Bitte aktivieren Sie JavaScript. Si prega di abilitare JavaScript. English EN. English en. Deutsch de. No suggestions found. Sign in. Results Packs. About us. Fact Sheet. Objective Cryptology is a foundation of information security in the digital world. In particular, this proposal questions the two pillars of public-key encryption: factoring and discrete logarithms.

Recently, the PI contributed to show that in some cases, the discrete logarithm problem is considerably weaker than previously assumed.

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We will study the generalization of the recent techniques and search for new algorithmic options with comparable or better efficiency. We will also study hardness assumptions based on codes and subset-sum, two candidates for post-quantum cryptography. We will consider the applicability of recent algorithmic and mathematical techniques to the resolution of the corresponding putative hard problems, refine the analysis of the algorithms and design new algorithm tools.

Should the security of these other assumptions become critical, they would be added to Almacrypt's scope. They could also serve to demonstrate other applications of our algorithmic progress. Programme s HEU. Activity type Higher or Secondary Education Establishments. Website Contact the organisation. Project information AlmaCrypt. Project website. Status Ongoing project. Start date 1 January End date 31 December This definable operator forms a "group" of finite length.

To add two points on an elliptic curve, we first need to understand that any straight line that passes through this curve intersects it at precisely three points. Now, say we define two of these points as u and v: we can then draw a straight line through two of these points to find another intersecting point, at w.

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We can then draw a vertical line through w to find the final intersecting point at x. This rule works, when we define another imaginary point, the Origin, or O, which exists at theoretically extreme points on the curve. As strange as this problem may seem, it does permit for an effective encryption system, but it does have its detractors. On the positive side, the problem appears to be quite intractable, requiring a shorter key length thus allowing for quicker processing time for equivalent security levels as compared to the Integer Factorization Problem and the Discrete Logarithm Problem.

On the negative side, critics contend that this problem, since it has only recently begun to be implemented in cryptography, has not had the intense scrutiny of many years that is required to give it a sufficient level of trust as being secure. This leads us to more general problem of cryptology than of the intractability of the various mathematical concepts, which is that the more time, effort, and resources that can be devoted to studying a problem, then the greater the possibility that a solution, or at least a weakness, will be found.

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Make it easier Each year, Computer Weekly announces the women who will be added to its Hall of Fame, a list of women recognised for their Now say we want to find the value of N, so that value is found by the following formula:. This is known as discrete exponentiation and is quite simple to compute.

However, the opposite is true when we invert it. If we are given P, a, and N and are required to find b so that the equation is valid, then we face a tremendous level of difficulty. This problem forms the basis for a number of public key infrastructure algorithms, such as Diffie-Hellman and EIGamal.

This problem has been studied for many years and cryptography based on it has withstood many forms of attacks. The Integer Factorization Problem: This is simple in concept. Say that one takes two prime numbers, P2 and P1, which are both "large" a relative term, the definition of which continues to move forward as computing power increases.

We then multiply these two primes to produce the product, N. The difficulty arises when, being given N, we try and find the original P1 and P2. The Rivest-Shamir-Adleman public key infrastructure encryption protocol is one of many based on this problem. To simplify matters to a great degree, the N product is the public key and the P1 and P2 numbers are, together, the private key. This problem is one of the most fundamental of all mathematical concepts.

It has been studied intensely for the past 20 years and the consensus seems to be that there is some unproven or undiscovered law of mathematics that forbids any shortcuts.

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## Protocols, cryptanalysis and mathematical cryptology

That said, the mere fact that it is being studied intensely leads many others to worry that, somehow, a breakthrough may be discovered. The Elliptic Curve Discrete Logarithm Problem: This is a new cryptographic protocol based upon a reasonably well-known mathematical problem. The properties of elliptic curves have been well known for centuries, but it is only recently that their application to the field of cryptography has been undertaken. First, imagine a huge piece of paper on which is printed a series of vertical and horizontal lines.

Each line represents an integer with the vertical lines forming x class components and horizontal lines forming the y class components. The intersection of a horizontal and vertical line gives a set of coordinates x,y.

## Mathematical Cryptology for Computer Scientists and Mathematicians by Wayne Patterson

In the highly simplified example below, we have an elliptic curve that is defined by the equation:. For the above, given a definable operator, we can determine any third point on the curve given any two other points. This definable operator forms a "group" of finite length. To add two points on an elliptic curve, we first need to understand that any straight line that passes through this curve intersects it at precisely three points. Now, say we define two of these points as u and v: we can then draw a straight line through two of these points to find another intersecting point, at w.

We can then draw a vertical line through w to find the final intersecting point at x. This rule works, when we define another imaginary point, the Origin, or O, which exists at theoretically extreme points on the curve. As strange as this problem may seem, it does permit for an effective encryption system, but it does have its detractors.

On the positive side, the problem appears to be quite intractable, requiring a shorter key length thus allowing for quicker processing time for equivalent security levels as compared to the Integer Factorization Problem and the Discrete Logarithm Problem. On the negative side, critics contend that this problem, since it has only recently begun to be implemented in cryptography, has not had the intense scrutiny of many years that is required to give it a sufficient level of trust as being secure.

This leads us to more general problem of cryptology than of the intractability of the various mathematical concepts, which is that the more time, effort, and resources that can be devoted to studying a problem, then the greater the possibility that a solution, or at least a weakness, will be found. Please check the box if you want to proceed. When using multiple cloud service providers, it's critical to consider your enterprise's cloud scope and the specifics of each