CryptRsaDecrypt(
		TPM2B               *dOut,          // OUT: the decrypted data
		TPM2B               *cIn,           // IN: the data to decrypt
		OBJECT              *key,           // IN: the key to use for decryption
		TPMT_RSA_DECRYPT    *scheme,        // IN: the padding scheme
		const TPM2B         *label          // IN: in case it is needed for the scheme
		)
{
    TPM_RC                 retVal;
    EVP_PKEY              *pkey = NULL;
    EVP_PKEY_CTX          *ctx = NULL;
    const EVP_MD          *md = NULL;
    const char            *digestname;
    size_t                 outlen;
    unsigned char         *tmp = NULL;
    unsigned char          buffer[MAX_RSA_KEY_BYTES];

    // Make sure that the necessary parameters are provided
    pAssert(cIn != NULL && dOut != NULL && key != NULL);
    // Size is checked to make sure that the encrypted value is the right size
    if(cIn->size != key->publicArea.unique.rsa.t.size)
        ERROR_RETURN(TPM_RC_SIZE);
    TEST(scheme->scheme);

    retVal = InitOpenSSLRSAPrivateKey(key, &pkey);
    if (retVal != TPM_RC_SUCCESS)
        return retVal;

    ctx = EVP_PKEY_CTX_new(pkey, NULL);
    if (ctx == NULL ||
        EVP_PKEY_decrypt_init(ctx) <= 0)
        ERROR_RETURN(TPM_RC_FAILURE);

    switch(scheme->scheme)
	{
	  case ALG_NULL_VALUE:  // 'raw' encryption
            if (EVP_PKEY_CTX_set_rsa_padding(ctx, RSA_NO_PADDING) <= 0)
                ERROR_RETURN(TPM_RC_FAILURE);
            break;
	  case ALG_RSAES_VALUE:
            if (EVP_PKEY_CTX_set_rsa_padding(ctx, RSA_PKCS1_PADDING) <= 0)
                ERROR_RETURN(TPM_RC_FAILURE);
            break;
	  case ALG_OAEP_VALUE:
            digestname = GetDigestNameByHashAlg(scheme->details.oaep.hashAlg);
            if (digestname == NULL)
                ERROR_RETURN(TPM_RC_VALUE);

            md = EVP_get_digestbyname(digestname);
            if (md == NULL ||
                EVP_PKEY_CTX_set_rsa_padding(ctx, RSA_PKCS1_OAEP_PADDING) <= 0 ||
                EVP_PKEY_CTX_set_rsa_oaep_md(ctx, md) <= 0)
                ERROR_RETURN(TPM_RC_FAILURE);

            if (label->size > 0) {
                tmp = malloc(label->size);
                if (tmp == NULL)
                    ERROR_RETURN(TPM_RC_FAILURE);
                memcpy(tmp, label->buffer, label->size);
            }

            if (EVP_PKEY_CTX_set0_rsa_oaep_label(ctx, tmp, label->size) <= 0)
                ERROR_RETURN(TPM_RC_FAILURE);
            tmp = NULL;
            break;
	  default:
            ERROR_RETURN(TPM_RC_SCHEME);
            break;
	}

    /* cannot use cOut->buffer */
    outlen = sizeof(buffer);
    if (EVP_PKEY_decrypt(ctx, buffer, &outlen,
                         cIn->buffer, cIn->size) <= 0)
        ERROR_RETURN(TPM_RC_FAILURE);

    if (outlen > dOut->size)
        ERROR_RETURN(TPM_RC_FAILURE);

    memcpy(dOut->buffer, buffer, outlen);
    dOut->size = outlen;

    retVal = TPM_RC_SUCCESS;

 Exit:
    EVP_PKEY_free(pkey);
    EVP_PKEY_CTX_free(ctx);
    free(tmp);

    return retVal;
}


CryptRsaSelectScheme(
		     TPMI_DH_OBJECT       rsaHandle,     // IN: handle of an RSA key
		     TPMT_RSA_DECRYPT    *scheme         // IN: a sign or decrypt scheme
		     )
{
    OBJECT              *rsaObject;
    TPMT_ASYM_SCHEME    *keyScheme;
    TPMT_RSA_DECRYPT    *retVal = NULL;
    // Get sign object pointer
    rsaObject = HandleToObject(rsaHandle);
    keyScheme = &rsaObject->publicArea.parameters.asymDetail.scheme;
    // if the default scheme of the object is TPM_ALG_NULL, then select the
    // input scheme
    if(keyScheme->scheme == TPM_ALG_NULL)
	{
	    retVal = scheme;
	}
    // if the object scheme is not TPM_ALG_NULL and the input scheme is
    // TPM_ALG_NULL, then select the default scheme of the object.
    else if(scheme->scheme == TPM_ALG_NULL)
	{
	    // if input scheme is NULL
	    retVal = (TPMT_RSA_DECRYPT *)keyScheme;
	}
    // get here if both the object scheme and the input scheme are
    // not TPM_ALG_NULL. Need to insure that they are the same.
    // IMPLEMENTATION NOTE: This could cause problems if future versions have
    // schemes that have more values than just a hash algorithm. A new function
    // (IsSchemeSame()) might be needed then.
    else if(keyScheme->scheme == scheme->scheme
	    && keyScheme->details.anySig.hashAlg == scheme->details.anySig.hashAlg)
	{
	    retVal = scheme;
	}
    // two different, incompatible schemes specified will return NULL
    return retVal;
}

tpm2: Fix output buffer parameter and size for RSA decryption

For the RSA decryption we have to use an output buffer of the size of the
(largest possible) RSA key for the decryption to always work.

This fixes a stack corruption bug that caused a SIGBUS and termination of
'swtpm'.

Signed-off-by: Stefan Berger <stefanb@linux.ibm.com>

int GetCachedRsaKey(OBJECT *key, RAND_STATE *rand);

#define GET_CACHED_KEY(key, rand)					\
    (s_rsaKeyCacheEnabled && GetCachedRsaKey(key, rand))

#define GET_CACHED_KEY(key, rand)


CryptRsaLoadPrivateExponent(
			    OBJECT          *rsaKey        // IN: the RSA key object
			    )
{
    BN_RSA_INITIALIZED(bnN, &rsaKey->publicArea.unique.rsa);
    BN_PRIME_INITIALIZED(bnP, &rsaKey->sensitive.sensitive.rsa);
    BN_RSA(bnQ);
    BN_PRIME(bnQr);
    BN_WORD_INITIALIZED(bnE, (rsaKey->publicArea.parameters.rsaDetail.exponent == 0)
			? RSA_DEFAULT_PUBLIC_EXPONENT
			: rsaKey->publicArea.parameters.rsaDetail.exponent);
    TPM_RC          retVal = TPM_RC_SUCCESS;
    if(!rsaKey->attributes.privateExp)
	{
	    TEST(ALG_NULL_VALUE);
	    // Make sure that the bigNum used for the exponent is properly initialized
	    RsaInitializeExponent(&rsaKey->privateExponent);
	    // Find the second prime by division
	    BnDiv(bnQ, bnQr, bnN, bnP);
	    if(!BnEqualZero(bnQr))
		ERROR_RETURN(TPM_RC_BINDING);
	    // Compute the private exponent and return it if found
	    if(!ComputePrivateExponent(bnP, bnQ, bnE, bnN,
				       &rsaKey->privateExponent))
		ERROR_RETURN(TPM_RC_BINDING);
	}
 Exit:
    rsaKey->attributes.privateExp = (retVal == TPM_RC_SUCCESS);
    return retVal;
}


