Riemann's hypothesis and the Internet (I)

In 1740, the Swiss mathematician Leonhard Euler (1707-1783) introduced the Zeta function:


The Zeta function is an infinite series that converges to every real number. s > 1 and Euler demonstrated that it expresses itself as a convergent infinite product, currently known as Euler's product,

where the product is taken over all prime numbers Pno. Thus we have:


In 1859 the German mathematician Bernhard Riemann, one of the pioneers of modern mathematics, treated the Zeta function as a function of a complex variable. z. For this reason, the function is known as Riemann's Zeta function.

The Zeta function has no zeros in the complex plane region where Re(z) 1; if Re(z) 0 your zeros are z = - 2, - 4, - 6,…; and it has infinite zeros, called nontrivial zeros, in the complex plane region 0 < Re(z) < 1.

Riemann conjectured that nontrivial zeros are on the straight line. Re(z) = ½.

This conjecture is called the Riemann Hypothesis. The demonstration of the Prime Number Theorem by Hadamard and de la Vallée-Poussin in 1896 had crucial reasoning showing that When Re(z) = 1, ie the Zeta function has no zeros in the line Re(z) = 1.

In 1914, the brilliant British mathematician Godfrey Hardy demonstrated that infinite zeros of the Zeta function are on the line. Re(z) = ½. It is known that the first 1.5.109 zeros in region 0 < Re(z) <1 are all straight Re(z) = ½. However, a demonstration of the Riemann Hypothesis is awaited.

This conjecture is considered to be one of the biggest and most interesting open-ended problems in all of mathematics because, in addition to revealing a deep understanding of prime number distribution, it is also related to one of today's most important problems: internet security.

When we write a letter of purchase instructions or make a bank transaction at an ATM, we rely on mathematical knowledge about the behavior of prime numbers to maintain system security.

On the other hand, a demonstration of the Riemann Hypothesis could lead to great achievements in investigating large number factorization and endanger the techniques currently used to maintain the security of the world wide web.

The issue of keeping a message secret, so that only the intended recipient can understand it, is an old problem, especially if we think of military, diplomatic, or commercial matters. The way found, so that someone unauthorized who has access to a message does not understand it, was to encrypt it.

Cryptology is the discipline that deals with sensitive systems and their origins date back to Classical Antiquity, when the Greeks proposed the following solution: a messenger slave had his hair shaved and the message copied on his scalp. After hair growth, he was sent to the destination of the message. The recipient was shaving the slave's hair and reading the message. Of course, there was always the possibility that the slave could be intercepted by the enemy.

Roman emperor Julius Caesar proposed another solution, now called Caesar's cipher, to make the messages he sent to his generals on missions in Europe a secret.

The written message was modified so that each letter of the message was transformed into the next three letters of the Latin alphabet and the last three letters corresponded to the first letters of the alphabet as follows: D would correspond to A, and E would correspond to B,…, Y would correspond to V,…, A would correspond to X,…, C would correspond to Z.

Thus, Julius Caesar's famous phrase “VENI.VIDI.VICI.” (“I came. Vi. Venci.”) Became “YHQL.YGLG.YLFL.” .

THE cryptography is a discipline within the cryptology dealing with the design and implementation of sensitive systems and the crypto-analysis It is the discipline that deals with the decryption of these sensitive systems.

Encrypt is the procedure of turning a message into a text encrypted. That is, the letters of the message are modified by means of a specified transformation. THE key determines a particular transformation from a set of possible transformations.

The reverse process of encrypt It is known as decipher or decipherO. The desired recipient of the message has the method for decrypting it. This process is different from the process that someone other than the intended recipient uses to make the message intelligible, a process called crypto-analysis.

a crypto-system consists of a set of admissible messages, a set of possible encrypted messages, a set of keyswhere each key specifies a encryption function particular and their corresponding decryption functions.

Due to the possibility of intercepting messages and deciphering them, scientists have made a great effort to develop secure methods. In general, in today's crypto systems, the encryption procedure consists of a computer program, or a chip, and a key consists of a secretly chosen number.

The key chosen is essential to encrypt the message and the resulting ciphertext can only be decrypted with the help of another secret and exclusive key of a single user of the crypto-system. This way, the encryption program can be used by many people and for a certain period of time, as security is “guaranteed” by the secret and exclusive key.

In the early keyed systems, people who wanted to communicate had a common key. This condition has some disadvantages. For example, business or bank transactions involving people from different parts of the globe proved unfeasible, because to keep the system secure, people had to agree on a common key, but how to communicate secretly to choose the key?

In the 1970s, the notion of cryptographic public key was introduced and with the development of crypto-public key systems great progress was made. The main features of this system are the public key, simplicity and efficiency, because the difficulty of violating an encrypted message is very difficult. This idea was proposed in 1976 by mathematicians Whitifield Diffie and Martin Hellman.

In public key encryption, two keys are required, one for encryption and one for decryption. Suppose a new user X gets the default program used by all members of a given network. User X generates two keys: a key to decrypt messages, which he keeps secret, and another key that he uses to encrypt messages to be sent to him by anyone else on the network. The latter he publishes in a directory of network users.

Therefore, to send a message to X, one has to look for his public key, encrypt the message using that key, and send it. To decipher the message is not enough to know the public key, which is even available to anyone. Knowledge of the deciphering key, which is known only to X, is also required.

Numerous methods have been developed to implement the idea of ​​Diffie and Hellman, but the one that has received the most support and remains the standard today has been obtained by Rivest, Shamir and Adleman. This crypto-system, named RSA due to its authors' initials, is the most widely used public-key cryptographic system.

Security is based on integer factorization. X chooses two prime numbers P and what, each having at least 100 digits. These cousins ​​are randomly generated by computers so that there is no system breach. The secret decryption key consists of these two prime numbers. The public key that encrypts is the product no = why of these cousins. Since there is no quick method to completely factor large integers, the system stays safe because, as no one can figure out P and whatsay in less than one year, allows time to end the transaction without interference from intruders.

In the next column we will return to this exciting theme when we introduce the modular arithmetic needed to understand RSA crypto-system in more detail.

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