. [9]
Prime factorisation of numbers is an computationally inefficient process. A
computer must try to divide an integer by prime numbers until it finds a
prime divisor. Therefore, finding the greatest common divisor of 2 integers
using prime factorisation is inefficient and so we use the Euclidean Algorithm
instead.
The Euclidean Algorithm applies the division algorithm in a recursive pro-
cess.
Given a, b ∈ Z, b > 0 find d = gcd(a, b).
Euclidean Algorithm Pseudocode:
a = x
b = y
while y ̸= 0
x = yq + r, 0 ≤ r < y
x = y
y = r
return x
6

8. Euclidean Algorithm Logic:
a = bq1 + r1 0 < r1 < b (1)
b = r1q2 + r2 0 < r2 < r1 (2)
r1 = r2q3 + r3 0 < r3 < r2 (3)
... (4)
rj = rj+1qj + rj 0 < rj < rj−1 (5)
rj−1 = rjqj+1 + 0 (6)
The greatest common divisor of a and b is rj, the last nonzero remainder in
the division algorithm.
Example gcd(745, 214)
745 = 214 × 3 + 103
214 = 103 × 2 + 8
103 = 8 × 12 + 7
8 = 7 × 1 + 1
7 = 1 × 7 + 0
(7)
Hence gcd(745, 214) = 1, which means that they are coprime.
2.3 Primes
Prime numbers are the building blocks of all natural numbers. Every integer
can be expressed as a product of primes, in a unique way. Prime numbers
have a special property called a trap-door function. It is very easy to find a
number, given its prime factors, however, it is much more difficult to find all
the prime factors of a given number. This principle is used in cryptosystems
to create a locking mechanism, allowing a message to be easily decrypted
with a key, but very difficult to decrypt by a brute force attack.
An integer p ∈ N is a prime number if there is no divisor d ∈ N of p where
1 < d < p. If an integer is not a prime, then it is a composite number.
The list of prime numbers seems to behave like a random list of numbers.
It is very difficult to say when the next prime will be, or whether a given
number is prime. There are models, however, to predict the behaviour and
density of primes. For example, one model is that the approximate density
of primes near any number is inversely proportional to how many digits the
number has.
7

9. Fundamental Theorem of Arithmetic:
The Fundamental Theorem of Arithmetic states that every integer n ∈ N
can be expressed uniquely as a product of primes.
Given an integer n, we can write n as a product of its prime factors:
n = pa1
1 pa2
2 ...par
r
where p1, p2, ..., pr are distinct primes and a1, a2, ..., ar > 0
Infinitude of Primes
The number of primes is infinite. This has been proven by many mathe-
maticians such as Euclid, Furstenberg and Goldbach [10]. Euclid’s theorem
is one of the more famous proofs for the infinitude of primes. It was proven
by Euclid in his work ”Elements”.
Euclid’s Theorem:
1. Assume the number of primes is finite. We have a finite list of primes:
p1, p2, ..., pn
2. Consider the product P of all the primes: P = p1p2...pn
3. Now consider the number P + 1. P + 1 must be either a prime number
or a composite number.
4. If P + 1 is prime, then we have another prime not in the finite list.
5. If P +1 is composite, then it must be divisible by some prime p. However,
P and P +1 are coprime, so they do not share any prime factors. Therefore,
the prime p cannot be in the finite list.
6. Both these options lead to a contradiction, proving that the number of
primes is not finite.
3 Modular Arithmetic
Modular arithmetic is a system of integer arithmetic, which considers the
remainder of a number after division by another number.
A common use of modular arithmetic is in a 24 hour clock system. If it is
15:00 now, then after 25 hours the time will be 16:00, but not 40:00. We
can write this as 40 ≡ 16 (mod 24).
8