RSADP(
      TPM2B           *inOut,        // IN/OUT: the value to encrypt
      OBJECT          *key           // IN: the key
      )
{
    BN_RSA_INITIALIZED(bnM, inOut);
    BN_RSA_INITIALIZED(bnN, &key->publicArea.unique.rsa);
    BN_RSA_INITIALIZED(bnP, &key->sensitive.sensitive.rsa);
    if(BnUnsignedCmp(bnM, bnN) >= 0)
	return TPM_RC_SIZE;
    // private key operation requires that private exponent be loaded
    // During self-test, this might not be the case so load it up if it hasn't
    // already done
    // been done
    if(!key->attributes.privateExp)
	CryptRsaLoadPrivateExponent(key);
    if(!RsaPrivateKeyOp(bnM, bnN, bnP, &key->privateExponent))
	FAIL(FATAL_ERROR_INTERNAL);
    BnTo2B(bnM, inOut, inOut->size);
    return TPM_RC_SUCCESS;
}


CryptRsaStartup(
		void
		)
{
    return TRUE;
}

#define CRYPT_RSA_C

extern int s_rsaKeyCacheEnabled;

#define GET_CACHED_KEY(key, rand)

int GetCachedRsaKey(OBJECT *key, RAND_STATE *rand);

#define GET_CACHED_KEY(key, rand)					\
    (s_rsaKeyCacheEnabled && GetCachedRsaKey(key, rand))


RSASSA_Decode(
	      TPM_ALG_ID       hashAlg,        // IN: hash algorithm to use for the encoding
	      TPM2B           *hIn,            // In: the digest to compare
	      TPM2B           *eIn             // IN: the encoded data
	      )
{
    BYTE             fail;
    const BYTE      *der;
    BYTE            *pe;
    INT32            derSize = CryptHashGetDer(hashAlg, &der);
    INT32            hashSize = CryptHashGetDigestSize(hashAlg);
    INT32            fillSize;
    TPM_RC           retVal;
    BYTE            *digest;
    UINT16           digestSize;
    pAssert(hIn != NULL && eIn != NULL);
    pe = eIn->buffer;
    // Can't use this scheme if the algorithm doesn't have a DER string
    // defined or if the provided hash isn't the right size
    if(derSize == 0 || (unsigned)hashSize != hIn->size)
	ERROR_RETURN(TPM_RC_SCHEME);
    // Make sure that this combination will fit in the provided space
    // Since no data movement takes place, can just walk though this
    // and accept nearly random values. This can only be called from
    // CryptValidateSignature() so eInSize is known to be in range.
    fillSize = eIn->size - derSize - hashSize - 3;
    // Start checking (fail will become non-zero if any of the bytes do not have
    // the expected value.
    fail = *pe++;                   // initial byte of zero
    fail |= *pe++ ^ 1;              // byte of 0x01
    for(; fillSize > 0; fillSize--)
	fail |= *pe++ ^ 0xff;       // bunch of 0xff
    fail |= *pe++;                  // another 0
    for(; derSize > 0; derSize--)
	fail |= *pe++ ^ *der++;    // match the DER
    digestSize = hIn->size;
    digest = hIn->buffer;
    for(; digestSize > 0; digestSize--)
	fail |= *pe++ ^ *digest++; // match the hash
    retVal = (fail != 0) ? TPM_RC_VALUE : TPM_RC_SUCCESS;
 Exit:
    return retVal;
}


CryptRsaValidateSignature(
			  TPMT_SIGNATURE  *sig,           // IN: signature
			  OBJECT          *key,           // IN: public modulus
			  TPM2B_DIGEST    *digest         // IN: The digest being validated
			  )
{
    TPM_RC          retVal;
    //
    // Fatal programming errors
    pAssert(key != NULL && sig != NULL && digest != NULL);
    switch(sig->sigAlg)
	{
	  case ALG_RSAPSS_VALUE:
	  case ALG_RSASSA_VALUE:
	    break;
	  default:
	    return TPM_RC_SCHEME;
	}
    // Errors that might be caused by calling parameters
    if(sig->signature.rsassa.sig.t.size != key->publicArea.unique.rsa.t.size)
	ERROR_RETURN(TPM_RC_SIGNATURE);
    TEST(sig->sigAlg);
    // Decrypt the block
    retVal = RSAEP(&sig->signature.rsassa.sig.b, key);
    if(retVal == TPM_RC_SUCCESS)
	{
	    switch(sig->sigAlg)
		{
		  case ALG_RSAPSS_VALUE:
		    retVal = PssDecode(sig->signature.any.hashAlg, &digest->b,
				       &sig->signature.rsassa.sig.b);
		    break;
		  case ALG_RSASSA_VALUE:
		    retVal = RSASSA_Decode(sig->signature.any.hashAlg, &digest->b,
					   &sig->signature.rsassa.sig.b);
		    break;
		  default:
		    return TPM_RC_SCHEME;
		}
	}
 Exit:
    return (retVal != TPM_RC_SUCCESS) ? TPM_RC_SIGNATURE : TPM_RC_SUCCESS;
}


CryptRsaEncrypt(
		TPM2B_PUBLIC_KEY_RSA        *cOut,          // OUT: the encrypted data
		TPM2B                       *dIn,           // IN: the data to encrypt
		OBJECT                      *key,           // IN: the key used for encryption
		TPMT_RSA_DECRYPT            *scheme,        // IN: the type of padding and hash
		//     if needed
		const TPM2B                 *label,         // IN: in case it is needed
		RAND_STATE                  *rand           // IN: random number generator
		//     state (mostly for testing)
		)
{
    TPM_RC                       retVal = TPM_RC_SUCCESS;
    TPM2B_PUBLIC_KEY_RSA         dataIn;
    //
    // if the input and output buffers are the same, copy the input to a scratch
    // buffer so that things don't get messed up.
    if(dIn == &cOut->b)
	{
	    MemoryCopy2B(&dataIn.b, dIn, sizeof(dataIn.t.buffer));
	    dIn = &dataIn.b;
	}
    // All encryption schemes return the same size of data
    cOut->t.size = key->publicArea.unique.rsa.t.size;
    TEST(scheme->scheme);
    switch(scheme->scheme)
	{
	  case ALG_NULL_VALUE:  // 'raw' encryption
	      {
		  INT32            i;
		  INT32            dSize = dIn->size;
		  // dIn can have more bytes than cOut as long as the extra bytes
		  // are zero. Note: the more significant bytes of a number in a byte
		  // buffer are the bytes at the start of the array.
		  for(i = 0; (i < dSize) && (dIn->buffer[i] == 0); i++);
		  dSize -= i;
		  if(dSize > cOut->t.size)
		      ERROR_RETURN(TPM_RC_VALUE);
		  // Pad cOut with zeros if dIn is smaller
		  memset(cOut->t.buffer, 0, cOut->t.size - dSize);
		  // And copy the rest of the value
		  memcpy(&cOut->t.buffer[cOut->t.size - dSize], &dIn->buffer[i], dSize);
		  // If the size of dIn is the same as cOut dIn could be larger than
		  // the modulus. If it is, then RSAEP() will catch it.
	      }
	      break;
	  case ALG_RSAES_VALUE:
	    retVal = RSAES_PKCS1v1_5Encode(&cOut->b, dIn, rand);
	    break;
	  case ALG_OAEP_VALUE:
	    retVal = OaepEncode(&cOut->b, scheme->details.oaep.hashAlg, label, dIn,
				rand);
	    break;
	  default:
	    ERROR_RETURN(TPM_RC_SCHEME);
	    break;
	}
    // All the schemes that do padding will come here for the encryption step
    // Check that the Encoding worked
    if(retVal == TPM_RC_SUCCESS)
	// Padding OK so do the encryption
	retVal = RSAEP(&cOut->b, key);
 Exit:
    return retVal;
}


CryptRsaValidateSignature(
			  TPMT_SIGNATURE  *sig,           // IN: signature
			  OBJECT          *key,           // IN: public modulus
			  TPM2B_DIGEST    *digest         // IN: The digest being validated
			  )
{
    TPM_RC          retVal;
    int             padding;
    EVP_PKEY       *pkey = NULL;
    EVP_PKEY_CTX   *ctx = NULL;
    const EVP_MD   *md;
    const char     *digestname;

