[wrong][spam] Cracking PGP

[Karl]

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'') = [[least common
> multiple|lcm]](''λ''(''p''),''λ''(''q'')), and since ''p'' and ''q'' are
> prime, ''λ''(''p'') = ''[[Euler totient function|φ]]''(''p'') = ''p'' − 1
> and likewise ''λ''(''q'') = ''q'' − 1. Hence ''λ''(''n'') = lcm(''p'' − 1,
> ''q'' − 1).
> #* ''λ''(''n'') is kept secret.
>

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 {{nowrap|1 < ''e'' < ''λ''(''n'')}}
> and {{nowrap|1=[[greatest common divisor|gcd]](''e'', ''λ''(''n'')) = 1}};
> that is, ''e'' and ''λ''(''n'') are [[coprime]].
> #* ''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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