10. 3.1 Congruences
If we have m ∈ N, m ̸= 0 such that m|(a−b), we say that a is congruent to
b modulo m. This means that they leave the same remainder upon division
by m [18]. We can write this as a ≡ b (mod m)
Properties of Congruences:
1. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m)
2. If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m)
3. If a ≡ b (mod m) and d|m, d > 0, then a ≡ b (mod d)
4. If a ≡ b (mod m) then ac ≡ bc (mod mc) for c > 0
3.2 The Chinese Remainder Theorem
The Chinese Remainder Theorem is used to solve simultaneous congruences,
where the moduli are coprime:
Find x that satisfies:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
...
x ≡ ar (mod mr)
(8)
Let m1, ..., mr represent r positive integers (moduli) that are coprime, and
let a1, ..., ar ∈ Z. Then the congruences have common solutions. If x0 is one
solution, then an integer x satisfies the congruences if and only if x = x0+km
for some integer k and where m = m1m2...mr [20].
For any j between 1 and r, we see that m
mj
∈ Z and that ( m
mj
, mj) = 1 as the
moduli are coprime. Therefore, for each j there is an integer bj such that
( m
mj
, mj)bj ≡ 1 (mod mj), and so ( m
mj
, mj)bj ≡ 0 (mod mi) if i ̸≡ j. Then:
x0 =
r
X
j=1
m
mj
bjaj
9

11. We can consider this number in modulo m.
x0 ≡
m
mi
biai ≡ ai (mod mi)
Therefore, x0 is a solution of the system (8). If x0 and x1 are two solutions
of the system (8), then x0 ≡ x1 (mod mi) for all the moduli and so x0 ≡
x1 (mod m) [20].
3.3 Rings
In modular arithmetic we work modulo some modulus m, and we can think
of all the possible remainders as a complete residue system:
Z/mZ = {0, 1, ..., m − 1} (9)
The ring Z/mZ is the ring of integers modulo m. We can to add and multiply
elements of this ring, then consider the result modulo m to obtain an element
in Z/mZ. This satisfies the closure condition for a group under an addition
or multiplication binary operation in modulo m [8].
Units of a Ring:
An element a ∈ Z/mZ has a a modular multiplicative inverse if and only if
gcd(a, m) = 1. The set of all numbers with modular inverses are denoted as
(Z/mZ)∗.
This is called the group of units modulo m; numbers which have inverses
are called units. A unit must be coprime with the modulus m [8]. This is
also sometimes called a reduced residue set modulo m.
For example:
(Z/15Z)∗ = {1, 2, 3, 4, 7, 8, 11, 13, 14} (10)
3.4 Fields
If every element in Z/mZ has a modular inverse, then it is classed as a
field [8]. A field is a ring in which division is possible and is usually denoted
as Fp. p represents the modulus as the only possible values for the modulus
of a field are prime numbers. A finite field (also known as a Galois field) is
a field with finite elements, such as those used in modular arithmetic.
10

12. Euler’s Totient Function:
Euler’s totient function returns the number of elements in the group of units
modulo m [14]. This can also be thought of as the number of natural numbers
smaller than m that are also coprime with m. Typically the symbol ”phi”
(ϕ) is used to denote the function:
ϕ(m) = #(Z/mZ)∗ = #{0 < a < m : gcd(a, m) = 1} (11)
To compute the Totient function of an integer n, it must first be decomposed
into its primes:
n = pa1
1 pa2
2 ...par
r (12)
where p1, p2, ..., pr are distinct primes and a1, a2, ..., ar > 0.
Now the following formula is used to compute the Totient function:
ϕ(n) = n
Y
p|n
(1 −
1
p
) (13)
Euler’s Formula:
If gcd(a, p) = 1 then:
aϕ(p)
≡ 1(mod p) (14)
Fermat’s Little Theorem:
ap−1
≡ 1(mod p) (15)
Euler’s formula is a special case of Fermat’s Little Theorem as ϕ(p) = p − 1.
4 RSA
RSA is the most widely used cryptosystem to this day, and it makes use
of some elementary principles in number theory, including Fermat’s Little
Theorem.
11

