BOX Index
BOX Index
BOX Index
BOX Index
NEM8
NEM8
NEM8
xBORDER DEVELOPMENT
xBORDER DEVELOPMENT
xBORDER CONCEPT DEVELOPMENT
RF
RF
NERD
NERD
NERD
CHAPTER ONE: THE ILLUSION OF REALITY.
Products  Books  Courses  Firewalls  Computer  Software  Trackers  Scanners  Bugs Tracers  Trackers  Volt  Wifi  Network  LTE  5G  GSM  Cameras  Advanced Spy Gear
THE UNION
GET REAL, WE LIVING IN ONE OF
AN LYNX && ASH  ORIGINAL XL PRESENTATION
xBORDER && CLOUDX @ THE BLOCKCHAIN RESEARCH INSTITUTE &&
IN  { AN UN_REAL_SOFTWARE @ ACIRRF POWERED  IOTENGINE
UNREAL SURREALITY
^^ REALITY , PRIVACY && PENETRATING TECHNOLOGY REALITY OF
THE HIGHEST LEVEL OF PROFFESIONNAL MILITAIRYGRADE HACKING,
TECHNOLOGY THAT IS UNKNOWN AT THE BIGGEST ENTERPRISES, =
GOOGLE, MICROSOFT, APPLE, PHONES OR IPHONES, FIREWALLS, VPN?
WELCOME TO  NERDRF TECHNOLOGY 
STARDATE NO.4 E.M. S.X 4  E.M. SPACEX
MILITAIR EMPOWERED HACKING TECHNOLOGY
VPN'S ? FIREWALLS ? VIRUSSCANNERS ?
WHAT ABOUT ACCESSING IPHONES / ANDROIDS TROUGH THE ELECTRIC POWER NET WHILE THERE CHARCHING ?
WELCOME TO NERD  RF,
TRY UNION_SOFTWARE FOR FREE,.
SAFE AND SECURE, THE NO.1 IN THE WORLD.
pW oLDENBURGER,, OLDENBURGER NEWS &&JOURNAL.
{ ALSO A JUNIOR DEVELOPER :p ISM. PRESS POINT.
.
xBORDER PROJECT 8: BACK_IT_UP
{ PROTECTING THE CORE }
PROJECT 8: CHAPTER ONE: THE ILLUSION OF REALITY.
N
PROJECT 8: CHAPTER ONE: THE ILLUSION OF REALITY.
The RSA is an Algorithm used by modern computers to encrypt/ing and decrypt/ing messages. It is an asymmetric cryptographic algorithm. Asymmetric means that there are two different keys. This is also called public key cryptography, because one of the keys can be given to anyone. The other key must be kept private. The algorithm is based on the fact that finding the factors of a large compositenumbers is difficult: when the factors are primenumbers, the problem is called primefactorization. It is also a key pair (public and private key) generator.

Loving the NERD

xBORDER CloudX
Operation
RSA involves a public key and private key. The public key can be known to everyone it is used to encrypt messages. Messages encrypted using the public key can only be decrypted with the private key. The keys for the RSA algorithm are generated the following way:

Choose two different large random prime numbers p {\displaystyle p\,} and q {\displaystyle q\,}

Calculate n = p q {\displaystyle n=pq\,}

n {\displaystyle n\,} is the modulus for the public key and the private keys

Calculate the totient: ϕ ( n ) = ( p − 1 ) ( q − 1 ) {\displaystyle \phi (n)=(p1)(q1)\,} .

Choose an integer e {\displaystyle e\,} such that 1 < e {\displaystyle e\,} < ϕ ( n ) {\displaystyle \phi (n)\,} , and e {\displaystyle e\,} is coprime to ϕ ( n ) {\displaystyle \phi (n)\,} i.e.: e {\displaystyle e\,} and ϕ ( n ) {\displaystyle \phi (n)\,} share no factors other than 1; gcd( e {\displaystyle e\,} , ϕ ( n ) {\displaystyle \phi (n)\,} ) = 1.

e {\displaystyle e\,} is released as the public key exponent


Compute d {\displaystyle d\,} to satisfy the congruence relation d e ≡ 1 ( mod ϕ ( n ) ) {\displaystyle de\equiv 1{\pmod {\phi (n)}}\,} i.e.: d e = 1 + x ϕ ( n ) {\displaystyle de=1+x\phi (n)\,} for some integer x {\displaystyle x\,} . (Simply to say : Calculate d = ( 1 + x ϕ ( n ) ) / e {\displaystyle d=(1+x\phi (n))/e\,} to be integer)

d {\displaystyle d\,} is kept as the private key exponent

Notes on the above steps:

Step 1: Numbers can be probabilistically tested for primality.


