This document discusses discrete logarithms. It begins with prerequisites of congruence modulo and primitive roots. It then defines congruence modulo, primitive roots and their properties. It proceeds to define discrete logarithms as the exponent that a primitive root must be raised to to get a given element modulo a prime. It provides examples of solving for discrete logarithms. It notes that while finding discrete logarithms is easy given the base and exponent, finding the exponent given the base and element is very difficult, forming the basis of cryptographic applications like Diffie-Hellman key exchange.

1. DISCRETE LOGARITHMS
By Nihal Jayachandran
2K20/CO/297

2. PRE-REQUISITES
1. CONGRUENCE MODULO
2. PRIMITIVE ROOTS

3. CONGRUENCE MODULO
● Congruence is an equivalence relation operation under modular arithmetic.
● It states that if given an integer n > 1, called a modulus, two integers a and b are said to be
congruent modulo n, if n is a divisor of their difference.
So there should be an integer k such that a − b = kn.
● Can be denoted as follows:-
Reflexivity: a ≡ a (mod n)
Symmetry: a ≡ b (mod n) if b ≡ a (mod n) for all a, b, and n.
Transitivity: If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)

4. PRIMITIVE ROOTS
● A primitive root of a prime number n is an integer g such that g(mod)n has a
multiplicative order of n-1. This is only true if g and n are relatively prime.
● That is an integer g is a primitive root of (mod n) if for every number relatively prime to n
there is an integer z such that
● Here, for every value of 1<=z<=(n-1) a distinct 𝛼 should be the resultant such that
1<=𝛼 <=(n-1)
● Multiplicative order- The smallest exponent e for which

5. DISCRETE LOGARITHMS
● If 𝛼 is an arbitrary integer relatively prime to n and g is a primitive root of n, then there
exists among the numbers 1, 2, ...,Φ(n), where Φ(n) is the totient function, exactly one
number z such that:-
● The number z is then called the discrete logarithm of a with respect to the base g modulo
n and can be denoted as: z = indg
a (mod n) OR z = logg
(𝛼).
● The reason it is called discrete logarithm is because its definition is analogous to that of
the usual logarithm.
● a ≡ g^z (mod m) is equivalent to logg
(a) ≡ z (mod k), where k is the order of g modulo m.
logb
(y) = x is equivalent to y = b^x.)

6. DISCRETE LOGARITHM PROBLEM
Question: Find the discrete logarithms of each unit modulo 11 to the base 2.
Solution: Since 2 is a primitive root modulo 11, we can write each unit as a power of
2. The simplest way to do this is, is to compute each of the values 2^0, 2^1, ... ,
2^10 modulo 11. The table of results is as follows:
Here, for example, 3 · 6 ≡ 7 (mod 11), and log2
(3)+log2
(6) ≡ log2
(7) (mod 10), since
10 is the order of 2 modulo 11.

7. DISCRETE LOGARITHM PROBLEM
Question: Calculate k in the equation 3^k ≡ 13 (mod 17) in the group (Z17
)*.
Solution: By doing the table similar to the last question we find that k=4.
According to Euler’s Theorem- 3^16 ≡ 1(mod 17)
Thus, the equation has infinitely many solutions of the form 4 + 16n, where
-∞<=n<∞

8. DISCRETE LOGARITHM PROBLEM
Question: Suppose G = (Z5641
)*. Then Calculate log3
37 in G.
Solution: The question can be rewritten as calculate x in 34 ≡ 3^x(mod 5641)
where x is the discrete logarithm of 34 with respect to base 3 modulo 5641.
As it can be seen it is very complex to find the discrete logarithm in this case using
the aforementioned tabular method. It is a known fact that for large prime numbers
it is in general very difficult to compute discrete logarithms using any method or
algorithms that are currently available.

9. DISCRETE LOGARITHM APPLICATION
● Here for given g, z, and n, it is a straightforward matter to calculate 𝛼. At the worst, we
must perform x repeated multiplications, and algorithms exists already for achieving
greater efficiency.
● However, given 𝛼, g, and n, it is, in general, very difficult to calculate z . The difficulty
seems to be on the same order of magnitude as that of factoring primes required for RSA.
● In fact there does not exist a polynomial time algorithm to solve the Discrete Logarithm
Problem and the fastest known algorithm works in exponential time.
● Discrete logarithms are fundamental to a number of public-key algorithms, such as
Diffie-Hellman key exchange and the digital signature algorithm (DSA).

10. LIST OF REFERENCES
● https://www.brainkart.com/article/Discrete-Logarithms_8433/
● https://www.hypr.com/security-encyclopedia/diffie-hellman-algorithim
● https://mathworld.wolfram.com/DiscreteLogarithm.html
● https://mathworld.wolfram.com/PrimitiveRoot.html
● https://www.whitman.edu/mathematics/higher_math_online/section03.01.html
● https://www.tutorialspoint.com/what-is-discrete-logarithmic-problem-in-informatio
n-security

11. THANK YOU