    //
    // Fatal programming errors
    pAssert(key != NULL && sig != NULL && digest != NULL);
    switch(sig->sigAlg)
	{
	  case ALG_RSAPSS_VALUE:
	    padding = RSA_PKCS1_PSS_PADDING;
	    break;
	  case ALG_RSASSA_VALUE:
	    padding = RSA_PKCS1_PADDING;
	    break;
	  default:
	    return TPM_RC_SCHEME;
	}
    // Errors that might be caused by calling parameters
    if(sig->signature.rsassa.sig.t.size != key->publicArea.unique.rsa.t.size)
	ERROR_RETURN(TPM_RC_SIGNATURE);
    TEST(sig->sigAlg);

    retVal = InitOpenSSLRSAPublicKey(key, &pkey);
    if (retVal != TPM_RC_SUCCESS)
        return retVal;

    digestname = GetDigestNameByHashAlg(sig->signature.any.hashAlg);
    if (digestname == NULL)
        ERROR_RETURN(TPM_RC_VALUE);

    md = EVP_get_digestbyname(digestname);
    ctx = EVP_PKEY_CTX_new(pkey, NULL);
    if (md == NULL || ctx == NULL ||
        EVP_PKEY_verify_init(ctx) <= 0)
        ERROR_RETURN(TPM_RC_FAILURE);

    if (EVP_PKEY_CTX_set_rsa_padding(ctx, padding) <= 0 ||
        EVP_PKEY_CTX_set_signature_md(ctx, md) <= 0)
        ERROR_RETURN(TPM_RC_FAILURE);

    if (EVP_PKEY_verify(ctx,
                        sig->signature.rsassa.sig.t.buffer, sig->signature.rsassa.sig.t.size,
                        digest->t.buffer, digest->t.size) <= 0)
        ERROR_RETURN(TPM_RC_SIGNATURE);

    retVal = TPM_RC_SUCCESS;

 Exit:
    EVP_PKEY_free(pkey);
    EVP_PKEY_CTX_free(ctx);

    return (retVal != TPM_RC_SUCCESS) ? TPM_RC_SIGNATURE : TPM_RC_SUCCESS;
}

extern int s_rsaKeyCacheEnabled;

int GetCachedRsaKey(OBJECT *key, RAND_STATE *rand);

#define GET_CACHED_KEY(key, rand)					\
    (s_rsaKeyCacheEnabled && GetCachedRsaKey(key, rand))

#define GET_CACHED_KEY(key, rand)


OaepDecode(
	   TPM2B           *dataOut,       // OUT: the recovered data
	   TPM_ALG_ID       hashAlg,       // IN: algorithm to use for padding
	   const TPM2B     *label,         // IN: null-terminated string (may be NULL)
	   TPM2B           *padded         // IN: the padded data
	   )
{
    UINT32       i;
    BYTE         seedMask[MAX_DIGEST_SIZE];
    UINT32       hLen = CryptHashGetDigestSize(hashAlg);
    BYTE         mask[MAX_RSA_KEY_BYTES];
    BYTE        *pp;
    BYTE        *pm;
    TPM_RC       retVal = TPM_RC_SUCCESS;
    // Strange size (anything smaller can't be an OAEP padded block)
    // Also check for no leading 0
    if((padded->size < (unsigned)((2 * hLen) + 2)) || (padded->buffer[0] != 0))
	ERROR_RETURN(TPM_RC_VALUE);
    // Use the hash size to determine what to put through MGF1 in order
    // to recover the seedMask
    CryptMGF1(hLen, seedMask, hashAlg, padded->size - hLen - 1,
	      &padded->buffer[hLen + 1]);
    // Recover the seed into seedMask
    pAssert(hLen <= sizeof(seedMask));
    pp = &padded->buffer[1];
    pm = seedMask;
    for(i = hLen; i > 0; i--)
	*pm++ ^= *pp++;
    // Use the seed to generate the data mask
    CryptMGF1(padded->size - hLen - 1, mask, hashAlg, hLen, seedMask);
    // Use the mask generated from seed to recover the padded data
    pp = &padded->buffer[hLen + 1];
    pm = mask;
    for(i = (padded->size - hLen - 1); i > 0; i--)
	*pm++ ^= *pp++;
    // Make sure that the recovered data has the hash of the label
    // Put trial value in the seed mask
    if((CryptHashBlock(hashAlg, label->size, (BYTE *)label->buffer,
		       hLen, seedMask)) != hLen)
	FAIL(FATAL_ERROR_INTERNAL);
    if(memcmp(seedMask, mask, hLen) != 0)
	ERROR_RETURN(TPM_RC_VALUE);
    // find the start of the data
    pm = &mask[hLen];
    for(i = (UINT32)padded->size - (2 * hLen) - 1; i > 0; i--)
	{
	    if(*pm++ != 0)
		break;
	}
    // If we ran out of data or didn't end with 0x01, then return an error
    if(i == 0 || pm[-1] != 0x01)
	ERROR_RETURN(TPM_RC_VALUE);
    // pm should be pointing at the first part of the data
    // and i is one greater than the number of bytes to move
    i--;
    if(i > dataOut->size)
	// Special exit to preserve the size of the output buffer
	return TPM_RC_VALUE;
    memcpy(dataOut->buffer, pm, i);
    dataOut->size = (UINT16)i;
 Exit:
    if(retVal != TPM_RC_SUCCESS)
	dataOut->size = 0;
    return retVal;
}


RSAEP(
      TPM2B       *dInOut,        // IN: size of the encrypted block and the size of
      //     the encrypted value. It must be the size of
      //     the modulus.
      // OUT: the encrypted data. Will receive the
      //      decrypted value
      OBJECT      *key            // IN: the key to use
      )
{
    TPM2B_TYPE(4BYTES, 4);
    TPM2B_4BYTES(e) = {{4, {(BYTE)((RSA_DEFAULT_PUBLIC_EXPONENT >> 24) & 0xff),
			    (BYTE)((RSA_DEFAULT_PUBLIC_EXPONENT >> 16) & 0xff),
			    (BYTE)((RSA_DEFAULT_PUBLIC_EXPONENT >> 8) & 0xff),
			    (BYTE)((RSA_DEFAULT_PUBLIC_EXPONENT)& 0xff)}}};
    //
    if(key->publicArea.parameters.rsaDetail.exponent != 0)
	UINT32_TO_BYTE_ARRAY(key->publicArea.parameters.rsaDetail.exponent,
			     e.t.buffer);
    return ModExpB(dInOut->size, dInOut->buffer, dInOut->size, dInOut->buffer,
		   e.t.size, e.t.buffer, key->publicArea.unique.rsa.t.size,
		   key->publicArea.unique.rsa.t.buffer);
}


