PROBLEMS IN CRYPTOGRAPHY: SOLUTIONS IN REALITY...
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PROBLEMS IN CRYPTOGRAPHY
Long story short: one way functions are likely inherent to the structure of reality and thus are likely to remain computationally secure. Quantum computing would be no solution to one way functions' opacity since the vectors of quanta or the positions of quanta, only, can be determined and thus quanta are in that regard also one-way. (quantum uncertainty). The uncertainty principle conforms to implications from Gödels theorem and the common intuition that one way functions exist. The quantum uncertainty principle states that we may determine either the position or the momentum of particles, but not both simultaneously. This is analogous to one way functions.
PROLOGUE
I. Some Mathematical Ideas: (skip if you already know it)
LEMMAS, PROBLEMATIQUES, AND HEURISTICS
The terms "lemma," "problematique," and "heuristic" each represent distinct problem-solving methods.
Lemma: In mathematics and logic, a lemma is a proven proposition or auxiliary theorem that is used as a stepping stone to prove a larger theorem. Lemmas are often introduced as intermediate results in mathematical proofs, helping to break down complex problems into more manageable components. Once a lemma is established, it can be used as a foundational element in subsequent arguments, leading to the eventual proof of the main theorem. In essence, a lemma serves as a building block in the logical structure of a mathematical proof, contributing to the overall coherence and validity of the argument.
Problematique: A problematique is a question presented, a question we seek to solve. The use of problematique to solve for a problem is rooted in the field of critical theory and refers to a framework for analyzing complex social, political, or philosophical issues. Coined by French philosopher Louis Althusser, a problematique involves identifying the underlying assumptions, power dynamics, and historical contingencies that shape a particular issue, with the goal of uncovering its inherent complexities and contradictions. Unlike a lemma, which serves a primarily analytical function within a specific domain, a problematique provides a broader perspective on the multifaceted nature of social and intellectual inquiries.
"Heuristic" denotes a general problem-solving strategy or guiding principle that is employed to facilitate decision-making or problem-solving in situations where there is no guaranteed algorithmic solution available. Heuristics are often used as mental shortcuts or rules of thumb to streamline complex decision-making processes and arrive at satisfactory outcomes, even in the absence of complete information or rigorous analysis. Unlike lemmas, which are specific to mathematical and logical reasoning, heuristics can be applied across a wide range of disciplines and contexts, including psychology, economics, and computer science. They are particularly useful in situations where time constraints or cognitive limitations prevent exhaustive exploration of all possible alternatives.
In sum, lemmas are intermediate results used in mathematical proofs to establish the validity of larger theorems, problematiques provide frameworks for analyzing complex social and intellectual issues, and heuristics are guiding principles or strategies employed to streamline decision-making and problem-solving processes across diverse domains.
II. PRIME FACTORIZATION: THE CANONICAL CIPHER EXAMPLE OF A ONE WAY FUNCTION
If you know nothing about cryptography and wish you weren't so stupid you need to read this.
Prime factorization is an example of a one-way function. A one-way function is a function that is easy to compute in one direction but difficult to invert in the opposite direction. In the case of prime factorization, given a positive integer n, it is relatively easy to compute its prime factors, but it is computationally infeasible to determine the original number n from its prime factors alone.
Prime factorization is a one-way function because it satisfies the following properties:
Easy to compute in one direction: Given a positive integer n, it is relatively easy to compute its prime factors using algorithms such as trial division, sieving, or modular exponentiation.
Difficult to invert in the opposite direction: Given a set of prime factors, it is computationally infeasible to determine the original number n. This is because the number of possible combinations of prime factors is exponentially large, and testing each combination would require an impractically large amount of computational resources.
Prime factorization is used in many cryptographic applications, such as RSA, Diffie-Hellman key exchange, and elliptic curve cryptography, where it serves as a one-way function to provide security against unauthorized access or tampering.
EPISTEMOLOGY: IDEAS AND MATHEMATICS
This is complex and abstract but the necessary background for the cryptographic argument. If you don't like philosophy just take my word on this at face value. 给我面子好不好.
I EIDETIC REALISM (FORMALIST REALISM)
Classical formalism was the idea that thoughts are prior, anterior, to facts, the material. This was the view taken up by Plato. His view could be criticized as superstitious magical thinking.
Modernity replaced classical formalism with the belief that mathematical ideas are purely abstract formalizations, which are not reflections of or reflect into, material facts of the real world.
