RSA public key exponent too long?

[hemml]

I'm trying to make ACME (RFC8555) client and always got "JWS verification error" from LetsEncrypt's server.
May be it because I'm constructing a wrong jwk in my request.
The following code:

(multiple-value-bind (priv-key pub-key) (ironclad:generate-key-pair :rsa :num-bits 4096)
  (print (ironclad:destructure-public-key pub-key)))

gives me something like:

(:E
 208085713664532760971160044664858453515928358938952263096107908738599293669243046165663481672665658831438261132368819790975618388615388814482557962887920548899877895925722585282327853629197113431705213579236181753719896978086489331618659800367374191972758732411952131514254997916693086837157948595405088665763217101230882063867377603769393470068081039685352174609157542776783920221098498533691024912388252839944675353915049625353166097918241314581876016439746483179626955546303256337901522365062568673049670238222822415120387686801894236279012907900417634049761766002129395269621625913887544460236370729242879332583372923675932492543059271216834389621638163764602702607225020049609564817890964587226225555294052892075399225352975057675874455782243385198369924188409538252612308137502658568930107954244763004226304719985151076344649633746314477132921172392255220583400570833361507385525575043927141735126168975220181187727034652911233864706872701111503782825310517644748733109249870663724953931009303834002334200382306489315365717859917278272096270168052147125730525305088904624397117712610111748932325807716901724004172273813651381741347232291552714200595971654261210128210008811942691869188818168919639363370647874725865035935485851
 :N
 883341698746903501464703537622227135044932644736638679968312155203622118126451734635967362407240214857554764403351993263863180051222571771542081362092560597011061650343993798256131731924879401236533110193063092256021109695236429214631260972568090817523012926382890236045404797306232897540999639365360591206741296934677942448598483888302346806834951285824748275966356671766111215325639136139446606943592186687731962347577584558469561626374319647216978003904112699745463354705636614762416376352862401704195920696448389380598743625387835559431755955129772874812521458380291129856778437605979534091540188673599682735602749703978221786843600777281892992237292454283393486078207029552871705295602135762730649544315168572868250020457368764675225223304466648555304243218335193402657282619777083024992415352799334266402541810712909109253277407192121969324866574791494508618836605762158595285367386491105291425724906736751621631967647386736461431221391017149259897582303187196979221156127192050475932346666676006354525052023709025782323922462774767420291174648982235763916784337183699250078987655983751925516990690801662882103794483119232502652698882185314167234294306169049163295742093586985764864288625609063327554740797018267536904880899727) 

Is the :E a public key exponent? In all examples I saw e is a much smaller integer.

[glv2]

Yes, E is the public exponent.
Many implementation of RSA use a small public exponent like 65537 (or even 3 or 17). Currently Ironclad is using a random exponent coprime with Phi.

[hemml]

Ok, thank you for the explanation!
I'm sorry if here is a wrong place for this question, but may be you can say what can be wrong in the following code to make a RS256 JOSE signature:

(multiple-value-bind (priv-key pub-key) (ironclad:generate-key-pair :rsa :num-bits 4096)
  (setf *acme-pub-key* pub-key)
  (setf *acme-priv-key* priv-key)))

(ironclad:sign-message
             *acme-priv-key* 
             (ironclad:digest-sequence
                  :sha256
                  (ironclad:ascii-string-to-byte-array 
                      (format nil "~A.~A" protected payload)))))

where protected and payload are url-safe-base64-encoded json strings.

[glv2]

According to https://datatracker.ietf.org/doc/html/rfc7518#section-3.3 it looks like RS256 JOSE signatures require the use of PKCS1-v1.5 padding of the message. However Ironclad only has PKCS1-v2.1 so far (OAEP and PSS), see issue #41.

I've never used it, but maybe https://github.com/fukamachi/jose has what you're looking for...

[hemml]

Thanks a lot! This may be an issue!