OaepEncode(
	   TPM2B       *padded,        // OUT: the pad data
	   TPM_ALG_ID   hashAlg,       // IN: algorithm to use for padding
	   const TPM2B *label,         // IN: null-terminated string (may be NULL)
	   TPM2B       *message,       // IN: the message being padded
	   RAND_STATE  *rand           // IN: the random number generator to use
	   )
{
    INT32        padLen;
    INT32        dbSize;
    INT32        i;
    BYTE         mySeed[MAX_DIGEST_SIZE];
    BYTE        *seed = mySeed;
    INT32        hLen = CryptHashGetDigestSize(hashAlg);
    BYTE         mask[MAX_RSA_KEY_BYTES];
    BYTE        *pp;
    BYTE        *pm;
    TPM_RC       retVal = TPM_RC_SUCCESS;
    pAssert(padded != NULL && message != NULL);
    // A value of zero is not allowed because the KDF can't produce a result
    // if the digest size is zero.
    if(hLen <= 0)
	return TPM_RC_VALUE;
    // Basic size checks
    //  make sure digest isn't too big for key size
    if(padded->size < (2 * hLen) + 2)
	ERROR_RETURN(TPM_RC_HASH);
    // and that message will fit messageSize <= k - 2hLen - 2
    if(message->size > (padded->size - (2 * hLen) - 2))
	ERROR_RETURN(TPM_RC_VALUE);
    // Hash L even if it is null
    // Offset into padded leaving room for masked seed and byte of zero
    pp = &padded->buffer[hLen + 1];
    if(CryptHashBlock(hashAlg, label->size, (BYTE *)label->buffer,
		      hLen, pp) != hLen)
	ERROR_RETURN(TPM_RC_FAILURE);
    // concatenate PS of k  mLen  2hLen  2
    padLen = padded->size - message->size - (2 * hLen) - 2;
    MemorySet(&pp[hLen], 0, padLen);
    pp[hLen + padLen] = 0x01;
    padLen += 1;
    memcpy(&pp[hLen + padLen], message->buffer, message->size);
    // The total size of db = hLen + pad + mSize;
    dbSize = hLen + padLen + message->size;
    // If testing, then use the provided seed. Otherwise, use values
    // from the RNG
    CryptRandomGenerate(hLen, mySeed);
    DRBG_Generate(rand, mySeed, (UINT16)hLen);
    // mask = MGF1 (seed, nSize  hLen  1)
    CryptMGF1(dbSize, mask, hashAlg, hLen, seed);
    // Create the masked db
    pm = mask;
    for(i = dbSize; i > 0; i--)
	*pp++ ^= *pm++;
    pp = &padded->buffer[hLen + 1];
    // Run the masked data through MGF1
    if(CryptMGF1(hLen, &padded->buffer[1], hashAlg, dbSize, pp) != (unsigned)hLen)
	ERROR_RETURN(TPM_RC_VALUE);
    // Now XOR the seed to create masked seed
    pp = &padded->buffer[1];
    pm = seed;
    for(i = hLen; i > 0; i--)
	*pp++ ^= *pm++;
    // Set the first byte to zero
    padded->buffer[0] = 0x00;
 Exit:
    return retVal;
}


RSAES_PKCS1v1_5Encode(
		      TPM2B       *padded,        // OUT: the pad data
		      TPM2B       *message,       // IN: the message being padded
		      RAND_STATE  *rand
		      )
{
    UINT32      ps = padded->size - message->size - 3;
    //
    if(message->size > padded->size - 11)
	return TPM_RC_VALUE;
    // move the message to the end of the buffer
    memcpy(&padded->buffer[padded->size - message->size], message->buffer,
	   message->size);
    // Set the first byte to 0x00 and the second to 0x02
    padded->buffer[0] = 0;
    padded->buffer[1] = 2;
    // Fill with random bytes
    DRBG_Generate(rand, &padded->buffer[2], (UINT16)ps);
    // Set the delimiter for the random field to 0
    padded->buffer[2 + ps] = 0;
    // Now, the only messy part. Make sure that all the 'ps' bytes are non-zero
    // In this implementation, use the value of the current index
    for(ps++; ps > 1; ps--)
	{
	    if(padded->buffer[ps] == 0)
		padded->buffer[ps] = 0x55;  // In the < 0.5% of the cases that the
	    // random value is 0, just pick a value to
	    // put into the spot.
	}
    return TPM_RC_SUCCESS;
}


RsaInitializeExponent(
		      privateExponent_t      *pExp
		      )
{
#if CRT_FORMAT_RSA == NO
    BN_INIT(pExp->D);
#else
    BN_INIT(pExp->Q);
    BN_INIT(pExp->dP);
    BN_INIT(pExp->dQ);
    BN_INIT(pExp->qInv);
#endif
}


RSASSA_Encode(
	      TPM2B               *pOut,      // IN:OUT on in, the size of the public key
	      //        on out, the encoded area
	      TPM_ALG_ID           hashAlg,   // IN: hash algorithm for PKCS1v1_5
	      TPM2B               *hIn        // IN: digest value to encode
	      )
{
    const BYTE      *der;
    BYTE            *eOut;
    INT32            derSize = CryptHashGetDer(hashAlg, &der);
    INT32            fillSize;
    TPM_RC           retVal = TPM_RC_SUCCESS;
    // Can't use this scheme if the algorithm doesn't have a DER string defined.
    if(derSize == 0)
	ERROR_RETURN(TPM_RC_SCHEME);
    // If the digest size of 'hashAl' doesn't match the input digest size, then
    // the DER will misidentify the digest so return an error
    if(CryptHashGetDigestSize(hashAlg) != hIn->size)
	ERROR_RETURN(TPM_RC_VALUE);
    fillSize = pOut->size - derSize - hIn->size - 3;
    eOut = pOut->buffer;
    // Make sure that this combination will fit in the provided space
    if(fillSize < 8)
	ERROR_RETURN(TPM_RC_SIZE);
    // Start filling
    *eOut++ = 0; // initial byte of zero
    *eOut++ = 1; // byte of 0x01
    for(; fillSize > 0; fillSize--)
	*eOut++ = 0xff; // bunch of 0xff
    *eOut++ = 0; // another 0
    for(; derSize > 0; derSize--)
	*eOut++ = *der++;   // copy the DER
    der = hIn->buffer;
    for(fillSize = hIn->size; fillSize > 0; fillSize--)
	*eOut++ = *der++;   // copy the hash
 Exit:
    return retVal;
}


CryptRsaSign(
	     TPMT_SIGNATURE      *sigOut,
	     OBJECT              *key,           // IN: key to use
	     TPM2B_DIGEST        *hIn,           // IN: the digest to sign
	     RAND_STATE          *rand           // IN: the random number generator
	     //      to use (mostly for testing)
	     )
{
    TPM_RC                retVal = TPM_RC_SUCCESS;
    UINT16                modSize;
    // parameter checks
    pAssert(sigOut != NULL && key != NULL && hIn != NULL);
    modSize = key->publicArea.unique.rsa.t.size;
    // for all non-null signatures, the size is the size of the key modulus
    sigOut->signature.rsapss.sig.t.size = modSize;
    TEST(sigOut->sigAlg);
    switch(sigOut->sigAlg)
	{
	  case ALG_NULL_VALUE:
	    sigOut->signature.rsapss.sig.t.size = 0;
	    return TPM_RC_SUCCESS;
	  case ALG_RSAPSS_VALUE:
	    retVal = PssEncode(&sigOut->signature.rsapss.sig.b,
			       sigOut->signature.rsapss.hash, &hIn->b, rand);
	    break;
	  case ALG_RSASSA_VALUE:
	    retVal = RSASSA_Encode(&sigOut->signature.rsassa.sig.b,
				   sigOut->signature.rsassa.hash, &hIn->b);
	    break;
	  default:
	    retVal = TPM_RC_SCHEME;
	}
    if(retVal == TPM_RC_SUCCESS)
	{
	    // Do the encryption using the private key
	    retVal = RSADP(&sigOut->signature.rsapss.sig.b, key);
	}
    return retVal;
}


