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Alfred J. Menezes coauthored the Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone. Its number-theoretic algorithms include computing modular square roots and implementing public-key cryptography. The presentation distinguishes computational problems, algorithmic procedures and correctness conditions, which is essential when a plausible-looking arithmetic operation is not by itself a secure cryptographic construction. ^[ref]

Andrew Yao is a computer scientist whose work spans analysis of algorithms, computational complexity theory, cryptography and quantum computing. His training included doctorates in both physics and computer science.^[ref]

With Danny Dolev, he developed formal models for the security of cryptographic protocols using public-key cryptography. Their Dolev-Yao model makes an active adversary's possible manipulations explicit while treating the underlying encryption and decryption symbolically. The resulting distinction is useful: a protocol can fail through its pattern of messages even if its encryption primitives behave as intended.^[ref]

A cryptographic key is a parameter selecting a particular transformation from a family of algorithms in cryptography. For an encryption family, correctness requiresfor every allowed plaintext $m$ and appropriate encryption/decryption keys. In symmetric-key cryptography, the communicating parties share a secret key. In public-key cryptography, the public encryption or verification key is paired with a private decryption or signing key.

For a one-bit example, $E_k(m)=m\oplus k$ and $D_k(c)=c\oplus k$ recover the message because applying exclusive or twice with the same bit cancels it. The one-time pad extends this to a uniformly random independent key as long as the message. Its secrecy depends on the key distribution and use, not on hiding the XOR algorithm. The number of possible keys gives a brute-force search space, but key length alone does not establish the security of a cryptographic construction.

Learning with errors asks for a secret vector $s$ from noisy modular linear observations. An observation has a known random vector $a$ andwhere $e$ is sampled from a specified small-error probability distribution. The search version recovers $s$; the decision version distinguishes these samples from uniformly random pairs. Parameters and the error distribution are essential parts of the problem.

For a tiny illustration, let $q=17$, $s=(3,5)$ and $a=(2,4)$. The exact inner product is $26\equiv 9(\mathrm{mod}17)$, while error $e=-1$ gives $b=8$. Without errors, enough independent equations admit ordinary modular linear solving; adding errors obstructs that direct approach. This tiny instance is not a secure parameter choice. Oded Regev's work connects appropriately parameterized instances to worst-case lattice problems and constructs a public-key cryptography from the assumption.^[ref]