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# Hellman

[hel-muhn]
Hellman, Lillian, 1905-84, American dramatist, b. New Orleans. Her plays, although often melodramatic, are marked by intelligence and craftsmanship. The Children's Hour (1934), her first drama, concerns the devastating effects of a child's malicious charge of lesbianism against two of her teachers. The Little Foxes (1939) and Another Part of the Forest (1946) constitute a chilling study of a wealthy and rapacious Southern family. Several of Hellman's dramas—notably Watch on the Rhine (1941) and The Searching Wind (1944)—treat international political themes such as isolationism and the rise of fascism. In 1952 she was called before the House Un-American Activities Committee. She has made several English adaptations of French plays and, with Richard Wilbur, wrote the libretto for a musical version of Voltaire's Candide (1955). Her other plays include Days to Come (1936), The Autumn Garden (1951), and Toys in the Attic (1960). In 1931 she met the writer Dashiell Hammett, who remained her constant companion until his death in 1961.

See her autobiographical works, An Unfinished Woman (1969) and Pentimento (1973); J. Mellen, Hellman and Hammett (1996).

(born June 20, 1905, New Orleans, La., U.S.—died June 30, 1984, Vineyard Haven, Martha's Vineyard, Mass.) U.S. playwright. After working as a book reviewer, press agent, and play reader, she began writing plays in the 1930s. Her first major success, The Children's Hour (1934), concerned two schoolteachers falsely accused of lesbianism. She examined family infighting in her hit The Little Foxes (1939) and political injustice in Watch on the Rhine (1941). All were made into successful films. Called before the House Un-American Activities Committee in 1952, she refused to testify. She wrote several memoirs and edited the works of her longtime companion, the novelist Dashiell Hammett.

(born June 20, 1905, New Orleans, La., U.S.—died June 30, 1984, Vineyard Haven, Martha's Vineyard, Mass.) U.S. playwright. After working as a book reviewer, press agent, and play reader, she began writing plays in the 1930s. Her first major success, The Children's Hour (1934), concerned two schoolteachers falsely accused of lesbianism. She examined family infighting in her hit The Little Foxes (1939) and political injustice in Watch on the Rhine (1941). All were made into successful films. Called before the House Un-American Activities Committee in 1952, she refused to testify. She wrote several memoirs and edited the works of her longtime companion, the novelist Dashiell Hammett.

Merkle-Hellman (MH) was one of the earliest public key cryptosystems and was invented by Ralph Merkle and Martin Hellman in 1978. Although its ideas are elegant, and far simpler than RSA, it has been broken.

## Description

Merkle-Hellman is an asymmetric-key cryptosystem, meaning that for communication, two keys are required: a public key and a private key. Furthermore, unlike RSA, it is one-way -- the public key is used only for encryption, and the private key is used only for decryption. Thus it is unusable for authentication by cryptographic signing.

The Merkle-Hellman system is based on the subset sum problem (a special case of the knapsack problem): given a list of numbers and a third number, which is the sum of a subset of these numbers, determine the subset. In general, this problem is known to be NP-complete. However, if the set of numbers (called the knapsack) is superincreasing -- that is, each element of the set is greater than the sum of all the numbers before it -- the problem is 'easy' and solvable in polynomial time with a simple greedy algorithm.

### Key generation

In Merkle-Hellman, the keys are comprised of knapsacks. The public key is a 'hard' knapsack, and the private key is an 'easy', or superincreasing, knapsack, combined with two additional numbers, a multiplier and a modulus, which were used to convert the superincreasing knapsack into the hard knapsack. These same numbers are used to transform the sum of the subset of the hard knapsack into the sum of the subset of the easy knapsack, which is solvable in polynomial time.

### Encryption

To encrypt a message, a subset of the hard knapsack is chosen by comparing it with a set of bits (the plaintext), equal in length to the key, and making each term in the public key that corresponds to a 1 in the plaintext an element of the subset, while ignoring the terms corresponding to 0 terms in the plaintext. The elements of this subset are added together, and the resulting sum is the ciphertext.

### Decryption

Decryption is possible because the multiplier and modulus used to transform the easy, superincreasing knapsack into the public key can also be used to transform the number representing the ciphertext into the sum of the corresponding elements of the superincreasing knapsack. Then, using a simple greedy algorithm, the easy knapsack can be solved using O(n) arithmetic operations, which decrypts the message.

## Mathematical Method

### Key generation

To encrypt n-bit messages, choose a superincreasing sequence

w = (w1, w2, ..., wn)

of n nonzero natural numbers. Pick a random integer q, such that

q>$sum_\left\{i = 1\right\}^n w_i$,

and a random integer, r, such that gcd(r,q) = 1.

q is chosen this way to ensure the uniqueness of the ciphertext. If it is any smaller, more than one plaintext may encrypt to the same ciphertext. r must be coprime to q or else it will not have an inverse mod q. The existence of the inverse of r is necessary so that decryption is possible.

Now calculate the sequence

β = (β1, β2, ..., βn)
where
βi = rwi mod q.
The public key is β, while the private key is (w, q, r).

### Encryption

To encrypt an n-bit message

α = (α1, α2, ..., αn),

where αi is the i-th bit of the message and αi $boldsymbol\left\{in\right\}$ {0, 1}, calculate

$c = sum_\left\{i = 1\right\}^n alpha_i beta_i$.
The cryptogram then is c.

### Decryption

In order to decrypt a ciphertext c a receiver has to find the message bits αi such that they satisfy
$c = sum_\left\{i = 1\right\}^n alpha_i beta_i$.
This would be a hard problem if the βi were random values because the receiver would have to solve an instance of the subset sum problem, which is known to be NP-hard. However, the values βi were chosen such that decryption is easy if the private key (w, q, r) is known.

The key to decryption is to find an integer s that is the modular inverse of r modulo q. That means s satisfies the equation s r mod q=1 or equivalently there exist an integer k such that sr = kq + 1. Since r was chosen such that gcd(r,q)=1 it is possible to find s and k by using the Extended Euclidean algorithm. Next the receiver of the ciphertext c computes

$c\text{'}equiv cs pmod\left\{q\right\}.$
Hence
$c\text{'} equiv cs equiv sum_\left\{i = 1\right\}^n alpha_i beta_i s pmod\left\{q\right\}.$
Because of rs mod q = 1 and βi = rwi mod q follows
$beta_i sequiv w_i r sequiv w_ipmod\left\{q\right\}.$
Hence
$c\text{'} equiv sum_\left\{i = 1\right\}^n alpha_i w_ipmod\left\{q\right\}.$
The sum of all values wi is smaller than q and hence $sum_\left\{i = 1\right\}^n alpha_i w_i,bmod, q$ is also in the interval [0,q-1]. Thus the receiver has to solve the subset sum problem
$c\text{'} = sum_\left\{i = 1\right\}^n alpha_i w_i.$
This problem is easy because w is a super-increasing sequence. Take the largest element in w, say wk. If wk > c' , then αk = 0, if wkc' , then αk = 1. Then, subtract wk×αk from c' , and repeat these steps until you have figured out α.

## References

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