CryptRsaGenerateKey(
		    OBJECT              *rsaKey,            // IN/OUT: The object structure in which
		    //          the key is created.
		    RAND_STATE          *rand               // IN: if not NULL, the deterministic
		    //     RNG state
		    )
{
    UINT32               i;
    BN_PRIME(bnP); // These four declarations initialize the number to 0
    BN_PRIME(bnQ);
    BN_RSA(bnD);
    BN_RSA(bnN);
    BN_WORD(bnE);
    UINT32               e;
    int                  keySizeInBits;
    TPMT_PUBLIC         *publicArea = &rsaKey->publicArea;
    TPMT_SENSITIVE      *sensitive = &rsaKey->sensitive;
    TPM_RC               retVal = TPM_RC_NO_RESULT;
    //
    // Need to make sure that the caller did not specify an exponent that is
    // not supported
    e = publicArea->parameters.rsaDetail.exponent;
    if(e == 0)
	e = RSA_DEFAULT_PUBLIC_EXPONENT;
    if(e < 65537)
	ERROR_RETURN(TPM_RC_RANGE);
    if(e != RSA_DEFAULT_PUBLIC_EXPONENT && !IsPrimeInt(e))
	ERROR_RETURN(TPM_RC_RANGE);
    BnSetWord(bnE, e);
    // Check that e is prime
    // check for supported key size.
    keySizeInBits = publicArea->parameters.rsaDetail.keyBits;
    if(((keySizeInBits % 1024) != 0)
       || (keySizeInBits > MAX_RSA_KEY_BITS)  // this might be redundant, but...
       || (keySizeInBits == 0))
	ERROR_RETURN(TPM_RC_VALUE);
    // Set the prime size for instrumentation purposes
    INSTRUMENT_SET(PrimeIndex, PRIME_INDEX(keySizeInBits / 2));
#if SIMULATION && USE_RSA_KEY_CACHE
    if(GET_CACHED_KEY(rsaKey, rand))
	return TPM_RC_SUCCESS;
#endif
    // Make sure that key generation has been tested
    TEST(ALG_NULL_VALUE);
#if USE_OPENSSL_FUNCTIONS_RSA          // libtpms added begin
    if (rand == NULL)
        return OpenSSLCryptRsaGenerateKey(rsaKey, e, keySizeInBits);
#endif                                 // libtpms added end
    // Need to initialize the privateExponent structure
    RsaInitializeExponent(&rsaKey->privateExponent);
    // The prime is computed in P. When a new prime is found, Q is checked to
    // see if it is zero.  If so, P is copied to Q and a new P is found.
    // When both P and Q are non-zero, the modulus and
    // private exponent are computed and a trial encryption/decryption is
    // performed.  If the encrypt/decrypt fails, assume that at least one of the
    // primes is composite. Since we don't know which one, set Q to zero and start
    // over and find a new pair of primes.
    for(i = 1; (retVal != TPM_RC_SUCCESS) && (i != 100); i++)
	{
	    if(_plat__IsCanceled())
		ERROR_RETURN(TPM_RC_CANCELED);
	    BnGeneratePrimeForRSA(bnP, keySizeInBits / 2, e, rand);
	    INSTRUMENT_INC(PrimeCounts[PrimeIndex]);
	    // If this is the second prime, make sure that it differs from the
	    // first prime by at least 2^100
	    if(BnEqualZero(bnQ))
		{
		    // copy p to q and compute another prime in p
		    BnCopy(bnQ, bnP);
		    continue;
		}
	    // Make sure that the difference is at least 100 bits. Need to do it this
	    // way because the big numbers are only positive values
	    if(BnUnsignedCmp(bnP, bnQ) < 0)
		BnSub(bnD, bnQ, bnP);
	    else
		BnSub(bnD, bnP, bnQ);
	    if(BnMsb(bnD) < 100)
		continue;
	    //Form the public modulus and set the unique value
	    BnMult(bnN, bnP, bnQ);
	    BnTo2B(bnN, &publicArea->unique.rsa.b,
		   (NUMBYTES)BITS_TO_BYTES(keySizeInBits));
	    // And the  prime to the sensitive area
	    BnTo2B(bnP, &sensitive->sensitive.rsa.b,
		   (NUMBYTES)BITS_TO_BYTES(keySizeInBits) / 2);
	    // Make sure everything came out right. The MSb of the values must be
	    // one
	    if(((publicArea->unique.rsa.t.buffer[0] & 0x80) == 0)
	       || ((sensitive->sensitive.rsa.t.buffer[0] & 0x80) == 0))
		FAIL(FATAL_ERROR_INTERNAL);
	    // Make sure that we can form the private exponent values
	    if(ComputePrivateExponent(bnP, bnQ, bnE, bnN, &rsaKey->privateExponent)
	       != TRUE)
		{
		    // If ComputePrivateExponent could not find an inverse for
		    // Q, then copy P and recompute P. This might
		    // cause both to be recomputed if P is also zero
		    if(BnEqualZero(bnQ))
			BnCopy(bnQ, bnP);
		    continue;
		}
	    retVal = TPM_RC_SUCCESS;
	    // Do a trial encryption decryption if this is a signing key
	    if(IS_ATTRIBUTE(publicArea->objectAttributes, TPMA_OBJECT, sign))
		{
		    BN_RSA(temp1);
		    BN_RSA(temp2);
		    BnGenerateRandomInRange(temp1, bnN, rand);
		    // Encrypt with public exponent...
		    BnModExp(temp2, temp1, bnE, bnN);
		    // ...  then decrypt with private exponent
		    RsaPrivateKeyOp(temp2, bnN, bnP, &rsaKey->privateExponent);
		    // If the starting and ending values are not the same,
		    // start over )-;
		    if(BnUnsignedCmp(temp2, temp1) != 0)
			{
			    BnSetWord(bnQ, 0);
			    retVal = TPM_RC_NO_RESULT;
			}
		}
	}
 Exit:
    if(retVal == TPM_RC_SUCCESS)
	rsaKey->attributes.privateExp = SET;
    return retVal;
}


PssDecode(
	  TPM_ALG_ID   hashAlg,        // IN: hash algorithm to use for the encoding
	  TPM2B       *dIn,            // In: the digest to compare
	  TPM2B       *eIn             // IN: the encoded data
	  )
{
    UINT32           hLen = CryptHashGetDigestSize(hashAlg);
    BYTE             mask[MAX_RSA_KEY_BYTES];
    BYTE            *pm = mask;
    BYTE            *pe;
    BYTE             pad[8] = {0};
    UINT32           i;
    UINT32           mLen;
    BYTE             fail;
    TPM_RC           retVal = TPM_RC_SUCCESS;
    HASH_STATE       hashState;
    // These errors are indicative of failures due to programmer error
    pAssert(dIn != NULL && eIn != NULL);
    pe = eIn->buffer;
    // check the hash scheme
    if(hLen == 0)
	ERROR_RETURN(TPM_RC_SCHEME);
    // most significant bit must be zero
    fail = pe[0] & 0x80;
    // last byte must be 0xbc
    fail |= pe[eIn->size - 1] ^ 0xbc;
    // Use the hLen bytes at the end of the buffer to generate a mask
    // Doesn't start at the end which is a flag byte
    mLen = eIn->size - hLen - 1;
    CryptMGF1(mLen, mask, hashAlg, hLen, &pe[mLen]);
    // Clear the MSO of the mask to make it consistent with the encoding.
    mask[0] &= 0x7F;
    pAssert(mLen <= sizeof(mask));
    // XOR the data into the mask to recover the salt. This sequence
    // advances eIn so that it will end up pointing to the seed data
    // which is the hash of the signature data
    for(i = mLen; i > 0; i--)
	*pm++ ^= *pe++;
    // Find the first byte of 0x01 after a string of all 0x00
    for(pm = mask, i = mLen; i > 0; i--)
	{
	    if(*pm == 0x01)
		break;
	    else
		fail |= *pm++;
	}
    // i should not be zero
    fail |= (i == 0);
    // if we have failed, will continue using the entire mask as the salt value so
    // that the timing attacks will not disclose anything (I don't think that this
    // is a problem for TPM applications but, usually, we don't fail so this
    // doesn't cost anything).
    if(fail)
	{
	    i = mLen;
	    pm = mask;
	}
    else
	{
	    pm++;
	    i--;
	}
    // i contains the salt size and pm points to the salt. Going to use the input
    // hash and the seed to recreate the hash in the lower portion of eIn.
    CryptHashStart(&hashState, hashAlg);
    // add the pad of 8 zeros
    CryptDigestUpdate(&hashState, 8, pad);
    // add the provided digest value
    CryptDigestUpdate(&hashState, dIn->size, dIn->buffer);
    // and the salt
    CryptDigestUpdate(&hashState, i, pm);
    // get the result
    fail |= (CryptHashEnd(&hashState, hLen, mask) != hLen);
    // Compare all bytes
    for(pm = mask; hLen > 0; hLen--)
	// don't use fail = because that could skip the increment and compare
	// operations after the first failure and that gives away timing
	// information.
	fail |= *pm++ ^ *pe++;
    retVal = (fail != 0) ? TPM_RC_VALUE : TPM_RC_SUCCESS;
 Exit:
    return retVal;
}