13. 4.1 A history of RSA
The RSA cryptosystem was invented by MIT colleagues Ron Rivest, Adi
Shamir and Leonard Adleman in 1977. Clifford Cock had already invented
the same system for GCHQ in 1973, but this remained secret until 1997.
Rivest and Shamir are computer scientists, who generated ideas for the one-
way function. Adelman is a rigorous mathematician and spotted the flaws
in the ideas from Rivest and Sharmir. They spent a year coming up with
ideas, with Adleman dismissing much of the ideas. In April 1977, Rivest was
returning to his house after spending Passover with Shamir and Adleman.
He couldn’t sleep so he lay in bed with a maths textbook, thinking about
this one way function he had been trying to find. In the middle of the night,
he had a breakthrough and spent the rest of the night writing up a paper.
Although the breakthrough came from Rivest, he could not have done it
without the help of Shamir and Adleman [7].
4.2 Encryption
Suppose Bob wants to send Alice a message. Alice generates 2 very large
primes p and q and multiplies them together, giving N = pq.
Alice then chooses the public exponent e, which should be coprime to ϕ(N)
(i.e. gcd(ϕ(N), e) = 1).
These two values of N and e are then made public as they are sent along a
public channel to Bob [15].
Bob then encrypts his message with RSA using N and e:
c ≡ me
(mod N) (16)
where c is the ciphertext and m is the plaintext.
4.3 Decryption
Once Bob’s encrypted message has arrived, Alice must decrypt it. First,
Alice calculates the totient function of N: ϕ(N) = (p − 1)(q − 1).
Then Alice calculates the modular multiplicative inverse of e (mod ϕ(N)),
12

14. denoted by d:
de ≡ 1 (mod ϕ(N)) (17)
d can be efficiently calculated using the Extended Euclidean Algorithm.
Now Alice can decrypt the ciphertext (denoted by c) into plaintext (denoted
by m):
m ≡ cd
(mod N) (18)
4.4 Prime Factorisation
The trapdoor function of RSA comes from the fact that prime factorisation
is very computationally difficult. To find the prime factors of a composite
number, a computer must brute force search through all the primes below
it, to see if any of those divide the original number. Attackers would need
to find the 2 prime factors p and q of N to compute ϕ(N).
4.5 Attacks
Since RSA’s first publication in 1977, it has undergone extensive study, and
innumerable attacks have been created that jeopardise its security [3]. The
RSA algorithm can be implemented incorrectly in a variety of ways, includ-
ing picking a public exponent that is too little, too large, or too close to p
and q numerically. These implementation flaws are not uncommon even in
large cryptography libraries (such as OpenSSL) used in the most important
systems in the world.
Hastad’s Broadcast Attack
Hastad’s Broadcast Attack is used when the public exponent is too small.
Sometimes, a user sends the same message to multiple people. In this case,
the public exponent stays the same (it is often standardised) but the public
key N may vary.
The attack can be used when the same message m is sent to at least e
people with the public modulii N1, N2, ..., Ne. This gives a system of linear
congruences.
13

15. For example, if e = 3, then we have the following 3 congruences:
me
≡ c1 (mod N1) (19)
me
≡ c2 (mod N2) (20)
me
≡ c3 (mod N3) (21)
The Chinese Remainder Theorem is used for solving a set of simultaneous
linear congruences where the moduli are coprime - in this example, it is a
way of finding me
(modN1N2N3) [3]. Note that m is less than the smallest
Nx we have, therefore me
< N1N2...Ne. This means we can simply take
the regular eth root of me
rather than the modular root as it is cannot be
greater than the modulus.
5 The Diffie-Hellman Key Exchange
The Diffie-Hellman Key Exchange (DHKE) is a protocol which allows 2 par-
ties to privately agree on a symmetric decryption key over a public channel
such as the internet. This means that files can be transferred over a public
channel using secure symmetric encryption methods (such as the AES), the
foremost symmetric cryptosystem. As the DHKE uses both asymmetric and
symmetric encryption, it is considered a hybrid cryptosystem [16].
5.1 A history of the Diffie-Hellman Key Exchange
The Diffie-Hellman Key Exchange was developed in 1976 as a result of Whit-
field Diffie and Martin Hellman’s collaboration. It was the first functional
protocol for creating a shared secret across an open communication channel.
The public key distribution work of Ralph Merkle had an large impact on
its discovery [16]. John Gill suggested application of the discrete logarithm
problem as a trapdoor function. An equivalent algorithm was initially de-
veloped by Malcolm Williamson of GCHQ in the UK several years prior,
but GCHQ decided to keep it a secret until 1997, by which time it had no
impact on academic study.
5.2 The Key Exchange Protocol
A large prime p and a generator g ∈ Fp, g ̸= 0 are made publicly known.
Alice and Bob choose their secret integers a and b respectively [4]. They
14