Step 3: changed in PKCS#1 (en) v2.0 to λ ( n ) = l c m ( p − 1 , q − 1 ) {\displaystyle \lambda (n)={\rm {lcm}}(p1,q1)\,} instead of ϕ ( n ) = ( p − 1 ) ( q − 1 ) {\displaystyle \phi (n)=(p1)(q1)\,} .

Step 4: A popular choice for the public exponents is e {\displaystyle e\,} = 216 + 1 = 65537. Some applications choose smaller values such as e {\displaystyle e\,} = 3, 5, or 35 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents may lead to greater security risks.

Steps 4 and 5 can be performed with the extended Euclidean algorithm (en); see modular arithmetic.
The public key is made of the modulus n {\displaystyle n\,} and the public (or encryption) exponent e {\displaystyle e\,} .
The private key is made of p,q and the private (or decryption) exponent d {\displaystyle d\,} which must be kept secret.

For efficiency a different form of the private key can be stored:

p {\displaystyle p\,} and q {\displaystyle q\,} : the primes from the key generation,

d mod ( p − 1 ) {\displaystyle d\mod (p1)\,} and d mod ( q − 1 ) {\displaystyle d\mod (q1)\,} : often called dmp1 and dmq1.

q − 1 mod ( p ) {\displaystyle q^{1}\mod (p)\,} : often called iqmp


All parts of the private key must be kept secret in this form. p {\displaystyle p\,} and q {\displaystyle q\,} are sensitive since they are the factors of n {\displaystyle n\,} , and allow computation of d {\displaystyle d\,} given e {\displaystyle e\,} . If p {\displaystyle p\,} and q {\displaystyle q\,} are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.

Although this form allows faster decryption and signing by using the Chinese Remainder Theorem (CRT) it is considerably less secure since it enables side channel attacks (en). This is a particular problem if implemented on smart cards, which benefit most from the improved efficiency. (Start with y = x e ( mod n ) {\displaystyle y=x^{e}{\pmod {n}}} and let the card decrypt that. So it computes y d ( mod p ) {\displaystyle y^{d}{\pmod {p}}} or y d ( mod q ) {\displaystyle y^{d}{\pmod {q}}} whose results give some value z {\displaystyle z} . Now, induce an error in one of the computations. Then gcd ( z − x , n ) {\displaystyle \gcd(zx,n)} will reveal p {\displaystyle p} or q {\displaystyle q} .)
Encrypting message
Alice gives her public key ( n {\displaystyle n\,} & e {\displaystyle e\,} ) to Bob and keeps her private key secret. Bob wants to send message M to Alice.
First he turns M into a number m {\displaystyle m} smaller than n {\displaystyle n} by using an agreedupon reversible protocol known as a padding scheme. He then computes the ciphertext c {\displaystyle c\,} corresponding to:
c = m e mod n {\displaystyle c=m^{e}\mod {n}}
This can be done quickly using the method of exponentiation by squaring. Bob then sends c {\displaystyle c\,} to Alice.
Decrypting message
Alice can recover m {\displaystyle m\,} from c {\displaystyle c\,} by using her private key d {\displaystyle d\,} in the following procedure:
m = c d mod n {\displaystyle m=c^{d}{\bmod {n}}}
Given m {\displaystyle m\,} , she can recover the original distinct prime numbers, applying the Chinese remainder theorem to these two congruences yields
m e d ≡ m mod p q {\displaystyle m^{ed}\equiv m{\bmod {pq}}} .
Thus,
c d ≡ m mod n {\displaystyle c^{d}\equiv m{\bmod {n}}} .
A working example
Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also use OpenSSL to generate and examine a real keypair.

Choose two random prime numbers.

: p = 61 {\displaystyle p=61} and q = 53 ; {\displaystyle q=53;} Compute n = p q {\displaystyle n=pq\,}

: n = 61 ∗ 53 = 3233 {\displaystyle n=61*53=3233}

Compute the totient ϕ ( n ) = ( p − 1 ) ( q − 1 ) {\displaystyle \phi (n)=(p1)(q1)\,}

: ϕ ( n ) = ( 61 − 1 ) ( 53 − 1 ) = 3120 {\displaystyle \phi (n)=(611)(531)=3120}

Choose e > 1 {\displaystyle e>1} coprime to 3120

: e = 17 {\displaystyle e=17}

Choose d {\displaystyle d\,} to satisfy d e mod ϕ ( n ) ≡ 1 {\displaystyle de{\bmod {\phi (n)}}\equiv 1\,}