The dominant contemporary position regarding mathematics characterizes it as a purely formal system. This perspective emphasizes the abstract nature of mathematical concepts, focusing on their logical structure and internal consistency rather than any inherent connection to physical reality.
At its core, the idea of mathematics as a purely formal system traces back to David Hilbert and Gottlob Frege in the late 19th and early 20th centuries. These mathematicians sought to establish mathematics on a rigorous, axiomatic basis, with all mathematical truths derived from a set of fundamental axioms through logical deduction. This approach, known as formalism, views mathematics as a game played with symbols according to predefined rules, much like a formal language or logical system.
One of the key tenets of formalism is the notion that mathematical objects are abstract formal entities devoid of any inherent meaning or reference to the material world. According to this view, numbers, geometric shapes, and other mathematical constructs exist purely as formal symbols to be manipulated according to internally consistent syntactic rules. For example, the number "2" is not seen as inherently linked to any specific collection of objects but rather as a placeholder in a mathematical system that follows certain rules of arithmetic.
Pure formalism displaced eidetic realist views that posit the existence of mathematical objects as pre-existing and independent of human thought or experience. For eidetic realists like Plato, reality is an imperfect reflection of ideas such as "circle". For the pure formalists there is no connection between the abstract ideas of maths and material reality because the ideas are purely abstract forms and neither reflections of nor reflect onto the material world. In other words, formalism posits that mathematical truths are "true", i.e. valid, by virtue of their logical consistency within a given formal system, rather than any correspondence to reality. This is a fundamental flaw in modern thought, which I expose in great detail in my book "Post Positivism".
The formalist perspective has had a significant impact on the practice of mathematics in contemporary times. It has led to the development of abstract branches of mathematics like set theory, logic, and category theory, which focus on the study of mathematical structures and relationships purely in terms of their formal properties. Mathematicians working within these disciplines often employ rigorous logical reasoning and symbolic manipulation to explore the consequences of different axiomatic systems.
The dominant contemporary position characterizes mathematics as a purely formal system, emphasizing its abstract nature and logical structure. This purely formalist perspective displaced eidetic realist views that are most easily seen in Platonism. However, pure formalism in mathematics is likely to be replaced by materialist views: science marches on.
II MATERIALISM
A monist and materialist perspective on the nature of reality entails viewing the world as fundamentally unified and composed solely of matter. This perspective combines ontological monism (the belief in a single fundamental reality as opposed to dualism) and epistemological materialism (the view that knowledge is derived from empirical observation). It challenges the notion that mathematics is a purely formal system divorced from the material world. Instead, it sees mathematics as "merely" a reflection of empirical facts, as a reflection of the underlying structure of the material world.
Monist epistemology and materialist ontology see the physical world is the ultimate reality: everything, including human thought and consciousness, emerges from physical processes: All of it can be connected to any of it, albeit sometimes only circuitously. Monism and materialism together reject the existence of abstract entities or transcendent realms beyond the material universe, including the purported field of pure mathematical forms posited by Platonism. Instead, monism and materialism see mathematical concepts as grounded in the empirical properties of the material world. Mathematical ideas arise from our observations and interactions with the world. For example: Perfect circles do not exist. What does exist are things which are nearly circles and which we class together and abstract into an idea existing only in our brain as "circle" or "blue" for that matter. These abstractions from observations are ideas and they are "mere" reflections of reality, but we use them to shape reality using material objects like muscles and machines. The formation of ideas results from the dialectical process of comparing our sensory experiences which we then form together into abstract ideas that we then compare with the thoughts of others through speech. This process of comparing sense experiences and ideas therefrom leads to a multifacted and more accurate view of reality, a much more accurate understanding of the world than we would have had in isolation. Plato's metaphor of the cave of ignorance is somewhat accurate, but not for the reason he supposed. It is our lack of comparing our ideas to facts, and then our own experiences and ideas to those of others that would leave us in the dark cave of ignorance.
From a monist materialist perspective mathematics is not merely a formal game played with symbols but rather descriptions of material facts.
For example, consider the relationship between mathematics and physics. Modern physics relies heavily on mathematical formalisms to describe the fundamental laws of nature, such as those governing the behavior of particles at the quantum level or the dynamics of celestial bodies in space. These mathematical models are not arbitrary constructions but are developed based on empirical observations and experimental data, with the aim of accurately representing the underlying physical reality. Their accuracy is judged using the scientific method, by comparing predictions of the (mathematical) theory with observations of actual facts (physics).