16. then compute the following integers A and B:
A = ga
(mod p) (22)
B = gb
(mod p) (23)
The integers A and B are exchanged between Alice and Bob over a public
channel, e.g. the internet. Once they receive each other’s respective integers,
they raise it to the power of their original integers. Alice computes Ba
and
Bob computes Ab
, both in modulo p.
Ab
= (ga
)b
= gab
= (gb
)a
= Ba
(mod p) (24)
This means that they have the same number (the key) once the protocol
terminates, without the secret key being shared.
5.3 The Discrete Logarithm Problem
The security of the Diffie-Hellman Key Exchange protocol comes from its
trapdoor function [4] called the discrete logarithm problem (DLP): the dif-
ficulty of computing x from:
gx
≡ a (mod p) (25)
In the DHKE protocol, the values of ga
(mod p) and gb
(mod p) are sent over
a public channel, but an attacker is unable to solve for a or b due to the
difficulty of solving the DLP.
5.4 Attacks - The Pohlig-Hellman Algorithm
Attacks on Diffie-Hellman rely on situations in which you can efficiently
compute the discrete logarithm problem. One such case is n the finite field
Fp, where p-1 is a smooth number, meaning it has many small factors.
The Pohlig-Hellman Algorithm can be used to efficiently solve the discrete
logarithm when such a prime is used [19].
Euler’s Theorem states that aϕ(n)
≡ 1(mod n). Let ϕ(n) = pq, where
gcd(p, q) = 1. We can now attempt to solve the DLP ax
≡ b(mod n).
15

17. Let x = a0 + a1p:
ax
≡ b (mod n) (26)
aqx
≡ bq
(mod n) (27)
aqa0+pqa1
≡ bq
(mod n) (28)
aqa0
· apqa1
≡ bq
(mod n) (29)
(aa0
)q
· (apq
)a1
≡ bq
(mod n) (30)
By Euler’s Theorem: apq
≡ 1 (mod n) (31)
(aa0
)q
· 1a1
≡ bq
(mod n) (32)
(aq
)a0
≡ bq
(mod n) (33)
Now we can brute force search for a0, since we already know what aq
and bq
are.
Once we find a0, since x = a0 + a1p we know that x ≡ a0(modp), which
gives a congruence. Repeating this the other way by raising both sides to
the power of p will give a second congruence modulo q. The system of linear
congruences can be solved for x by using the Chinese Remainder Theorem.
The smoother a prime is, the smaller the factors of p1. The Pohlig-Hellman
Algorithm is more efficient for solving the DLP in this case as the smaller
the prime factors of ϕ(n), the faster the calculations can be computed. can
be made.
6 Elliptic Curve Cryptography
6.1 What is an Elliptic Curve?
An elliptic curve is a smooth, projective, algebraic curve of genus one. It is
in the form:
y2
= x3
+ ax + b (34)
where a and b are integers.
On an elliptic curve, all of the points with rational x and y values are con-
sidered rational points. An interesting property of these curves is that the
rational points of these curves all form a group, which means they are all
connected by an operation.
16

18. Figure 2: Here is an example of the elliptic curve y2
= x3
− 5x + 10
17

19. If we choose any two rational points, P and Q, a line between them intersects
the curve at a third point R. If we reflect R in the x-axis, we get the point R
and we say this point R is the result of the point addition of P and Q [21].
Point addition is the binary operation of the group; this is not the same as
regular addition.
Figure 3: Point addition of P + Q (Image courtesy of Joseph Silverman)
If we wish to compute P + P, instead of drawing a line between 2 points we
draw a tangent to the curve at P [21], which also intersects at a third point.
We then reflect this in the x-axis to calculate P + P.
Figure 4: Point addition of P + P (Image courtesy of Joseph Silverman)
The numbers in the rational group can get infinitely large and so square
rooting these numbers would be computationally unfeasible. In practice, we
18