: d = 2753 {\displaystyle d=2753}

: 17 ∗ 2753 = 46801 = 1 + 15 ∗ 3120 {\displaystyle 17*2753=46801=1+15*3120} .
The public key is ( n = 3233 {\displaystyle n=3233} , e = 17 {\displaystyle e=17} ). For a padded message m {\displaystyle m\,} the encryption function c = m e mod n {\displaystyle c=m^{e}{\bmod {n}}} becomes:
c = m 17 mod 3 233 {\displaystyle c=m^{17}{\bmod {3}}233\,}
The private key is ( n = 3233 {\displaystyle n=3233} , d = 2753 {\displaystyle d=2753} ). The decryption function m = c d mod n {\displaystyle m=c^{d}{\bmod {n}}} becomes:
m = c 2753 mod 3 233 {\displaystyle m=c^{2753}{\bmod {3}}233\,}
For example, to encrypt m = 123 {\displaystyle m=123} , we calculate
c = 123 17 mod 3 233 = 855 {\displaystyle c=123^{17}{\bmod {3}}233=855}
To decrypt c = 855 {\displaystyle c=855} , we calculate
m = 855 2753 mod 3 233 = 123 {\displaystyle m=855^{2753}{\bmod {3}}233=123}
Both of these calculations can be computed efficiently using the squareandmultiply algorithm for modular exponentiation (en).
Deriving RSA equation from Euler's theorem
RSA can easily be derived using Euler's theorem and Euler's totient function.
Starting with Euler's theorem,
m φ ( n ) ≡ 1 ( mod n ) {\displaystyle m^{\varphi (n)}\equiv 1{\pmod {n}}}
Using Modular arithmetic, the equation can be rewritten as
Multiply both sides by m
m ≡ m φ ( n ) + 1 ( mod n ) {\displaystyle m\equiv m^{\varphi (n)+1}{\pmod {n}}}
Euler's totient function has the property for some values p and q,
φ ( p × q ) = φ ( p ) × φ ( q ) {\displaystyle \varphi (p\times q)=\varphi (p)\times \varphi (q)}
Therefore, the equation can be rewritten as
m ≡ m φ ( p × q ) + 1 ( mod p × q ) {\displaystyle m\equiv m^{\varphi (p\times q)+1}{\pmod {p\times q}}}
Furthermore, for all integers k, the following relation still holds:
m ≡ m k × φ ( p × q ) + 1 ( mod p × q ) {\displaystyle m\equiv m^{k\times \varphi (p\times q)+1}{\pmod {p\times q}}}
The value d is derived from the following equation, where e is some large prime number and for some k such that d is an integer:
d = k × φ ( p × q ) + 1 e {\displaystyle d={k\times {\varphi (p\times q)+1} \over {e}}}
And by exponent rules, the following statement holds:
m = m e k × φ ( p × q ) + 1 e {\displaystyle m=m^{e^{k\times {\varphi (p\times q)+1} \over {e}}}}
As the e's will cancel out. So when encrypting the ciphertext m,
c ≡ m e ( mod p × q ) {\displaystyle c\equiv m^{e}{\pmod {p\times q}}}
The original value of m can be deduced from c by raising c to d.
m ≡ c d ( mod p × q ) ≡ m e d ( mod p × q ) ≡ m e k × φ ( p × q ) + 1 e ( mod p × q ) {\displaystyle m\equiv c^{d}{\pmod {p\times q}}\equiv m^{e^{d}}{\pmod {p\times q}}\equiv m^{e^{k\times {\varphi (p\times q)+1} \over {e}}}{\pmod {p\times q}}}
The equation shows the equivalence holds.and the progress
Padding schemes
When used in practice, RSA must be combined with some form of padding scheme, so that no values of M result in insecure ciphertexts. RSA used without padding may have some problems:

The values m = 0 or m = 1 always produce ciphertexts equal to 0 or 1 respectively, due to the properties of exponentiation.

When encrypting with small encryption exponents (e.g., e = 3) and small values of the m, the (nonmodular) result of m e {\displaystyle m^{e}} may be strictly less than the modulus n. In this case, ciphertexts may be easily decrypted by taking the eth root of the ciphertext with no regard to the modulus.

