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Let's now examine public key asymmetric algorithms.

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Let's start with their operation scheme.

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There's the plain text that will be encrypted using a key.

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This time the key used for encryption will be called the public key.

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The output of this operation is the ciphertext But unlike with symmetric key algorithms the reverse

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encryption and to read the ciphertext a different key will be needed.

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The private key asymmetric cryptography makes use of two different keys although they're related to

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each other you can extract either from the other if you have a public key you will not be able to generate

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a corresponding private key.

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What are the benefits of replacing a shared key scheme with a pair of keys.

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First of all exchange is not the problem anymore.

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The user needs only two keys regardless of the number of other users they want to exchange data with

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a private key and a public key.

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The public doesn't have or even shouldn't be protected.

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The name is self-explanatory.

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The key is public and should be shared with all users in the system.

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A private key on the other hand is a user's exclusive property if it's shared the entire system is compromised

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secure data exchange in public key cryptography systems with 50 users requires 100 keys to keys per

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user symmetric key systems required as many as 1225.

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What's more asymmetric algorithms allow users to establish a trust relationship.

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They have one strength that symmetric algorithms lacked the so-called non repudiation.

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You can use this feature to establish the authenticity of the ciphertext.

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We'll show you how in a moment and also to prove to a message sender that the message truly comes from

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them and that its integrity has not been breached.

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This solution has some shortcomings as well.

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Public key cryptography is extremely slow as you've seen the basic operation used in symmetric ciphers

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was the exclusive disjunction ex-SO are this operation is quick and equivalent of the X-O are an asymmetric

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cryptography is a Madugalle operation.

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The result of modular exponentiation can be a large prime this operation has a high computational cost

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and other weaknesses the asymmetric ciphers have not been designed for encrypting long messages.

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This flows naturally out of the first problem.

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Several well documented attacks on metric ciphers have been developed that rely on excessive key use

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the situation involves using a key to encrypt messages that are longer than necessary.

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As a rule asymmetric encryption should be applied to data that is of similar length to the used key

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kids should be a bit longer than the plaintext.

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The strength of a 512 bit asymmetric key is more or less equivalent to the strength of a 60 bit symmetric

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key as you know 60 bits does not provide adequate security.

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512 bits is likewise insufficient.

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Today even a 1024 bit strength can proved not enough people do but the.

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We'll talk about this at greater length later in the module.

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There are also problems that don't stream from any cryptographic solutions if asymmetric algorithms

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allow you to establish trust relationships.

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You need to keep in mind that ciphers and keys are used by software not by users themselves.

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You shouldn't transpose this trust freely.

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The starting point for asymmetric encryption depends on finding some one way functions one way functions

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or isn't sufficient to compute one direction but computationally hard to invert it's difficult to establish

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the exact computational complexity of these functions.

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Whether it's an MP a complete problem or the complexity is simply over exponential is determined by

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the development of mathematics.

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A good example of this is the controversy surrounding the RSA algorithm which we'll talk about soon.

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The RSA algorithm security is based on the fact that multiplying is relatively easy it's fast.

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While the decomposition of the object into factors factorization has high computational complexity just

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how complex is it.

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Factoring problems were analyzed by Turing one of the greatest mathematicians of the 20th century Turing's

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paper on factorization remains classified confidential.

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The UK has been keeping and handling this document as an exclusive possession.

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What is the reason for this.

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It could be that in fact efficient factoring methods exist after all.

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I can risk saying this because there is a peculiar correlation mathematicians that examined this problem

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and published works on factorization and did disappear from the scientific limelight in scientific life

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altogether.

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They are probably working now for agencies that pay better and beginning from that point.

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It is

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this is relevant since practically all facets of our security depend on the assumed high computational

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complexity of factorization many critically important systems have been based on the RSA algorithm.

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If a simple laptop would be enough to crack the algorithm we'd be in a tight spot.

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RSA is not the only asymmetric algorithm El-Gamal is just as popular and perhaps even more secure.

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The two algorithms are designed for encryption and decryption while the TSA algorithm is used for digital

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signatures El-Gamal NTSA are based on a discrete logarithms problem.

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The logarithm is the inverse operation of exponentiation.

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The exponentiation can be modular Ahmadullah is counting to 20 on ten fingers.

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The characteristic trait of the Modelo operation is that while something is lost not all information

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is lost.

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The same was true for the axle or operation.

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Since you can perform exponentiation over a MONDOUBLEAU you can calculate Ahmadullah logarithm.

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As we mentioned El-Gamal and DSA are based on the computational simplicity of modular exponentiation.

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This is the logarithmic complexity class inverting this operation seems to require a brute force search

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of all possible exponent values.