20. use modular arithmetic with ECC. We work over a finite field Fp to reduce
the x and y coordinates of the points in modulo p. The operation of point
addition and the structure of the group still holds for elliptic curves over a
finite field (or Galois field), which is denoted as E(Fp).
6.2 The Key Exchange Protocol
If Alice and Bob wish to generate a shared secret, they first choose an
elliptic curve E(Fp). The curve used is often a standard curve known to
public. For example, the elliptic curve used by Bitcoin, Ethereum, and most
other cryptocurrencies is called secp256k1. The equation for the secp256k1
curve is y2
= x3
+ 7. Making the curve public does not compromise the
security of the exchange.
They choose a rational point (called the generator point) G ∈ Fp, which is
made public.
Now Alice and Bob generate a private number each, denoted by a and b
respectively. Alice then calculates the rational point A, which is done by
adding the generator G to itself a times, i.e. A = aG. Bob does the same
with his private number to calculate B = bG. Once this is done, Alice sends
A to Bob and Bob sends B back [12].
Note that the notation bG does not denote multiplication, it denotes b series
of point additions (not regular addition) of G to itself.
Once Alice receives B from Bob, Alice can now compute a point additions
of G and add that to B: Sa = B + aG = bG + aG = (b + a)G. Bob also
computes Sb = A + bG = aG + bG = (a + b)G.
This means that Sa = Sb so Alice and Bob end up with the same number,
which is a shared secret. The x coordinate of the point S can be used
as a secret key. Then, the Elliptic Curve Cryptosystem can be used as a
hybrid cryptosystem by using this secret key as a private key for a symmetric
cryptosystem such as AES [12].
6.3 The Elliptic Curve Discrete Logarithm Problem
The The Elliptic Curve Discrete Logarithm Problem is defined as follows:
Given G, A ∈ E(Fp) and A = aG, calculate a.
19

21. This problem is harder to solve than it seems - the problem uses point
addition instead of multiplication. This essentially the same problem as the
DLP from the DHKE, just with a different operation - point addition rather
than multiplication [23]. The problem is abstracted into the group, and the
generic Pohlig-Hellman algorithm can also be used on the curve if its order
is smooth.
7 Comparing Cryptosystems
7.1 Security
As shown in previous sections, the RSA cryptosystem is vulnerable to many
attacks. This makes RSA very difficult to implement well; even major li-
braries such as OpenSSL(the most commonly-used cryptographic library)
can fail to do it correctly.
For example, in 1998, Daniel Bleichenbacher discovered a vulnerability in the
implementation of RSA in SSL. SSL stands for Secure Sockets Layer and was
the main protocol for establishing encrypted channels between computers
on a network, before it was succeeded by TLS (Transport Layer Security)
in 1999. Bleichenbacher’s Attack crafted around 1 million messages and
analysed error codes. This variant of attack has been refined and now Is
called the ROBOT attack. Large websites such as Facebook have been
shown to be vulnerable to this type of attack, from as recently as 2017.
In contrast, protocols such as the Diffie-Hellman Key Exchange and Elliptic
Curve Cryptography are vulnerable to far fewer attacks. Mathematical at-
tacks on the protocols are easy to prevent – to prevent the Pohlig-Hellman
attack, the public prime p is made a safe prime, a prime number in the form
p = 2q + 1 where q is also a prime.
7.2 Efficiency
To encrypt a message (called plaintext), a number of bits of data is to
encrypt the message. This produces which produces a number of bits of
security. Bit security represents the amount of computational resources,
such as time, required to break an algorithm’s security [17]. If an algorithm
has bit security of n, then it would take a maximum of 2n
operations to
break the security.
20

