[wrong][spam] Cracking PGP

[Karl]

Maybe clean it up some

On Fri, Jun 18, 2021, 2:00 PM Karl <gmkarl at gmail.com> wrote:

> I'll go through it a little.
>
> On Fri, Jun 18, 2021, 1:58 PM Karl <gmkarl at gmail.com> wrote:
>
>> ===Key generation===
>> The keys for the RSA algorithm are generated in the following way:
>> # Choose two distinct [[prime number]]s ''p'' and ''q''.
>>
>
> #* ''p'' and ''q'' are kept secret.
>> # Compute 'n'' = ''pq''
>>
>
> public n = private p * private q
> n is keylength bits long.
>
> # Compute ''λ''(''n''), where ''λ'' is Carmichael's totient function.
>> Since ''n'' = ''pq'', ''λ''(''n'') = lcm(''λ''(''p''),''λ''(''q'')), and
>> since ''p'' and ''q'' are prime, ''λ''(''p'') = ''φ''(''p'') = ''p'' − 1
>> where φ is the Euler totient function, and likewise ''λ''(''q'') = ''q''
>> − 1. Hence ''λ''(''n'') = lcm(''p'' − 1, ''q'' − 1)
>>
>
> private lambda(n) is calculated and has a number of relevant algebraic
> identities associated
>
> #* The lcm may be calculated through the Euclidean algorithm, since
>> lcm(''a'',''b'') = ''|ab|''/gcd(''a'',''b'').
>> # Choose an integer ''e'' such that 1 < ''e'' < ''λ''(''n'') and
>> gcd(''e'', ''λ''(''n'')) = 1; that is, ''e'' and ''λ''(''n'') are coprime.
>>
>
Organised a little up to here.

#* ''e'' having a short [[bit-length]] and small [[Hamming weight]] results
>> in more efficient encryption {{snd}} the most commonly chosen value for
>> ''e'' is {{nowrap|1=2<sup>16</sup> + 1 = 65,537}}. The smallest (and
>> fastest) possible value for ''e'' is 3, but such a small value for ''e''
>> has been shown to be less secure in some settings.<ref name="Boneh99">
>> {{cite journal
>>  | url = http://crypto.stanford.edu/~dabo/abstracts/RSAattack-survey.html
>>  | last = Boneh | first = Dan
>>  | title = Twenty Years of attacks on the RSA Cryptosystem
>>  | journal = [[Notices of the American Mathematical Society]]
>>  | volume = 46 | issue = 2
>>  | pages = 203–213 | year = 1999
>> }}</ref>
>> #*''e'' is released as part of the public key.
>> # Determine ''d'' as {{nowrap|''d'' ≡ ''e''<sup>−1</sup> (mod
>> ''λ''(''n''))}}; that is, ''d'' is the [[modular multiplicative inverse]]
>> of ''e'' modulo ''λ''(''n'').
>> #* This means: solve for ''d'' the equation {{nowrap|''d''⋅''e'' ≡ 1 (mod
>> ''λ''(''n''))}}; ''d'' can be computed efficiently by using the [[Extended
>> Euclidean algorithm]], since, thanks to ''e'' and ''λ''(''n'') being
>> coprime, said equation is a form of [[Bézout's identity]], where ''d'' is
>> one of the coefficients.
>> #* ''d'' is kept secret as the ''private key exponent''.
>> The ''public key'' consists of the modulus ''n'' and the public (or
>> encryption) exponent ''e''. The ''private key'' consists of the private (or
>> decryption) exponent ''d'', which must be kept secret. ''p'', ''q'', and
>> ''λ''(''n'') must also be kept secret because they can be used to calculate
>> ''d''. In fact, they can all be discarded after ''d'' has been
>> computed.<ref>Applied Cryptography, John Wiley & Sons, New York, 1996.
>> [[Bruce Schneier]], p. 467</ref>
>>
>
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