Advances in fields such as differential geometry, topology, and algebraic geometry also reveal connections between mathematics and the physical world. Concepts such as manifolds, symmetry groups, and tensor calculus play crucial roles in modern theories of relativity, quantum mechanics, and particle physics. The fact that mathematical structures developed purely for their formal properties turn out to have profound implications for our understanding of the universe suggests a close correspondence between mathematics and the material world.
The application of mathematics to engineering, technology, and other empirical domains also demonstrates its connection to the material world. Mathematical models and algorithms are used to design and optimize everything from bridges and buildings to computer chips and communication networks, with tangible effects on the physical environment and human experience.
A monist-materialist perspective on the nature of reality challenges the idea that mathematics is a purely formal system divorced from empirical facts. Instead, mathematics is a reflection of the material world and serves as a reflection of its underlying structure. From this viewpoint, mathematics emerges from our observations and interactions with the physical universe, and its postulates and theorems are grounded in the empirical facts. By recognizing that mathematics is a reflection of the material world we can derive from maths hypotheses for use as in other fields such as physics, material sciences, and human sciences.
III PLATO GOT IT BACKWARDS: IDEAS ARE REFLECTIONS OF MATERIAL FACTS, NOT THE OTHER WAY ROUND LIKE HE THOUGHT.
Plato's theory of Forms, or eidos, posits that the material world is but a shadow or imperfect reflection of a higher realm of perfect, eternal Forms. According to Plato, these Forms are the true reality, and the objects and phenomena we perceive in the physical world are mere imperfect copies or instantiations of these ideal Forms. Modernity, at least in mathematics, rejected Platonic formalism replacing the idea of forms with the idea that mathematics is a purely formal system, an abstract "glass bead game" (See: Hesse).
However, from a monist materialist perspective Plato and Modernity both were mistaken, but in different ways. To monist-materialists Plato got it backwards: rather than reality being a reflection of Forms, our ideas about reality are reflections of material facts, which we then use to influence material reality. Mathematics is not a purely formal system, nor is it anterior or a priori to real facts. Instead, mathematical ideas are reflections of the material world.
From a contemporary standpoint, influenced by scientific inquiry and empirical observation, the idea of a realm of perfect Forms existing independently of the material world is untenable. Our ideas about reality are constructed through our interactions with the physical world and shaped by our sensory experiences, cognitive processes, and social interactions. Our ideas are abstractions and integrations of these various sensory experiences and then they are compared with others' experiences through speech and writing to attain an even better understanding of the world. That comparison and exchange of ideas, which are resolved to a higher level of understanding is dialectics.
The relationship between our ideas such as mathematics and material facts is empirical. Empiricism holds that knowledge is derived from sensory experience and observation of the external world. According to this perspective, our understanding of reality is built upon our perceptions of material phenomena and the patterns we discern through systematic observation and experimentation. Thereby mathematical ideas are ultimately reflections of, and abstractions of, real material facts. Plato got it backwards.
For example, consider the development of scientific theories to explain natural phenomena. Scientists formulate hypotheses based on empirical evidence and then test these hypotheses through experimentation and observation. Why should or how could mathematics be otherwise? The success of scientific theories in explaining and predicting the behavior of the natural world rests on their ability to accurately reflect material facts and observations.
Advances in fields such as cognitive science and psychology have also shed light on the ways in which our ideas and beliefs about reality are influenced by our perceptual and cognitive processes. Concepts such as schema theory and embodied cognition suggest that our understanding of the world, including our ideas such as mathematics, is shaped by our sensory-motor experiences and interactions with the environment. In other words, our ideas about reality are not merely passive reflections of an external realm of Forms but are actively constructed through our engagement with the material world.
The practical application of knowledge to manipulate and transform the material world also underlines the relationship between our ideas and material facts. From technology and engineering to medicine and agriculture, human endeavors are driven by our ability to understand and manipulate the physical environment to achieve desired outcomes. It is reality which forms our ideas, but we can then use our ideas about reality to influence and change reality, at least to some extent, even if only exceptionally. Mao Zedong describes the dialectical formation and refinement of ideas in his essay "On Practice".
In sum, Plato's theory of Forms posits that reality is a reflection of ideal, eternal Forms. His view was wrong, and to empiricists and skeptics rather obviously so. The correct view is that reality is not a reflection of ideas, but that ideas are a reflection of material facts and empirical observations. Our ideas may then shape reality through practical application of material forces of production such as muscles and tools. Our ideas about reality are not passive reflections of a transcendent realm but are actively constructed through our interactions with the physical world. By recognizing the dynamic relationship between our ideas and material facts, we gain a deeper understanding of the nature of knowledge and our place in the world
FROM IDEAS TO PROOFS
Ideas are reflections of the material world.
