New Semi-Prime Factorization and Application in Large RSA Key Attacks

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

Prime factorization using quantum variational imaginary time evolution

The road to computing on quantum devices has been accelerated by the promises that come from using Shor’s algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as $$O(n^{5}d)$$ O ( n 5 d ) , where n is the bit-length of the number to be factorized and d is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9-qubit Hamiltonians. We also verify the method’s performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware.

An Investigation Into the Mathematics of Decryption Techniques in RSA Encryption, With an Implementation in Python

This study explores the mathematics of two different techniques that can be used to access the decryption key in RSA encryption including semi-prime factorization and a logarithmic method. The study then presents a Python program, written by the author, that automates the calculations for either of the decryption techniques and also calculates the number of iterations required to determine the decryption key in either circumstance. Most importantly, the program utilizes only values of the RSA encryption algorithm that would be made publicly available in actual circumstances to calculate the decryption key so as to mimic real-life occurrences with as much integrity and accuracy as possible.

Collective Signature Protocols for Signing Groups based on Problem of Finding Roots Modulo Large Prime Number

Generally, digital signature algorithms are based on a single difficult computational problem like prime factorization problem, discrete logarithm problem, elliptic curve problem. There are also many other algorithms which are based on the hybrid combination of prime factorization problem and discrete logarithm problem. Both are true for different types of digital signatures like single digital signature, group digital signature, collective digital signature etc. In this paper we propose collective signature protocols for signing groups based on difficulty of problem of finding roots modulo large prime number. The proposed collective signatures protocols have significant merits one of which is connected with possibility of their practical using on the base of the existing public key infrastructures.

Symmetries of abelian Chern-Simons theories and arithmetic

We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including U(1)k Chern-Simons theory and (ℤk)ℓ gauge theories. For example, we prove that U(1)k Chern-Simons theory is time-reversal invariant if and only if −1 is a quadratic residue modulo k, which happens if and only if all the prime factors of k are Pythagorean (i.e., of the form 4n + 1), or Pythagorean with a single additional factor of 2. Many distinct non-abelian finite symmetry groups are found.

Optical framed knots as information carriers

Modern beam shaping techniques have enabled the generation of optical fields displaying a wealth of structural features, which include three-dimensional topologies such as Möbius, ribbon strips and knots. However, unlike simpler types of structured light, the topological properties of these optical fields have hitherto remained more of a fundamental curiosity as opposed to a feature that can be applied in modern technologies. Due to their robustness against external perturbations, topological invariants in physical systems are increasingly being considered as a means to encode information. Hence, structured light with topological properties could potentially be used for such purposes. Here, we introduce the experimental realization of structures known as framed knots within optical polarization fields. We further develop a protocol in which the topological properties of framed knots are used in conjunction with prime factorization to encode information.