RSA encryption is a deterministic encryption algorithm. It has no random component. Therefore, an attacker can successfully launch a chosen plaintext attack against the cryptosystem. They can make a dictionary by encrypting likely plaintexts under the public key, and storing the resulting ciphertexts. The attacker can then observe the communication channel. As soon as they see ciphertexts that match the ones in their dictionary, the attackers can then use this dictionary in order to learn the content of the message.
In practice, the first two problems can arise when short ASCII messages are sent. In such messages, m might be the concatenation of one or more ASCIIencoded character(s). A message consisting of a single ASCII NUL character (whose numeric value is 0) would be encoded as m = 0, which produces a ciphertext of 0 no matter which values of e and N are used. Likewise, a single ASCII SOH (whose numeric value is 1) would always produce a ciphertext of 1. For systems which conventionally use small values of e, such as 3, all single character ASCII messages encoded using this scheme would be insecure, since the largest m would have a value of 255, and 2553 is less than any reasonable modulus. Such plaintexts could be recovered by simply taking the cube root of the ciphertext.
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that m does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts. The latter property can increase the cost of a dictionary attack beyond the capabilities of a reasonable attacker.
Standards such as PKCS have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with some number of additional bits, the size of the unpadded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks. This may be made easier by a predictable message structure. Early versions of the PKCS standard used adhoc constructions, which were later found vulnerable to a practical adaptive chosen ciphertext attack. Modern constructions use secure techniques such as Optimal Asymmetric Encryption Padding (OAEP) to protect messages while preventing these attacks. The PKCS standard also has processing schemes designed to provide additional security for RSA signatures, e.g., the Probabilistic Signature Scheme for RSA (RSAPSS).
Signing messages
Suppose Alice uses Bob's public key to send him an encrypted message. In the message, she can claim to be Alice but Bob has no way of verifying that the message was actually from Alice since anyone can use Bob's public key to send him encrypted messages. So, in order to verify the origin of a message, RSA can also be used to sign a message.
Suppose Alice wishes to send a signed message to Bob. She produces a hash value of the message, raises it to the power of d mod n (just like when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he raises the signature to the power of e mod n (just like encrypting a message), and compares the resulting hash value with the message's actual hash value. If the two agree, he knows that the author of the message was in possession of Alice's secret key, and that the message has not been tampered with since.
Note that secure padding schemes such as RSAPSS are as essential for the security of message signing as they are for message encryption, and that the same key should never be used for both encryption and signing purposes.
References
Other websites

The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, U.S. Patent 4,405,829 .

PKCS #1: RSA Cryptography Standard (RSA Laboratories website)

The PKCS #1 standard "provides recommendations for the implementation of publickey cryptography based on the RSA algorithm, covering the following aspects: cryptographic primitives; encryption schemes; signature schemes with appendix; ASN.1 syntax for representing keys and for identifying the schemes".


Example of an RSA implementation with PKCS#1 padding (GPL source code)

An animated explanation of RSA with its mathematical background by CrypTool

Hacking Secret Ciphers with Python, Chapter 24, Public Key Cryptography and the RSA Cipher

Grime, James. "RSA Encryption". Numberphile. Brady Haran.

Prime Numbers, Factorization, and their Relationship with Encryption
N
PROJECT 8: CHAPTER ONE: THE ILLUSION OF REALITY.

Step 1: Numbers can be probabilistically tested for primality.


Step 3: changed in PKCS#1 (en) v2.0 to λ ( n ) = l c m ( p − 1 , q − 1 ) {\displaystyle \lambda (n)={\rm {lcm}}(p1,q1)\,} instead of ϕ ( n ) = ( p − 1 ) ( q − 1 ) {\displaystyle \phi (n)=(p1)(q1)\,} .

Step 4: A popular choice for the public exponents is e {\displaystyle e\,} = 216 + 1 = 65537. Some applications choose smaller values such as e {\displaystyle e\,} = 3, 5, or 35 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents may lead to greater security risks.

Steps 4 and 5 can be performed with the extended Euclidean algorithm (en); see modular arithmetic.
The public key is made of the modulus n {\displaystyle n\,} and the public (or encryption) exponent e {\displaystyle e\,} .
The private key is made of p,q and the private (or decryption) exponent d {\displaystyle d\,} which must be kept secret.

For efficiency a different form of the private key can be stored:

p {\displaystyle p\,} and q {\displaystyle q\,} : the primes from the key generation,

d mod ( p − 1 ) {\displaystyle d\mod (p1)\,} and d mod ( q − 1 ) {\displaystyle d\mod (q1)\,} : often called dmp1 and dmq1.

q − 1 mod ( p ) {\displaystyle q^{1}\mod (p)\,} : often called iqmp


All parts of the private key must be kept secret in this form. p {\displaystyle p\,} and q {\displaystyle q\,} are sensitive since they are the factors of n {\displaystyle n\,} , and allow computation of d {\displaystyle d\,} given e {\displaystyle e\,} . If p {\displaystyle p\,} and q {\displaystyle q\,} are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.

