Exponent
Exponentiation is a mathematical operation, written as b raised to the power of n, involving two numbers, the base b and the exponent (or power) n.
Modulo
The modulo (sometimes called modulus) operation finds the remainder of division of one number by another
Number lower bit
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"1024 bit private exponent large number lower than the modulo" from question Checking RSA key pairs error |
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"In rsa signing a message m means exponentiation with the private exponent d the result r is the smallest integer 0 and smaller than the modulo n so that" from question RSA signature size? |
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"I assumed the biased exponent of the modulo is lower than the one of the number to be operated on as in the other case the mask is all ones and the number is not affected by the mask" from question Are there any numbers that enable fast modulo calculation on floats? |
Rsa shorter public
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"I must also add that designing the rsa key so that the private exponent is substantially shorter than the modulo to speed up operations is a security risk if the exponent is smaller than 29 of the modulo length then the key can be cracked" from question A factor that impacts RSA encryption/decryption time |
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"So no choice of the public exponent for this modulo is better than 19 using the public exponent to decrypt will work for at least half of the messages when eâ² 9 16 and in many cases for almost all the messages when eâ² 1 16" from question RSA public exponent is also decrypting message |
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"When i generate rsa key pairs by openssl it seems like private key private exponent is always less than public key modulo" from question Is RSA private key less than public? |
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"The rsa private exponent may actually be shorter than the modulo" from question Does RSA2048 key modulus have to be exactly 2048 bits long? |
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"The private exponent is always smaller than the modulo so you should be able to encrypt it using the raw rsa operation if you make sure to remove the prepended zero" from question How to encrypt a RSAKey using another RSAKey? "Edit as gregs points out in the comments you cannot be sure that the private exponent of the key you want to encrypt is smaller than the modulo of the key you want to use to encrypt with" from question How to encrypt a RSAKey using another RSAKey? |
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"The private exponent must not be longer than the modulo" from question Microsoft RSA CSP key size |
Java.security.invalidkeyexception files der
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"I can t create a a new .jks file using instructions here create new jks it gives me error java.security.invalidkeyexception exponent is larget than modulo make sure the ca files are in der format" from question WebLogicServer doesn't create DemoIdentity.jks |
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"Generating a certificate with common name test01.local.lan and key strength 2048 issued by ca with certificate from c weblogicser wlserver server lib certgenca.der file and key from c weblogicser wlserver server lib certgencakey.der file failed to generate the certificate java.security.invalidkeyexception exponent is larger than modulo make sure the ca files are in der format" from question * (no title is found for this review) |
Others
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Little wonder you get errors the exponent is normally shorter than the modulo which is always the same size as the key size from question RSA BadPaddingException when decrypting |
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Using the pow function and passing a modulo value is faster than computing the full exponent and then taking the modulo because the modulo can be applied to the partial products at each stage of the calculation which stops the value from getting too large 10 6 to the power of 10 6 has 6 million decimal digits with a modulo applied at each step the values never have to grow larger than the size of the modulo - about 13 digits in this example from question Finding remainder mod involving exponent and division involving huge numbers |