In the end of the election, all the votes (ciphertext) are operated to obtain a ciphertext of the sum of all votes. Then, to verify the validity of a particular public key, the chain of trust from the root CA to the final user is used, and every certificate in the chain is verified with the public key of the issuer. Explain the principle of Public key cryptography. If a (possibly dishonest) prover P can make a honest verifier V accept a conversation (a,c,t) with probability greater than 1/q, then there exists a commitment a such that P can compute t1 and t2 for some different c1 and c2 such that both (a,c1,t1) and (a,c2,t2) are accepted by V. For deterministic encryption schemes, the two notions are equivalent, with Rerand(pk,c)=c, since the ciphertexts contain no randomness. In some cases, a CA can enforce security by asking a user to prove in zero-knowledge that she knows the corresponding secret key. Output s=(a,t). Compute t=(m-xr)k-1 mod p-1. In addition, digital signatures can be used to avoid impersonation attacks like the man-in-the-middle attack, by adding signatures of the users' public keys by a trusted authority (see below). Given a modulus MMM, only the numbers that are relatively prime to MMM have multiplicative inverses in (modM)\pmod M(modM). Parse s=(a,t). Indeed: gt1=ayc1 and gt2=ayc2 ⇒ Assuming that the prover's public key pk is known to the verifier, the prover starts the protocol sending some committing information a. P samples a random r∈Zq and computes the commitment a=gr. A signature can authenticate some extra information like place, time and purpose of the signature, as in conventional handwritten signatures, by appending in an unambiguous way the extra information to the original message. Output 1 if gm ≡ rtyr (mod p), and 0 otherwise. Digital signature as one of the applications of public key cryptography ensures the identity of the signer and integrity of the signed data, hence the security of the private key is crucial. Because the numbers in this example are quite small, an attacker can feasibly compute the discrete log to retrieve either XBX_BXB​ or XAX_AXA​. From a digital signature, it is easy to build an identification scheme. Then the other key is used as a decryption key to decrypt this cipher text so that the recipient can read the original message. For M=10M = 10M=10, for example, 2, 5, 6, and 8 do not have multiplicative inverses. One issue with RSA is that the algorithm is deterministic. There exists some security proofs in the Random Oracle Model that require some extra assumptions. A MODEL FOR NETWORK SECURITY A message is to be transferred from one party to another across some sort of internet. Choose a prime number q of λ bits. If the prover's private input is (pk,sk) and the verifier's private input is (pk) and both parties follow the protocol honestly, then the verifier accepts the conversation with probability 1. This algorithm is only used for digital signature, not encryption or key exchange. 1.Asymmetric algorithms rely on one key for encryption and a different but related key for decryption. Parse s=(r,t). This inverse is ddd, the private key. For example, 2+8(mod10)=02 + 8 \pmod{10} = 02+8(mod10)=0, because 10÷1010 \div 1010÷10 divides evenly whereas 3+8(mod10)=13 + 8 \pmod {10} = 13+8(mod10)=1 because 11÷1011 \div 1011÷10 yields a remainder of 111. Bucket Brigade Attack, Man in the Middle (MIM). The security assumption in Diffie-Hellman is that finding the discrete logarithm is infeasible given a very large, prime qqq. ∀(pk,sk)∈K, ∀m1,m2∈Mpk, the vectors (c1,c2,Rerand(pk,c1*c2)) and (c1,c2,Enc(pk,m1+m2), where c1=Enc(pk,m1) and c2=Enc(pk,m2), are identically distributed random variables. Additionally, the sole purpose of Diffie-Hellman is key exchange. Output (pk,sk)=((param,y),x), where param=(G,g,q). Key distribution under symmetric key encryption requires either (1) that two communicants already share a key, which someone has been distributed to them or (2) the use of a key distribution center. Public Key Cryptography: First International Workshop on Practice and Theory in Public Key Cryptography, PKC'98 Public Key Encryption from a Hardcore Predicate, ElGamal (1984) and Pointcheval-Stern (1996). Like all asymmetric cryptosystems, the Rabin system uses a key pair: a public key for encryption and a private key for decryption. In the Diffie-Hellman key exchange algorithm, there are two publicly known numbers qqq and α\alphaα. The private key is d,n{d, n}d,n. The bible for people who want to implement cryptograpy. Then the verifier can check whether the signature is valid with the corresponding public key pk. This signature scheme is vulnerable in the same way as the basic RSA signature. With the spread of more unsecure computer networks in last few decades, a genuine need was felt to use cryptography at larger scale. In modular exponentiation, xy(modn)=xy(modϕ(n))(modn)x^y \pmod n = x^{y \pmod{\phi(n)}} \pmod nxy(modn)=xy(modϕ(n))(modn). V choses a random challenge c∈Zq, and sends it back to P. Parse sk=(x). KeyGen(λ): Next, we compute n=p∗qn = p * qn=p∗q, and ϕ(n)=(p−1)∗(q−1)\phi(n) = (p - 1) * (q - 1)ϕ(n)=(p−1)∗(q−1). Sig(pk,sk,m): For instance, the party can show that she knows the secret key sk corresponding to a public key pk (without revealing sk). There is no way for Alice to know that the message that she has received is from Bob, and vice versa. A Verification algorithm, Ver, that given a public key pk, a message m and a signature s, it outputs 1 if s is a valid signature for pk and m, or 0 otherwise. Public-key cryptography, where different keys are used for encryption and decryption. Compute the bit sequence bj=h(xj-1) for j=1,...,n. Every voter encrypts 1 (for “yes”) or 0 (for “no”) and sends the resulting ciphertext to the board. The generation of such keys depends on cryptographic algorithms based on mathematical problems t This public key gets stored in the Notes Certificate and the Certificate is stored in User ID and in the Lotus Domino Directory. A public key is used for encryption and a private key is used for decryption. Sig(pk,sk,m): Notes on 'Applied Cryptography, Chapter 1 - Foundations' Classic book by Bruce Schneier. Similarly, user BBB selects a random number XB