Although this form allows faster decryption and signing by using the Chinese Remainder Theorem (CRT) it is considerably less secure since it enables side channel attacks (en). This is a particular problem if implemented on smart cards, which benefit most from the improved efficiency. (Start with y = x e ( mod n ) {\displaystyle y=x^{e}{\pmod {n}}} and let the card decrypt that. So it computes y d ( mod p ) {\displaystyle y^{d}{\pmod {p}}} or y d ( mod q ) {\displaystyle y^{d}{\pmod {q}}} whose results give some value z {\displaystyle z} . Now, induce an error in one of the computations. Then gcd ( z − x , n ) {\displaystyle \gcd(zx,n)} will reveal p {\displaystyle p} or q {\displaystyle q} .)
Encrypting message
Alice gives her public key ( n {\displaystyle n\,} & e {\displaystyle e\,} ) to Bob and keeps her private key secret. Bob wants to send message M to Alice.
First he turns M into a number m {\displaystyle m} smaller than n {\displaystyle n} by using an agreedupon reversible protocol known as a padding scheme. He then computes the ciphertext c {\displaystyle c\,} corresponding to:
c = m e mod n {\displaystyle c=m^{e}\mod {n}}
This can be done quickly using the method of exponentiation by squaring. Bob then sends c {\displaystyle c\,} to Alice.
Decrypting message
Alice can recover m {\displaystyle m\,} from c {\displaystyle c\,} by using her private key d {\displaystyle d\,} in the following procedure:
m = c d mod n {\displaystyle m=c^{d}{\bmod {n}}}
Given m {\displaystyle m\,} , she can recover the original distinct prime numbers, applying the Chinese remainder theorem to these two congruences yields
m e d ≡ m mod p q {\displaystyle m^{ed}\equiv m{\bmod {pq}}} .
Thus,
c d ≡ m mod n {\displaystyle c^{d}\equiv m{\bmod {n}}} .
A working example
Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also use OpenSSL to generate and examine a real keypair.

Choose two random prime numbers.

: p = 61 {\displaystyle p=61} and q = 53 ; {\displaystyle q=53;} Compute n = p q {\displaystyle n=pq\,}

: n = 61 ∗ 53 = 3233 {\displaystyle n=61*53=3233}

Compute the totient ϕ ( n ) = ( p − 1 ) ( q − 1 ) {\displaystyle \phi (n)=(p1)(q1)\,}

: ϕ ( n ) = ( 61 − 1 ) ( 53 − 1 ) = 3120 {\displaystyle \phi (n)=(611)(531)=3120}

Choose e > 1 {\displaystyle e>1} coprime to 3120

: e = 17 {\displaystyle e=17}

Choose d {\displaystyle d\,} to satisfy d e mod ϕ ( n ) ≡ 1 {\displaystyle de{\bmod {\phi (n)}}\equiv 1\,}

: d = 2753 {\displaystyle d=2753}

: 17 ∗ 2753 = 46801 = 1 + 15 ∗ 3120 {\displaystyle 17*2753=46801=1+15*3120} .
The public key is ( n = 3233 {\displaystyle n=3233} , e = 17 {\displaystyle e=17} ). For a padded message m {\displaystyle m\,} the encryption function c = m e mod n {\displaystyle c=m^{e}{\bmod {n}}} becomes:
c = m 17 mod 3 233 {\displaystyle c=m^{17}{\bmod {3}}233\,}
The private key is ( n = 3233 {\displaystyle n=3233} , d = 2753 {\displaystyle d=2753} ). The decryption function m = c d mod n {\displaystyle m=c^{d}{\bmod {n}}} becomes:
m = c 2753 mod 3 233 {\displaystyle m=c^{2753}{\bmod {3}}233\,}
For example, to encrypt m = 123 {\displaystyle m=123} , we calculate
c = 123 17 mod 3 233 = 855 {\displaystyle c=123^{17}{\bmod {3}}233=855}
To decrypt c = 855 {\displaystyle c=855} , we calculate
m = 855 2753 mod 3 233 = 123 {\displaystyle m=855^{2753}{\bmod {3}}233=123}
Both of these calculations can be computed efficiently using the squareandmultiply algorithm for modular exponentiation (en).
Deriving RSA equation from Euler's theorem
RSA can easily be derived using Euler's theorem and Euler's totient function.
Starting with Euler's theorem,
m φ ( n ) ≡ 1 ( mod n ) {\displaystyle m^{\varphi (n)}\equiv 1{\pmod {n}}}
Using Modular arithmetic, the equation can be rewritten as
Multiply both sides by m
m ≡ m φ ( n ) + 1 ( mod n ) {\displaystyle m\equiv m^{\varphi (n)+1}{\pmod {n}}}
Euler's totient function has the property for some values p and q,
φ ( p × q ) = φ ( p ) × φ ( q ) {\displaystyle \varphi (p\times q)=\varphi (p)\times \varphi (q)}
Therefore, the equation can be rewritten as
m ≡ m φ ( p × q ) + 1 ( mod p × q ) {\displaystyle m\equiv m^{\varphi (p\times q)+1}{\pmod {p\times q}}}
Furthermore, for all integers k, the following relation still holds:
m ≡ m k × φ ( p × q ) + 1 ( mod p × q ) {\displaystyle m\equiv m^{k\times \varphi (p\times q)+1}{\pmod {p\times q}}}
The value d is derived from the following equation, where e is some large prime number and for some k such that d is an integer:
d = k × φ ( p × q ) + 1 e {\displaystyle d={k\times {\varphi (p\times q)+1} \over {e}}}
And by exponent rules, the following statement holds:
m = m e k × φ ( p × q ) + 1 e {\displaystyle m=m^{e^{k\times {\varphi (p\times q)+1} \over {e}}}}
As the e's will cancel out. So when encrypting the ciphertext m,
c ≡ m e ( mod p × q ) {\displaystyle c\equiv m^{e}{\pmod {p\times q}}}
The original value of m can be deduced from c by raising c to d.
m ≡ c d ( mod p × q ) ≡ m e d ( mod p × q ) ≡ m e k × φ ( p × q ) + 1 e ( mod p × q ) {\displaystyle m\equiv c^{d}{\pmod {p\times q}}\equiv m^{e^{d}}{\pmod {p\times q}}\equiv m^{e^{k\times {\varphi (p\times q)+1} \over {e}}}{\pmod {p\times q}}}
The equation shows the equivalence holds.and the progress
Padding schemes
When used in practice, RSA must be combined with some form of padding scheme, so that no values of M result in insecure ciphertexts. RSA used without padding may have some problems:

The values m = 0 or m = 1 always produce ciphertexts equal to 0 or 1 respectively, due to the properties of exponentiation.

When encrypting with small encryption exponents (e.g., e = 3) and small values of the m, the (nonmodular) result of m e {\displaystyle m^{e}} may be strictly less than the modulus n. In this case, ciphertexts may be easily decrypted by taking the eth root of the ciphertext with no regard to the modulus.

RSA encryption is a deterministic encryption algorithm. It has no random component. Therefore, an attacker can successfully launch a chosen plaintext attack against the cryptosystem. They can make a dictionary by encrypting likely plaintexts under the public key, and storing the resulting ciphertexts. The attacker can then observe the communication channel. As soon as they see ciphertexts that match the ones in their dictionary, the attackers can then use this dictionary in order to learn the content of the message.
In practice, the first two problems can arise when short ASCII messages are sent. In such messages, m might be the concatenation of one or more ASCIIencoded character(s). A message consisting of a single ASCII NUL character (whose numeric value is 0) would be encoded as m = 0, which produces a ciphertext of 0 no matter which values of e and N are used. Likewise, a single ASCII SOH (whose numeric value is 1) would always produce a ciphertext of 1. For systems which conventionally use small values of e, such as 3, all single character ASCII messages encoded using this scheme would be insecure, since the largest m would have a value of 255, and 2553 is less than any reasonable modulus. Such plaintexts could be recovered by simply taking the cube root of the ciphertext.
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that m does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts. The latter property can increase the cost of a dictionary attack beyond the capabilities of a reasonable attacker.
Standards such as PKCS have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with some number of additional bits, the size of the unpadded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks. This may be made easier by a predictable message structure. Early versions of the PKCS standard used adhoc constructions, which were later found vulnerable to a practical adaptive chosen ciphertext attack. Modern constructions use secure techniques such as Optimal Asymmetric Encryption Padding (OAEP) to protect messages while preventing these attacks. The PKCS standard also has processing schemes designed to provide additional security fo
N
THE RISE OF NERD
NETHERLANDS< AMSTERDAM. {2:THE EXPERIMENT OF NERD.}
#THE EXPERIMENT OF NERD: PRERELEASING THE xBORDER PLATFORM.
\
TAKE THIS UNIQUE CHAN
ACKING ALL FAMOUS VIRUSSCANNERS WHITEPAPER / PDF
NETHERLANDS< AMSTERDAM. {2:THE EXPERIMENT OF NERD.}
#THE EXPERIMENT OF NERD: PRERELEASING THE xBORDER PLATFORM.
\
TAKE THIS UNIQUE CHAN
EPISODE 1:
2019
THE RISE OF NERD
IN THE WORLD OF IA SOFTWARE IS CHANGING, VIRUSSCANNERS
PROTECTING YOUR SYSTEMS 'BRAIN" IN A VM CKOUD, PINPELINED TO YOU ENCRYPTED BY OUR x11 NIGHTSHADE 524SHA BLOCKCHAIN. PROTOCOL. THE FUTRUE OF MALWARE PROTECTION. STORED IN AMSTERDAM, DATACENTER OF THE WORLD.
INJECTING DLL
DOWNLOAD INJECTOR
SPECTRUM ANALYSING
NERD SPECTRUM ANALYSER + SOFTWARE
WIFI, VLAN, WLAN, GSM EXPLAINED.
NETHERLANDS< AMSTERDAM.
B
PRE_RELEASE OF THE xBORDER PROJECT8.
W
WE OFFER A PRERELEASE IN THE NETHERLANDS, TO JOIN OUR SYNDICATE. WE
PROGRAMMERS, PHYTHON,C++, VISUALB, VISUAL ARTISTS, GEEKS, NERDS, 3D DESIGNERS, GOVERNMENT WORKERS<,
WORLDS FIRST BLOCKCHAIN POWERED, APIINTELLIGENT ECOMMERCE NETWORK TO BE RELEASED CROWDFUNDED WORLDWIDE.
THE DEAL :
EXCLUSIVE PRE_RELEASED IN THE NETHERLANDS, THE MAYOR RELEASE, OF ALL 8 PROJECTS IS END JANUAR,
