I think Pedersen commitments are a gentle starting point for teaching cryptography to anyone!
A **commitment** is a sealed envelope with a message m in it such that:
1. no one can tell what m is in it (hiding)
2. no one can open it to a different m (binding)
Pedersen below 👇
Commitments are very useful: e.g., randomness beacons.
- Each player commits to a random message m_i
- All players exchange their commitments C_i
- All players open their commitments or they're penalized
Final random value = sum of all opened m_i's
(Problematic design, I know)
My point? Pedersen commitments have many applications.
They are also v. simple:
You need:
1. a prime-order group (e.g., elliptic curve)
2. two elements G, H in this group s.t. nobody knows the relation between them
i.e., the discrete logarithm \tau of H w.r.t. G 👇
First, it is very easy to explain why the envelope is "binding": i.e., attacker cannot open it in two different ways.
=> it would allow the attacker to compute \tau, which is supposed to be very hard in (say) elliptic curves 👇
Note that binding holds only under a *computational* assumption.
i.e., it is in fact *possible* to open the envelope to two different messages
it's just *hard* to: such an algorithm would immediately give an algorithm that solves discrete logarithms faster than currently known.
Second, it is very easy to explain why the envelope is "hiding": i.e., no one can tell what m is inside.
=> because, really, Pedersen commitments are just polynomial commitments:
committing to m using randomness r <=> committing to f(X) = m + Xr as C = f(tau) G 👇
Crucially, C hides f(X), and therefore hides (m, r), because there are a bazillion other polynomials f'(X) = m' + X r' with m' != m such that f'(tau) = f(tau).
So, it could be that:
- C commits to m as C = f(tau) G
- C commits to m' also as C = f'(tau) G = f(tau) G
Can't tell!
(FYI: "Bazillion" is a technical term for p, where p is the order of the group.)
You can also guess what my next point was going to be...
Pedersen commitments are a gateway drug into KZG commitments! ()
But that's for another time!
8.24 ألف
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المحتوى الوارد في هذه الصفحة مُقدَّم من أطراف ثالثة. وما لم يُذكَر خلاف ذلك، فإن OKX ليست مُؤلِّفة المقالة (المقالات) المذكورة ولا تُطالِب بأي حقوق نشر وتأليف للمواد. المحتوى مٌقدَّم لأغراض إعلامية ولا يُمثِّل آراء OKX، وليس الغرض منه أن يكون تأييدًا من أي نوع، ولا يجب اعتباره مشورة استثمارية أو التماسًا لشراء الأصول الرقمية أو بيعها. إلى الحد الذي يُستخدَم فيه الذكاء الاصطناعي التوليدي لتقديم مُلخصَّات أو معلومات أخرى، قد يكون هذا المحتوى الناتج عن الذكاء الاصطناعي غير دقيق أو غير مُتسِق. من فضلك اقرأ المقالة ذات الصِلة بهذا الشأن لمزيدٍ من التفاصيل والمعلومات. OKX ليست مسؤولة عن المحتوى الوارد في مواقع الأطراف الثالثة. والاحتفاظ بالأصول الرقمية، بما في ذلك العملات المستقرة ورموز NFT، فيه درجة عالية من المخاطر وهو عُرضة للتقلُّب الشديد. وعليك التفكير جيِّدًا فيما إذا كان تداوُل الأصول الرقمية أو الاحتفاظ بها مناسبًا لك في ظل ظروفك المالية.