Gödels incompleteness theorem is an idea.
Thus, Gödels theorems may be used as an heuristic to infer answers to other similar phenomena such as the proof of the existence of one way functions or the proof of p != np.
I. HYPOTHESIS (PROBLEMATIQUE): Gödels incompleteness theorem implies that one way functions exist and also that P !=NP
Although Gödel's incompleteness theorem does not itself prove the existence of one way functions it does imply they likely do exist and thus we should assume they exist and then seek to backward chain (rationalize) from the presupposition of the existence of one way functions to arguments and ultimately a proof they exist using the same or similar methods as Gödel used. For the fact that a function MUST be incomplete implies also that it is in fact "one way".
II. AN ARGUMENT BY ANALOGY FROM GÖDELS THEOREMS TO SUPPORT THE HYPOTHESIS THAT ONE WAY FUNCTIONS EXIST
Formal systems, as per Gödel's theorems, are either incomplete or inconsistent. This, incidentally, parallels the quantum uncertainty principle.
An incomplete formal system lacks the ability to prove every true statement that can be made within the system.
An inconsistent formal system allows for both a statement and its negation to be provable, rendering it meaningless.
Similarly, a one-way function can be seen as a "formal system" that takes an input (the plaintext) and produces an output (the ciphertext) according to a set of fixed rules.
Just as a formal system can be either incomplete or inconsistent, a one-way function can be viewed as having similar limitations.
An incomplete one-way function would allow for some inputs to be transformed into outputs that cannot be inferred from the original input, effectively creating a "black box" that cannot be reverse-engineered.
An inconsistent one-way function, on the other hand, would produce conflicting outputs for the same input, making it impossible to determine the correct output for a given input.
Therefore, just as Gödel's theorems demonstrate the limitations of formal systems, we can analogously argue that one-way functions must also have inherent limitations due to their similarity to formal systems.
These limitations suggest that one-way functions cannot be perfectly secure, as there will always be a risk of exploiting the function's inconsistencies or incompletenesses to deduce the original plaintext from the ciphertext.
III. RELATIONSHIPS AMONG GÖDEL'S INCOMPLETENESS THEOREM, THE PROOF OR DISPROOF OF THE EXISTENCE OF ONE WAY FUNCTIONS, AND THE P NP PROBLEM
There are some indirect connections between Gödel's Incompleteness Theorem, one-way functions, and the P vs. NP problem.
Gödel's Incompleteness Theorem: This theorem states that any formal system that is powerful enough to describe basic arithmetic is either incomplete (there are statements that cannot be proved or disproved within the system) or inconsistent (the system can prove both a statement and its negation). The theorem has far-reaching implications for the foundations of mathematics and computer science.
One-way functions: A one-way function is a mathematical function that is easy to compute in one direction but computationally infeasible to invert in the other direction. Their existence has neither been proven nor refuted: it is a currently unsolved problem in mathematics, just as the P NP problem is also currently unsolved.
Formally, a function f is one-way if it satisfies the following properties:
Easy to compute: Given an input x, computing f(x) is efficient and straightforward.
Hard to invert: Given an output y, computing x such that f(x) = y is computationally infeasible.
One-way functions have numerous applications in cryptography, secure protocols, and other areas where data privacy and security are essential.
P vs. NP problem: The P vs. NP problem is a fundamental question in computer science that deals with the relationship between two classes of computational problems:
P (Polynomial Time): A problem is in P if a proposed solution can be verified efficiently (in polynomial time). Examples include sorting algorithms or finding the shortest path in a graph.
NP (Nondeterministic Polynomial Time): A problem is in NP if a proposed solution can be verified efficiently, but the actual solution might take longer than polynomial time to find. Examples include the traveling salesman problem or the knapsack problem.
The P vs. NP problem asks whether every problem in NP can be solved quickly by a deterministic Turing machine (i.e., in polynomial time). In other words, the problem asks whether NP=P. If someone were to come up with a fast algorithm for solving all NP problems, they would win a million-dollar prize offered by the Clay Mathematics Institute, since it would imply that P=NP.
