- Naccache–Stern knapsack cryptosystem
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Note: this is not to be confused with the Naccache–Stern cryptosystem based on the higher residuosity problem.
The Naccache–Stern Knapsack Cryptosystem is an atypical public-key cryptosystem developed by David Naccache and Jacques Stern in 1997. This cryptosystem is deterministic, and hence is not semantically secure. This system also lacks provable security.
Contents
System Overview
This system is based on a type of knapsack problem. Specifically, the underlying problem is this: given integers c,n,p and v0,...,vn, find a vector such that
The idea here is that when the vi are relatively prime and much smaller than the modulus p this problem can be solved easily. It is this observation which allows decryption.
Key Generation
To generate a public/private key pair
- Pick a large prime modulus p.
- Pick a positive integer n.
- For i from 0 to n, set pi to be the ith prime, starting with p0 = 2.
- Pick a secret integer s < p-1, such that gcd(p-1,s) = 1.
- Set .
The public key is then p,n and v0,...,vn. The private key is s.
Encryption
To encrypt an n-bit long message m, calculate
where mi is the ith bit of the message m.
Decryption
To decrypt a message c, calculate
This works because the fraction
is 0 or 1 depending on whether pi divides cs mod p.
See also
- Merkle–Hellman knapsack cryptosystem
- Graham–Shamir knapsack cryptosystem
References
Public-key cryptography Algorithms Benaloh · Blum–Goldwasser · Cayley–Purser · CEILIDH · Cramer–Shoup · Damgård–Jurik · DH · DSA · EPOC · ECDH · ECDSA · EKE · ElGamal (encryption · signature scheme) · GMR · Goldwasser–Micali · HFE · IES · Lamport · McEliece · Merkle–Hellman · MQV · Naccache–Stern · NTRUEncrypt · NTRUSign · Paillier · Rabin · RSA · Okamoto–Uchiyama · Schnorr · Schmidt–Samoa · SPEKE · SRP · STS · Three-pass protocol · XTR
Theory Standardization ANS X9F1 · CRYPTREC · IEEE P1363 · NESSIE · NSA Suite B
Topics Digital signature · OAEP · Fingerprint · PKI · Web of trust · Key size
Cryptography Categories:- Asymmetric-key cryptosystems
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