RSAES_Decode(
	     TPM2B       *message,       // OUT: the recovered message
	     TPM2B       *coded          // IN: the encoded message
	     )
{
    BOOL        fail = FALSE;
    UINT16      pSize;
    fail = (coded->size < 11);
    fail = (coded->buffer[0] != 0x00) | fail;
    fail = (coded->buffer[1] != 0x02) | fail;
    for(pSize = 2; pSize < coded->size; pSize++)
	{
	    if(coded->buffer[pSize] == 0)
		break;
	}
    pSize++;
    // Make sure that pSize has not gone over the end and that there are at least 8
    // bytes of pad data.
    fail = (pSize >= coded->size) | fail;
    fail = ((pSize - 2) < 8) | fail;
    if((message->size < (UINT16)(coded->size - pSize)) || fail)
	return TPM_RC_VALUE;
    message->size = coded->size - pSize;
    memcpy(message->buffer, &coded->buffer[pSize], coded->size - pSize);
    return TPM_RC_SUCCESS;
}


ImagingConvertTransparent(Imaging imIn, const char *mode, int r, int g, int b) {
    ImagingSectionCookie cookie;
    ImagingShuffler convert;
    Imaging imOut = NULL;
    int y;

    if (!imIn) {
        return (Imaging)ImagingError_ModeError();
    }

    if (!((strcmp(imIn->mode, "RGB") == 0 || strcmp(imIn->mode, "1") == 0 ||
           strcmp(imIn->mode, "I") == 0 || strcmp(imIn->mode, "L") == 0) &&
          strcmp(mode, "RGBA") == 0))
#ifdef notdef
    {
        return (Imaging)ImagingError_ValueError("conversion not supported");
    }
#else
    {
        static char buf[100];
        snprintf(
            buf,
            100,
            "conversion from %.10s to %.10s not supported in convert_transparent",
            imIn->mode,
            mode);
        return (Imaging)ImagingError_ValueError(buf);
    }
#endif

    if (strcmp(imIn->mode, "RGB") == 0) {
        convert = rgb2rgba;
    } else {
        if (strcmp(imIn->mode, "1") == 0) {
            convert = bit2rgb;
        } else if (strcmp(imIn->mode, "I") == 0) {
            convert = i2rgb;
        } else {
            convert = l2rgb;
        }
        g = b = r;
    }

    imOut = ImagingNew2Dirty(mode, imOut, imIn);
    if (!imOut) {
        return NULL;
    }

    ImagingSectionEnter(&cookie);
    for (y = 0; y < imIn->ysize; y++) {
        (*convert)((UINT8 *)imOut->image[y], (UINT8 *)imIn->image[y], imIn->xsize);
        rgbT2rgba((UINT8 *)imOut->image[y], imIn->xsize, r, g, b);
    }
    ImagingSectionLeave(&cookie);

    return imOut;
}

Use snprintf instead of sprintf


convert(
    Imaging imOut, Imaging imIn, const char *mode, ImagingPalette palette, int dither) {
    ImagingSectionCookie cookie;
    ImagingShuffler convert;
    int y;

    if (!imIn) {
        return (Imaging)ImagingError_ModeError();
    }

    if (!mode) {
        /* Map palette image to full depth */
        if (!imIn->palette) {
            return (Imaging)ImagingError_ModeError();
        }
        mode = imIn->palette->mode;
    } else {
        /* Same mode? */
        if (!strcmp(imIn->mode, mode)) {
            return ImagingCopy2(imOut, imIn);
        }
    }

    /* test for special conversions */

    if (strcmp(imIn->mode, "P") == 0 || strcmp(imIn->mode, "PA") == 0) {
        return frompalette(imOut, imIn, mode);
    }

    if (strcmp(mode, "P") == 0 || strcmp(mode, "PA") == 0) {
        return topalette(imOut, imIn, mode, palette, dither);
    }

    if (dither && strcmp(mode, "1") == 0) {
        return tobilevel(imOut, imIn, dither);
    }

    /* standard conversion machinery */

    convert = NULL;

    for (y = 0; converters[y].from; y++) {
        if (!strcmp(imIn->mode, converters[y].from) &&
            !strcmp(mode, converters[y].to)) {
            convert = converters[y].convert;
            break;
        }
    }

    if (!convert) {
#ifdef notdef
        return (Imaging)ImagingError_ValueError("conversion not supported");
#else
        static char buf[100];
        snprintf(buf, 100, "conversion from %.10s to %.10s not supported", imIn->mode, mode);
        return (Imaging)ImagingError_ValueError(buf);
#endif
    }

    imOut = ImagingNew2Dirty(mode, imOut, imIn);
    if (!imOut) {
        return NULL;
    }

    ImagingSectionEnter(&cookie);
    for (y = 0; y < imIn->ysize; y++) {
        (*convert)((UINT8 *)imOut->image[y], (UINT8 *)imIn->image[y], imIn->xsize);
    }
    ImagingSectionLeave(&cookie);

    return imOut;
}

Use snprintf instead of sprintf


MagickExport MagickBooleanType SetResampleFilterVirtualPixelMethod(
  ResampleFilter *resample_filter,const VirtualPixelMethod method)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);
  assert(resample_filter->image != (Image *) NULL);
  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);
  resample_filter->virtual_pixel=method;
  if (method != UndefinedVirtualPixelMethod)
    (void) SetCacheViewVirtualPixelMethod(resample_filter->view,method);
  return(MagickTrue);
}

#define DEBUG_ELLIPSE 0       /* output ellipse info for debug */

struct _ResampleFilter
{
  CacheView
    *view;

  Image
    *image;

  ExceptionInfo
    *exception;

  MagickBooleanType
    debug;

  /* Information about image being resampled */
  ssize_t
    image_area;

  PixelInterpolateMethod
    interpolate;

  VirtualPixelMethod
    virtual_pixel;

  FilterType
    filter;

  /* processing settings needed */
  MagickBooleanType
    limit_reached,
    do_interpolate,
    average_defined;

  PixelInfo
    average_pixel;

  /* current ellipitical area being resampled around center point */
  double
    A, B, C,
    Vlimit, Ulimit, Uwidth, slope;

#if FILTER_LUT
  /* LUT of weights for filtered average in elliptical area */
  double
    filter_lut[WLUT_WIDTH];
#else
  /* Use a Direct call to the filter functions */
  ResizeFilter
    *filter_def;

  double
    F;
#endif

  /* the practical working support of the filter */
  double
    support;

  size_t
    signature;
}


MagickExport ResampleFilter *DestroyResampleFilter(
  ResampleFilter *resample_filter)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);
  assert(resample_filter->image != (Image *) NULL);
  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);
  resample_filter->view=DestroyCacheView(resample_filter->view);
  resample_filter->image=DestroyImage(resample_filter->image);
#if ! FILTER_LUT
  resample_filter->filter_def=DestroyResizeFilter(resample_filter->filter_def);
#endif
  resample_filter->signature=(~MagickCoreSignature);
  resample_filter=(ResampleFilter *) RelinquishMagickMemory(resample_filter);
  return(resample_filter);
}

#define FILTER_LUT 1          /* Use a LUT rather then direct filter calls */

#define DEBUG_ELLIPSE 0       /* output ellipse info for debug */

struct _ResampleFilter
{
  CacheView
    *view;

  Image
    *image;

  ExceptionInfo
    *exception;

  MagickBooleanType
    debug;

  /* Information about image being resampled */
  ssize_t
    image_area;

  PixelInterpolateMethod
    interpolate;

  VirtualPixelMethod
    virtual_pixel;

  FilterType
    filter;

  /* processing settings needed */
  MagickBooleanType
    limit_reached,
    do_interpolate,
    average_defined;

  PixelInfo
    average_pixel;