 DAILY MEETUPS IN AMSTERDAM  YOUR INPUT IS VALUED  GENERATE MONRE DIGISHARES  JOIN THE RELEASE TEAM.
WORKING IN SYNDICATE OUR TEAM WILL DEPLOY THE PROJECTS ONE BY ONE, AND SET THE DATE OF THE CROWDFUND, RELEASED
IN THE UNITED STATES, SILIV
2019
THE RISE OF NERD
NETHERLANDS< AMSTERDAM.
2013, First Philosophy,
After having been conceived as a science of the Artificial Intelligent,..
2016, First Contact,
While working on my Projects, First Contact was made,..
2017, First Contact,
Contact has been confirmed, Governments are Interested in the Platform,..
2018, MultiLayers of APIIntelligent SignCods, or Blockchains.
INERD was looking in MEETUPS and other Interesting places: NERD saw mostly
MEETUPS reffering BlockChain as a Currency, after Suckerberg and his magic
idea of deploying Libra, (retard) , NERD, wntsto show the Worlds what Blockchain
really is: Small Intelligent SIGNING of CODE. ( =010) 01,111,010, S, Like exatoshi, ment the
Blockchain Satoshi=fouder of Blockchain Japan,
Blockchainj is just a 'HYPE WORD'for Intelligent Code Signing.
Intelligent, in the Chain, Means There has to be an Response.
Signing=Respons mjust be Included.
(else it would be Smart Code, or
Blockchain or Intelligent Code Signing is the very Basic of a a New Age.
o
T
Like Electriciry (or ELectra TESLA, Light EDISON  Biggest
Movement of Europeans to America. ) (Light) and before that Steam Pressure. (Carnagy Travel, Train))
Followed by Mass Railroad Deployment East West US, foloowed by the Industrial Revolution.
Transformation TESLA, Electrra  TESLA  LIGHBULB EDISON  Future Leap.
: E
Internet, Computer Chip Software  2015/2019 DIscovery: Quantum Physica. \
Intelligent Code Signing  Nano Technology  Quantum Physica  Cybernetic Age
Cybernetics  Bionic Technologies  StarWars and Inifinite Life.
Human will not go FASTER than Light : He Will Bend Space Time, So the Result
will be FASTER than LIGHT , but he will NOT travel faster than Light... GET IT ?
op
THere is no awnser for that . sorry. HA Hawkings says that everything will stop
6 PATENTS @ UNITED STATES TRADEMARK&PATENT OFFICE USPTO, CODE 6. (READY Kyoto. FOR DEPLOY W)
https://trademarks.justia.com/owners/xborderltd3727681/
https://trademarks.justia.com/877/35/xvote87735175.html
WORLDS FIRST BLOCKCHAIN POWERED, APIINTELLIGENT ECOMMERCE NETWORK TO BE RELEASED CROWDFUNDED WORLDWIDE.
THE DEAL :
EXCLUSIVE PRE_RELEASED IN THE NETHERLANDS, THE MAYOR RELEASE, OF ALL 8 PROJECTS IS END JANUAR,
 DAILY MEETUPS IN AMSTERDAM  YOUR INPUT IS VALUED  GENERATE MONRE DIGISHARES  JOIN THE RELEASE TEAM.
WORKING IN SYNDICATE OUR TEAM WILL DEPLOY THE PROJECTS ONE BY ONE, AND SET THE DATE OF THE CROWDFUND, RELEASED
IN THE UNITED STATES, SILIV
BOX Project 8
xBorder, a Blockchain based APIcontrollable (APIX) Cloud Network (CloudX) that contains multiple configurable Nodes, whom are controlled by its Nodemasters.
Project Xborder contains 8 Projectlayers that are userconfigurable and will be interconnected by Blockchain based SkyX contracts.
BOX = BLOCKX + CLOUDX + APIX + SANDBx
BLOCKX, Blockchain Based Interconnecting Ruling System.
CLOUDX, Secured Cloud Network.
APIX, APINetwork that configures, deploys, runs & utilizes EndPoints, Apps, authorizes contracts, websites.
SANDBx, MultiManager SANDBx, operates as a VPN Manager, creates virtual secured pipelines.
I would like to meet Geeks & Nerds whom are interested in Cloud and Crypto Layers, Technological Development and Startrek. in the xBORDER Network, Virtual Application that is Bendable in many configurations & (Bender!) applications
My mission is to Rise and Expand this project, mediawise (TV, Events, etc.) with the Grace of a CrowdFund open a Flagship Store in Amsterdam and Show the world that xBorder Platform was Born to be Global.
With 5 patents pending at USPTO Patents US and an Energy source from the Future, xBORDER has the Power and Creativity to become a Visionary in Blockchain based platforms.
