Public Key Generation Principles Impact Cybersecurity

Abstract

A R T I C L E I N F O : RECEIVED: 22 JUNE 2020 REVISED: 08 SEP 2020 ONLINE: 22 SEP 2020 K E Y W O R D S : public key cryptography, Miller–Rabin primality test improvement, cybersecurity Creative Commons BY-NC 4.0 Introduction Nowadays the growing use of online communications over the Internet and the associated threats to the data we exchange, requires sufficient and reliable protection of the information exchanged. One of the most reliable and basic method to make information secure, when two communicating parties don't know each other, is public key cryptography. Hard-to-solve mathematical problems are used to realize the mathematical foundations of the existing algorithms using public key cryptography. 2 In this mathematics, integer operations with large numbers are used and are based on modulo calculations of large prime numbers. Large prime numbers are also used to produce the user’s cryptographic keys (public and private). N. Stoianov & A. Ivanov, ISIJ 47, no. 2 (2020): 249-260 250 We could state that the security of the exchanged data protected by public key cryptography is due to two main facts: the difficulty of solving a mathematical algorithm and the reliability of the generated prime numbers used as keys in such a system. In this paper we will consider deterministic and probabilistic primality tests and will focus over the most widely used in practice an algorithm for testing prime numbers, that of Miller-Rabin 3, 4 and we will propose a new addition to it, which will increase the reliability of the estimation that this probabilistic algorithm gives. Public Key Generation and The Importance of The Prime Numbers Public key cryptography algorithms are based on two main things: difficult to solve mathematical problems and prime numbers with big values that serve as user private keys. If that prime numbers are generated not according to the prescribed rules or are not reliably confirmed as such, the security strength of protected data could be not enough. In an effort to ensure better protection of information, new algorithms and rules for generating private keys and the prime numbers involved in their compilation are created and proposed. 6 The more than 85% from certificate authorities (CA) based their root certificate security by using RSA encryption and signing scheme. Approximately of 10% of CA combinate both RSA and ECDSA cryptographic schemes to protect their public key infrastructure (PKI). This statement is based on our study in which we analysed the certificates stored into Windows, Android and Linux operating systems (OS) certificates stores. These operating systems are the most commonly used worldwide. We can say that a reliable estimate of divisibility of numbers is essential. Connected with this we will consider algorithms for primality testing. In practice, they are divided into two main types. Deterministic and probabilistic algorithms. Deterministic Primality Testing The most elementary approach to primality proving is trial division. If attempt to divide p by every integer n ≤ ⌊√p⌋ and no such n divides p, then p is prime. But this task will take O (√p M(log p)) time complexity, which is impractical for large values of p. That is why the most practical algorithms have to be used to deter the big numbers divisibility to factors. An algorithm created in 2002, AKS (Agrawal, Kayal, and Saxena), falls into the group of tests that give an unambiguous assessment of divisibility of numbers. At the heart of AKS algorithm is Fermat's Little Theorem. The Fermat's Little Theorem states that: if a number p is prime, a ∈ Z, p ∈ N and GCD(a, p) = 1 then a ≡ a mod p The primality test by using this theorem fails for a specific class of numbers, known as pseudoprimes, which include the Carmichael numbers. Primarily based on a polynomial generalization of the Fermat’s Little Theorem the AKS algorithm state that: the number p is prime if and only (x + a) ≡ (x + a) mod p Public Key Generation Principles Impact Cybersecurity 251 where a ∈ Z, p ∈ N. The time complexity here would be Ω(n) which is not polynomial time. To reduce complexity, we can divide both sides by (x − 1). Therefore, for a chosen r the number of computations needed to be performed is less. Hence, the main objective now is to choose an appropriately small r and test if the equation: (x + a) ≡ (x + a) mod GCD(x − 1, p) is satisfied for sufficient number of a’s. The algorithm proposed by Agrawal, Kayal and Saxena 8,9 for primality testing has following steps: (1) if p = a for a ∈ N, b > 1 output COMPOSIT (2) find smallest r such that Or(p) > log p (3) if 1 < GCD(a, p) < p for some a ≤ r then output COMPOSIT (4) if n ≤ r output PRIME (5) for each a ∈ 1. . ⌊√φ(p) log p⌋ (6) if (x + a) ≠ (x + a) mod GCD(x − 1, p) (7) output COMPOSIT (8) output PRIME The complexity of execution of that algorithm is ?