Relationships between these concepts:
a. Gödel's Incompleteness Theorem and one-way functions: While Gödel's Incompleteness Theorem does not directly address one-way functions, the idea of undecidability (i.e., the existence of statements that cannot be proved or disproved within a formal system) has implications for the study of one-way functions. Specifically, certain results related to one-way functions, such as the existence of collision-resistant hash functions, rely on the assumption that certain mathematical problems are hard to solve, which is closely related to the ideas in Gödel's Incompleteness Theorem. However, this connection is indirect, and one-way functions can be studied independently of Gödel's work.
b. One-way functions and the P vs. NP problem: One-way functions are relevant to the P vs. NP problem because many encryption schemes and secure protocols rely on the hardness of certain problems in NP to ensure their security. For example, RSA encryption relies on the difficulty of factorizing large composite numbers, which is believed to be an NP-hard problem. If someone were to come up with a fast algorithm for factorizing large numbers, they could break the RSA encryption scheme and potentially compromise the security of many online transactions. In this sense, the P vs. NP problem has significant implications for the design and analysis of secure protocols, which often rely on one-way functions.
c. Gödel's Incompleteness Theorem and the P vs. NP problem: There is no direct connection between Gödel's Incompleteness Theorem and the P vs. NP problem. However, some researchers have attempted to use insights from Gödel's work to shed light on the P vs. NP problem. For instance, Stephen Cook, who first showed that Satisfiability (SAT) is NP-complete, was inspired by Gödel's work when developing his proof. Additionally, some attempts to resolve the P vs. NP problem involve using techniques from proof theory, model theory, or other areas of logic, which are connected to Gödel's work. Nonetheless, the connection between Gödel's Incompleteness Theorem and the P vs. NP problem remains indirect and speculative.
Gödel's Incompleteness Theorem, one-way functions, and the P vs. NP problem are three important concepts in computer science and mathematics, their connections are mostly indirect. Gödels theorems are proven. The questions whether one-way functions exist and whether P != NP are unsolved Each has its unique significance and plays a distinct role in understanding different aspects of computability, complexity, and information security but appear to be facets of an underlying structure reflected also in the quantum uncertainty principle. Since Gödels theorems and the quantum uncertainty principle appear consistent it is likelier that one-way functions in fact do exist and that P != NP.
IV. OBJECTIONS TO THE HYPOTHESIS:
Gödel's Incompleteness Theorem does not directly imply that there exist one-way functions or that P≠NP. The theorem only shows that any formal system that is powerful enough to describe basic arithmetic is either incomplete or inconsistent. However, a thing which is difficult or impossible to determine, namely the position and location and velocity of a particle or a complete and consistent formal systemable to explain arithmetic exhibits computational intractibility similar to that require for a one-way function.
An informal statement of Gödels more famous theorem goes like this:
"Any formal system that is strong enough to describe basic arithmetic is either incomplete (there are statements that cannot be proved or disproved within the system) or inconsistent (the system can prove both a statement and its negation)."
The proof of Gödel's Incompleteness Theorem involves constructing a formula that asserts its own consistency, and then showing that if the formula is consistent, it must be unprovable within the system. This leads to a contradiction, which implies that the system must be either incomplete or inconsistent.
Gödel's Incompleteness Theorem has profound implications for the foundations of mathematics and computer science but it does not directly address the question of one-way functions or the relationship between P and NP. To see why, consider what a one-way function would look like in computational terms:
A one-way function f takes an input x and produces an output y such that it is computationally infeasible to compute x from y. In other words, given y, it is hard to find x without knowing the underlying algorithm or structure that generated y.
However, this property of one-wayness does not rely on the consistency or completeness of any particular formal system. Instead, it relies on the computational difficulty of reversing the function, which could still hold even if the underlying formal system were inconsistent or incomplete.
Therefore, while Gödel's Incompleteness Theorem establishes limits on the power of formal systems, it does not directly impact the existence or non-existence of one-way functions or the P vs. NP problem. Indirectly however it can help us form hypothesese and develop heuristics for determining whether one-way functions exist and whether P !=NP.
CONCLUSION:
Despite objections, if mathematical ideas are in fact reflections of empirical reality then we may make probabalistic inferences from them to determine heuristics to solve contentious ideas. Since Gödels theorems are reflections of the material world they may be used to imply solutions for similar problems such as whether one way functions exist and whether P != NP. If Gödels theorems are reflections of material reality then it would be logically consistent for one way functions to exist and for P != NP and thus one way cryptographic functions such as public key cryptography are likely secure. A "postulated quantum computer" would be no solution to the problem of cryptography since quantum phenonmena, consistent with Gödels theorems, are also uncertain.
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