22. The number of bits of data used in encryption is often called the key/block
size. For example, RSA-2048 has 2048 bits for the key size. This is different
from the bit security – RSA-2048 only has 112 bits of security, despite having
a key size of 2048. This difference comes from the progress made in attacking
the maths behind the cryptographic primitives. As more efficient algorithms
are developed to attack a cryptosystem, its bit security decreases. For ex-
ample, improvements to prime factorisation algorithms undermines the bit
security of the RSA cryptosystem.
Why does key size matter?
In general, the longer the key size for a given cryptosystem, the higher the
bit security. The time complexity of solving a cryptosystem varies with the
type of cryptographic primitive used.
However, a longer key size requires more computational resources to en-
crypt and decrypt. In addition, a longer key size requires more data to be
transferred over a network, which may increase data transfer times. Modern
implementations of cryptographic primitives typically use enough bits for
key size for a high level of security, but increasing the number of bits would
be unnecessary [5].
How many bits of security do we need?
The predecessor of AES (a symmetric cryptosystem) was DES, which used
56 bits. 56-bit security is not sufficiently secure for modern computing.
It requires 25
6 operations, which is approximately 7.2e16. If we use 1000
processors with clock speed (no. operations per second) of 2GHz, then DES
can be cracked in a maximum of 7.2e4 seconds, or 20 hours.
The general consensus among cryptographers is that 112-bit security is suffi-
cient until 2030, and 128-bit security is sufficient until a major breakthrough
in maths or computing. For this reason, NIST (National Institute of Stan-
dards and Technology) recommends that more than 2048 bits for RSA is
unnecessary.
How is efficiency measured?
Adversary defence efficiency is a measure of the ratio between key size and
bit security. In this regard, RSA falls behind other cryptosystems such as
ECC. Here is a table of recommended key sizes (in bits) according to the
21

23. NIST [2]:
Bit Security RSA Size ECC Key Size DHKE Size
80 1024 160 1024
112 2048 224 2048
128 3072 256 3072
192 7680 384 7680
256 15360 521 15360
(35)
7.3 Effect of Large Adversaries
Many large adversaries such as the NSA (National Security Agency) and
GCHQ (Government Communications Headquarters) are suspected to have
intellectual property that allows them to break encryption algorithms. For
example, the RSA and DHKE algorithms were developed at GCHQ in secret
years before they were discovered by the outside cryptography community.
Eliptic Curve Cryptography is a relatively new cryptosystem, and many
cryptographers are skeptical of its rigidity against unknown attacks. Cryp-
tographic primitives are difficult to implement well and so many be vulner-
able to attacks as shown with RSA implementations. In addition, many
cryptographers are concerned that the NSA has inserted a kleptographic
backdoor into one or more elliptic curve pseudo random generator, after
Edward Snowden leaked internal memos from the NSA [11].
For this reason, many cryptographers prefer to minimise the risk of this and
use RSA, even if it has a higher key size. During my interview with cryp-
tography researcher Dr Daniel Gardham (at University of Surrey), he stated
that he prefers to use RSA-2048, and sometimes even RSA-4096 when he is
feeling paranoid. This is because RSA is a well documented and researched
cryptosystem, so he is certain that breaking RSA-4096 is unfeasible, at least
for the next 30 years. On the other hand, although ECC is theoretically
very secure, he has concerns that it could be broken by large adversaries.
8 Conclusion
In conclusion, RSA is an outdated crpytosystem. It suffers from problems
such as its performance inefficiency, as well as being very difficult to im-
plement correctly. Protocols such as TLS have already recognised this and
begun to phase out RSA in favour of better alternatives, such as ECC.
22

24. However, ECC protocols are relatively new to market and their attacks and
weaknesses have not been researched and exploited to the same extent as
they have for RSA protocols. In addition to this, there are concerns that
the NSA has implemented kleptographic backdoors into implementations of
the ECC cryptosystem, which means that cryptographers are unaware of its
real practical security.
It may take several decades for the attacks and weaknesses of ECC to be
fully discovered. But with the advent of quantum computing, it will be
feasible solve the DLP and the ECDLP in polynomial time using Shor’s Al-
gorithm. Therefore, current existing cryptosystems will have to be replaced
by new crpytosystems. Research should be directed towards developing a
new mathematical primitive resistant to post-quantum attacks. This should
be the top priority as such a crpytosystem needs to exist before quantum
computing becomes usable.
For now, using RSA is still the safest choice, as it is the most widely re-
searched algorithm. Although it is not the most efficient algorithm, crpy-
tographers still use it for its safety against large adversaries. In the near
future, RSA will continue to be used, until a crpytosystem resistant to post-
quantum attacks is found.
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