  /* current ellipitical area being resampled around center point */
  double
    A, B, C,
    Vlimit, Ulimit, Uwidth, slope;

#if FILTER_LUT
  /* LUT of weights for filtered average in elliptical area */
  double
    filter_lut[WLUT_WIDTH];
#else
  /* Use a Direct call to the filter functions */
  ResizeFilter
    *filter_def;

  double
    F;
#endif

  /* the practical working support of the filter */
  double
    support;

  size_t
    signature;
}


static inline void ClampUpAxes(const double dux,
			       const double dvx,
			       const double duy,
			       const double dvy,
			       double *major_mag,
			       double *minor_mag,
			       double *major_unit_x,
			       double *major_unit_y,
			       double *minor_unit_x,
			       double *minor_unit_y)
{
  /*
   * ClampUpAxes takes an input 2x2 matrix
   *
   * [ a b ] = [ dux duy ]
   * [ c d ] = [ dvx dvy ]
   *
   * and computes from it the major and minor axis vectors [major_x,
   * major_y] and [minor_x,minor_y] of the smallest ellipse containing
   * both the unit disk and the ellipse which is the image of the unit
   * disk by the linear transformation
   *
   * [ dux duy ] [S] = [s]
   * [ dvx dvy ] [T] = [t]
   *
   * (The vector [S,T] is the difference between a position in output
   * space and [X,Y]; the vector [s,t] is the difference between a
   * position in input space and [x,y].)
   */
  /*
   * Output:
   *
   * major_mag is the half-length of the major axis of the "new"
   * ellipse.
   *
   * minor_mag is the half-length of the minor axis of the "new"
   * ellipse.
   *
   * major_unit_x is the x-coordinate of the major axis direction vector
   * of both the "old" and "new" ellipses.
   *
   * major_unit_y is the y-coordinate of the major axis direction vector.
   *
   * minor_unit_x is the x-coordinate of the minor axis direction vector.
   *
   * minor_unit_y is the y-coordinate of the minor axis direction vector.
   *
   * Unit vectors are useful for computing projections, in particular,
   * to compute the distance between a point in output space and the
   * center of a unit disk in output space, using the position of the
   * corresponding point [s,t] in input space. Following the clamping,
   * the square of this distance is
   *
   * ( ( s * major_unit_x + t * major_unit_y ) / major_mag )^2
   * +
   * ( ( s * minor_unit_x + t * minor_unit_y ) / minor_mag )^2
   *
   * If such distances will be computed for many [s,t]'s, it makes
   * sense to actually compute the reciprocal of major_mag and
   * minor_mag and multiply them by the above unit lengths.
   *
   * Now, if you want to modify the input pair of tangent vectors so
   * that it defines the modified ellipse, all you have to do is set
   *
   * newdux = major_mag * major_unit_x
   * newdvx = major_mag * major_unit_y
   * newduy = minor_mag * minor_unit_x = minor_mag * -major_unit_y
   * newdvy = minor_mag * minor_unit_y = minor_mag *  major_unit_x
   *
   * and use these tangent vectors as if they were the original ones.
   * Usually, this is a drastic change in the tangent vectors even if
   * the singular values are not clamped; for example, the minor axis
   * vector always points in a direction which is 90 degrees
   * counterclockwise from the direction of the major axis vector.
   */
  /*
   * Discussion:
   *
   * GOAL: Fix things so that the pullback, in input space, of a disk
   * of radius r in output space is an ellipse which contains, at
   * least, a disc of radius r. (Make this hold for any r>0.)
   *
   * ESSENCE OF THE METHOD: Compute the product of the first two
   * factors of an SVD of the linear transformation defining the
   * ellipse and make sure that both its columns have norm at least 1.
   * Because rotations and reflexions map disks to themselves, it is
   * not necessary to compute the third (rightmost) factor of the SVD.
   *
   * DETAILS: Find the singular values and (unit) left singular
   * vectors of Jinv, clampling up the singular values to 1, and
   * multiply the unit left singular vectors by the new singular
   * values in order to get the minor and major ellipse axis vectors.
   *
   * Image resampling context:
   *
   * The Jacobian matrix of the transformation at the output point
   * under consideration is defined as follows:
   *
   * Consider the transformation (x,y) -> (X,Y) from input locations
   * to output locations. (Anthony Thyssen, elsewhere in resample.c,
   * uses the notation (u,v) -> (x,y).)
   *
   * The Jacobian matrix of the transformation at (x,y) is equal to
   *
   *   J = [ A, B ] = [ dX/dx, dX/dy ]
   *       [ C, D ]   [ dY/dx, dY/dy ]
   *
   * that is, the vector [A,C] is the tangent vector corresponding to
   * input changes in the horizontal direction, and the vector [B,D]
   * is the tangent vector corresponding to input changes in the
   * vertical direction.
   *
   * In the context of resampling, it is natural to use the inverse
   * Jacobian matrix Jinv because resampling is generally performed by
   * pulling pixel locations in the output image back to locations in
   * the input image. Jinv is
   *
   *   Jinv = [ a, b ] = [ dx/dX, dx/dY ]
   *          [ c, d ]   [ dy/dX, dy/dY ]
   *
   * Note: Jinv can be computed from J with the following matrix
   * formula:
   *
   *   Jinv = 1/(A*D-B*C) [  D, -B ]
   *                      [ -C,  A ]
   *
   * What we do is modify Jinv so that it generates an ellipse which
   * is as close as possible to the original but which contains the
   * unit disk. This can be accomplished as follows:
   *
   * Let
   *
   *   Jinv = U Sigma V^T
   *
   * be an SVD decomposition of Jinv. (The SVD is not unique, but the
   * final ellipse does not depend on the particular SVD.)
   *
   * We could clamp up the entries of the diagonal matrix Sigma so
   * that they are at least 1, and then set
   *
   *   Jinv = U newSigma V^T.
   *
   * However, we do not need to compute V for the following reason:
   * V^T is an orthogonal matrix (that is, it represents a combination
   * of rotations and reflexions) so that it maps the unit circle to
   * itself. For this reason, the exact value of V does not affect the
   * final ellipse, and we can choose V to be the identity
   * matrix. This gives
   *
   *   Jinv = U newSigma.
   *
   * In the end, we return the two diagonal entries of newSigma
   * together with the two columns of U.
   */
  /*
   * ClampUpAxes was written by Nicolas Robidoux and Chantal Racette
   * of Laurentian University with insightful suggestions from Anthony
   * Thyssen and funding from the National Science and Engineering
   * Research Council of Canada. It is distinguished from its
   * predecessors by its efficient handling of degenerate cases.
   *
   * The idea of clamping up the EWA ellipse's major and minor axes so
   * that the result contains the reconstruction kernel filter support
   * is taken from Andreas Gustaffson's Masters thesis "Interactive
   * Image Warping", Helsinki University of Technology, Faculty of
   * Information Technology, 59 pages, 1993 (see Section 3.6).
   *
   * The use of the SVD to clamp up the singular values of the
   * Jacobian matrix of the pullback transformation for EWA resampling
   * is taken from the astrophysicist Craig DeForest.  It is
   * implemented in his PDL::Transform code (PDL = Perl Data
   * Language).
   */
  const double a = dux;
  const double b = duy;
  const double c = dvx;
  const double d = dvy;
  /*
   * n is the matrix Jinv * transpose(Jinv). Eigenvalues of n are the
   * squares of the singular values of Jinv.
   */
  const double aa = a*a;
  const double bb = b*b;
  const double cc = c*c;
  const double dd = d*d;
  /*
   * Eigenvectors of n are left singular vectors of Jinv.
   */
  const double n11 = aa+bb;
  const double n12 = a*c+b*d;
  const double n21 = n12;
  const double n22 = cc+dd;
  const double det = a*d-b*c;
  const double twice_det = det+det;
  const double frobenius_squared = n11+n22;
  const double discriminant =
    (frobenius_squared+twice_det)*(frobenius_squared-twice_det);
  /*
   * In exact arithmetic, discriminant can't be negative. In floating
   * point, it can, because of the bad conditioning of SVD
   * decompositions done through the associated normal matrix.
   */
  const double sqrt_discriminant =
    sqrt(discriminant > 0.0 ? discriminant : 0.0);
  /*
   * s1 is the largest singular value of the inverse Jacobian
   * matrix. In other words, its reciprocal is the smallest singular
   * value of the Jacobian matrix itself.
   * If s1 = 0, both singular values are 0, and any orthogonal pair of
   * left and right factors produces a singular decomposition of Jinv.
   */
  /*
   * Initially, we only compute the squares of the singular values.
   */
  const double s1s1 = 0.5*(frobenius_squared+sqrt_discriminant);
  /*
   * s2 the smallest singular value of the inverse Jacobian
   * matrix. Its reciprocal is the largest singular value of the
   * Jacobian matrix itself.
   */
  const double s2s2 = 0.5*(frobenius_squared-sqrt_discriminant);
  const double s1s1minusn11 = s1s1-n11;
  const double s1s1minusn22 = s1s1-n22;
  /*
   * u1, the first column of the U factor of a singular decomposition
   * of Jinv, is a (non-normalized) left singular vector corresponding
   * to s1. It has entries u11 and u21. We compute u1 from the fact
   * that it is an eigenvector of n corresponding to the eigenvalue
   * s1^2.
   */
  const double s1s1minusn11_squared = s1s1minusn11*s1s1minusn11;
  const double s1s1minusn22_squared = s1s1minusn22*s1s1minusn22;
  /*
   * The following selects the largest row of n-s1^2 I as the one
   * which is used to find the eigenvector. If both s1^2-n11 and
   * s1^2-n22 are zero, n-s1^2 I is the zero matrix.  In that case,
   * any vector is an eigenvector; in addition, norm below is equal to
   * zero, and, in exact arithmetic, this is the only case in which
   * norm = 0. So, setting u1 to the simple but arbitrary vector [1,0]
   * if norm = 0 safely takes care of all cases.
   */
  const double temp_u11 =
    ( (s1s1minusn11_squared>=s1s1minusn22_squared) ? n12 : s1s1minusn22 );
  const double temp_u21 =
    ( (s1s1minusn11_squared>=s1s1minusn22_squared) ? s1s1minusn11 : n21 );
  const double norm = sqrt(temp_u11*temp_u11+temp_u21*temp_u21);
  /*
   * Finalize the entries of first left singular vector (associated
   * with the largest singular value).
   */
  const double u11 = ( (norm>0.0) ? temp_u11/norm : 1.0 );
  const double u21 = ( (norm>0.0) ? temp_u21/norm : 0.0 );
  /*
   * Clamp the singular values up to 1.
   */
  *major_mag = ( (s1s1<=1.0) ? 1.0 : sqrt(s1s1) );
  *minor_mag = ( (s2s2<=1.0) ? 1.0 : sqrt(s2s2) );
  /*
   * Return the unit major and minor axis direction vectors.
   */
  *major_unit_x = u11;
  *major_unit_y = u21;
  *minor_unit_x = -u21;
  *minor_unit_y = u11;
}