Set: xBORDER will work fully transparent and fully open source.
Set: Many developing Teams are currently working on improved, fast and NonEnergyConsuming setups of "Scripting of Contracts", or making the secured calculations with aslessaspossibleenergy, however, for commercial deployment these Calculations are not fast enough. Making a Payment or Transcript, a Vote ( XVOTE !), an Authorization, or whatever have to be Instant, that is, for commercial use.
xBORDER Project 8,
combines, experiments, develops & integrates, multilayers or Cloud Based Software Programs like Acronis System Sync, NORD VPN's Network, WIX's CORVID Api Based CodeHTML Platform, and many more to come.
Experimenting, Negotiating, Deploying, small pieces of software, stepbystep an Ultiumate Intelligent Cloud based APINetwork will Rise and with the Platforms so we can build the Ultimate Intelligent Smart API Network. continue to implemetations of Softwarewe'll try to..
Visions, Visions of rockets that will reach for the Stars..
P. Oldenburger
INTELLIGENT NEXT LEVEL ECOMMERCE
APIX
APIX
WELCOME TO THE FUTURE OF BUSINESS
xBORDER Project 8  Chapter 1 Visions
Welcome, أ , ברוך הבWillkommen, هلا بك, Welkom, 欢迎欢迎, ようこそ, Bienvenue, 어서 오십시오,
All my Life I had Visions, Visions of building Robots that would reach for the Stars
Peter Willem Jacub Oldenburger
xBORDER Project 8, Project Infinity.
xBORDER focuses on deploying Business Concepts with Advance to realize dreams of selfreliance, entrepreneurship, creativity, and to independence. Fully APIEnhanced Intelligent Networks deployed in Virtual Cloud Environments. Controllable from the xBORDER Network by its Admins and Controllers, every Project is configurable through Block chain blockchain contracts deployed in CloudX and Xvoted by its owners. Projects are configurable by its owners, where ever you are, you can run a multi  million dollar emporium trough the xBORDER Network. The Ultimate APIEnhanced Intelligent Platform.
xBORDER is a leader in search engine optimization and placement.
xBORDER Search Agency offers cutting edge search engine marketing and placement solutions.
xBORDER search solution captures an individual at the moment they view their query result, creating profound and immediate opportunities.
xBORDER provides a full array of Web design and eCommerce solutions including Google SERP, SEA, SEO, search engine marketing (SEM), payperclick (PPC) management, affiliate program management, call center services, backend order management, print design, and consulting.
xBORDER devises efficient systems that integrate workers, machines, materials, information, and energy to make a product. Clothing, Cosmetics, Supermarket Assortments and more.
xBORDER Project Platform, Stay UpToDay with our Interactive Project Platform where ever you are and when you want. You can Adjust, Add, Delete, Change and Configurate your Project Online.
Contact us for an Introduction and what we can do for your Business.
Xborder Project
BlockChain
Crypto Network
API Network
Events
Blog
SiteIndex
Contact
Language
FAQ
CrowdFunding
CloudX
Cryptocurrency
Patents US
News
Forum
Whitepaper
Phone
Light
Font Big
Login
Whitepaper Project 8
Click links for demo's, read Whitepaper to white the paper.
The Singularity
TRise of the Machines
XBo
XBo
XBo
Past, Present, Future
xBorder Index
True Events
Blog Daily
Forum Talk
Hot News
Member Login
Platform Sneak Peak
2020: xBorder Project8
xBorder Network Platform
2021: xBorder xBlockchain
xBorder's CloudX
Megastore.com
BookGo.com
SexShop.app
Sandbx
KiX
xBorder Index
True Events
Blog Daily
Forum Talk
Hot News
Member Login
Platform Sneak Peak
2020: xBorder Project8
xBorder Network Platform
2021: xBorder xBlockchain
xBorder's CloudX
Megastore.com
BookGo.com
SexShop.app
Sandbx
KiX
Xborder Small Product Box
Available Now, will be implemented