Tags: eulertheorem rsa
Rating:
We are given the following RSA parameters
```
n = 19574201286059123715221634877085223155972629451020572575626246458715199192950082143183900970133840359007922584516900405154928253156404028820410452946729670930374022025730036806358075325420793866358986719444785030579682635785758091517397518826225327945861556948820837789390500920096562699893770094581497500786817915616026940285194220703907757879335069896978124429681515117633335502362832425521219599726902327020044791308869970455616185847823063474157292399830070541968662959133724209945293515201291844650765335146840662879479678554559446535460674863857818111377905454946004143554616401168150446865964806314366426743287
s = 3737620488571314497417090205346622993399153545806108327860889306394326129600175543006901543011761797780057015381834670602598536525041405700999041351402341132165944655025231947620944792759658373970849932332556577226700342906965939940429619291540238435218958655907376220308160747457826709661045146370045811481759205791264522144828795638865497066922857401596416747229446467493237762035398880278951440472613839314827303657990772981353235597563642315346949041540358444800649606802434227470946957679458305736479634459353072326033223392515898946323827442647800803732869832414039987483103532294736136051838693397106408367097
c = 7000985606009752754441861235720582603834733127613290649448336518379922443691108836896703766316713029530466877153379023499681743990770084864966350162010821232666205770785101148479008355351759336287346355856788865821108805833681682634789677829987433936120195058542722765744907964994170091794684838166789470509159170062184723590372521926736663314174035152108646055156814533872908850156061945944033275433799625360972646646526892622394837096683592886825828549172814967424419459087181683325453243145295797505798955661717556202215878246001989162198550055315405304235478244266317677075034414773911739900576226293775140327580
e = 0x10001
```
We know `s = (557p - 127q)^(n-p-q) mod n` and `phi(n) = n - p - q + 1` which means `n - p - q = phi(n) - 1`
Using Euler's theorem, we get
`(557p - 127q) congruent to inverse(s,n) mod n`. Now, let's make an assumption that `557p - 127q = inverse(s,n)` i.e. `557p - 127n/p = inverse(s,n)`
We can obtain a quadratic equation: `557p^2 - inverse(s,n)*p - 127n = 0`, On solving this, we get p, and therefore q since we have n. This gives away the decryption exponent d, and we can decode the ciphertext c to obtain our flag.
```
Flag: DUCTF{e4sy_RSA_ch4ll_t0_g3t_st4rt3d}
```