̃?(logn) time. Hence the execution time will be proportional to (log n) if p grows larger. This is a polynomial time function, which although not as fast as the probabilistic tests used nowadays, has the advantage of being fully deterministic. Probabilistic Testing of Prime Numbers. Miller-Rabin Primality Test We know two mathematical ways to prove that a number p is composite: a. number p factorization, where: p = a. b and a, b > 1 (2.2.1) b. Exhibit a Fermat witness for p, i.e. find a number x satisfying: x ≢ 1 mod p (2.2.2) The speed of these algorithms, which certainly determine whether a number is divisible, is unsatisfactory. This requires some of the probabilistic algorithms for primality test to be more widely used. The Miller-Rabin 10,11 test is based on a third way to prove that a number is composite. c. Exhibit a no square root of 1 mod p. That means to find a number x such that: x ≡ 1 mod p and x ≢ ±1 mod p (2.2.3) N. Stoianov & A. Ivanov, ISIJ 47, no. 2 (2020): 249-260 252 The Miller-Rabin test is the most widely used probabilistic primality test. This algorithm was proposed in 70’s. Miller and Rabin gave two versions of the same algorithm to test whether a number p is prime or not. Rabin’s algorithm works with a randomly chosen x ∈ Zp, and is therefore a randomized one. Correctness of Miller’s algorithm depends on correctness of Extended Riemann Hypothesis. In his test method it is need to tests deterministically for all x’s, where 1 < x < 4. logp. If x is a witness for an integer p, then p must be composite and we say that x witnesses the compositeness of p. Prime numbers clearly have no witnesses. If we picked up enough count of x’s (100 or more depends on size of p) and no one is a witness, we can accept that number p is probably prime. The algorithm realization steps are: (1) Denote p − 1 = s. 2, where s mod 2 = 1 (2) randomly picked up x ∈ Zp (3) if x ≢ 1 mod p output COMPOSITE (p is definitely composite) (4) b = x mod p (5) if b ≡ ±1 mod p , output PRIME (x is not a witness, p could be prime) (6) Loop i ∈0..m-1 (7) b ← b mod p (8) if b ≡ −1 mod p then (9) output PRIME (x is not a witness, p could be prime) (10) output COMPOSITE x is a witness p, is definitely not prime The time complexity of that algorithm is ?̃?(y. logn) where the y is the count of the iterations i.e. the different values of randomly chosen x. Subgroup Extending by New Generating Number of a Ring is Gained. In this part of the paper we will considering primality test based on two criteria and an idea of a method of transitioning (without intersection) or extending different multiplicative subgroups formed by their generator integer. To describe this method of transitioning/extending multiplicative subgroups formed by number p, we will use linear Diophantine equation: dx . dy − p. k = dz (2.3.1) where di = g i mod p. Every di ∈ Zp and it is part of a ring generated by number g and has ring order #O(g,p) or smaller. In the particular case when dx = dy −1 mod p, value of dz = 1. In our practice dealing with number rings we saw that if dz = 1 quadratic reciprocity ( dx p ) = ( dy p ), and when size of #O(g,p) < p − 1 then number k could highly has different quadratic reciprocity, i.e. ( dx p ) ≠ ( k p ). In cases when that is not true, we can just do that with new value of dx ← k and in few steps of repeating that we can reach a value of k which has different quadratic reciprocity Public Key Generation Principles Impact Cybersecurity 253 to p than the initial dx. We saw more important thing, that the rings formed with generators dx and k, have very often different elements on its sets. If we use q = dx . k mod p as a ring generator its #O(q,p) is different then #O(dx,p) and #O(k,p). To show that we will use two examples. Into the first we will use a prime number with value 1117 and in the second one composite number 21421 = 11 . 1931 . Example 1 p = 1117 , g = 430 , #O(p,p) = 372 , ( 43