#define EWA 1                 /* Normal EWA handling - raw or clamped */

#define EWA_CLAMP 1           /* EWA Clamping from Nicolas Robidoux */

#define FILTER_LUT 1          /* Use a LUT rather then direct filter calls */

#define DEBUG_ELLIPSE 0       /* output ellipse info for debug */

#define DEBUG_HIT_MISS 0      /* output hit/miss pixels (as gnuplot commands) */

#define DEBUG_NO_PIXEL_HIT 0  /* Make pixels that fail to hit anything - RED */

#define WLUT_WIDTH 1024       /* size of the filter cache */

struct _ResampleFilter
{
  CacheView
    *view;

  Image
    *image;

  ExceptionInfo
    *exception;

  MagickBooleanType
    debug;

  /* Information about image being resampled */
  ssize_t
    image_area;

  PixelInterpolateMethod
    interpolate;

  VirtualPixelMethod
    virtual_pixel;

  FilterType
    filter;

  /* processing settings needed */
  MagickBooleanType
    limit_reached,
    do_interpolate,
    average_defined;

  PixelInfo
    average_pixel;

  /* current ellipitical area being resampled around center point */
  double
    A, B, C,
    Vlimit, Ulimit, Uwidth, slope;

#if FILTER_LUT
  /* LUT of weights for filtered average in elliptical area */
  double
    filter_lut[WLUT_WIDTH];
#else
  /* Use a Direct call to the filter functions */
  ResizeFilter
    *filter_def;

  double
    F;
#endif

  /* the practical working support of the filter */
  double
    support;

  size_t
    signature;
}


MagickExport ResampleFilter *AcquireResampleFilter(const Image *image,
  ExceptionInfo *exception)
{
  ResampleFilter
    *resample_filter;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  assert(exception != (ExceptionInfo *) NULL);
  assert(exception->signature == MagickCoreSignature);
  resample_filter=(ResampleFilter *) AcquireCriticalMemory(sizeof(
    *resample_filter));
  (void) memset(resample_filter,0,sizeof(*resample_filter));
  resample_filter->exception=exception;
  resample_filter->image=ReferenceImage((Image *) image);
  resample_filter->view=AcquireVirtualCacheView(resample_filter->image,
    exception);
  resample_filter->debug=IsEventLogging();
  resample_filter->image_area=(ssize_t) (image->columns*image->rows);
  resample_filter->average_defined=MagickFalse;
  resample_filter->signature=MagickCoreSignature;
  SetResampleFilter(resample_filter,image->filter);
  (void) SetResampleFilterInterpolateMethod(resample_filter,image->interpolate);
  (void) SetResampleFilterVirtualPixelMethod(resample_filter,
    GetImageVirtualPixelMethod(image));
  return(resample_filter);
}

#define FILTER_LUT 1          /* Use a LUT rather then direct filter calls */

#define DEBUG_ELLIPSE 0       /* output ellipse info for debug */

#define WLUT_WIDTH 1024       /* size of the filter cache */

struct _ResampleFilter
{
  CacheView
    *view;

  Image
    *image;

  ExceptionInfo
    *exception;

  MagickBooleanType
    debug;

  /* Information about image being resampled */
  ssize_t
    image_area;

  PixelInterpolateMethod
    interpolate;

  VirtualPixelMethod
    virtual_pixel;

  FilterType
    filter;

  /* processing settings needed */
  MagickBooleanType
    limit_reached,
    do_interpolate,
    average_defined;

  PixelInfo
    average_pixel;

  /* current ellipitical area being resampled around center point */
  double
    A, B, C,
    Vlimit, Ulimit, Uwidth, slope;

#if FILTER_LUT
  /* LUT of weights for filtered average in elliptical area */
  double
    filter_lut[WLUT_WIDTH];
#else
  /* Use a Direct call to the filter functions */
  ResizeFilter
    *filter_def;

  double
    F;
#endif

  /* the practical working support of the filter */
  double
    support;

  size_t
    signature;
}


MagickExport MagickBooleanType SetResampleFilterInterpolateMethod(
  ResampleFilter *resample_filter,const PixelInterpolateMethod method)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);
  assert(resample_filter->image != (Image *) NULL);
  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);
  resample_filter->interpolate=method;
  return(MagickTrue);
}


MagickExport void SetResampleFilter(ResampleFilter *resample_filter,
  const FilterType filter)
{
  ResizeFilter
     *resize_filter;

  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);

  resample_filter->do_interpolate = MagickFalse;
  resample_filter->filter = filter;

  /* Default cylindrical filter is a Cubic Keys filter */
  if ( filter == UndefinedFilter )
    resample_filter->filter = RobidouxFilter;

  if ( resample_filter->filter == PointFilter ) {
    resample_filter->do_interpolate = MagickTrue;
    return;  /* EWA turned off - nothing more to do */
  }

  resize_filter = AcquireResizeFilter(resample_filter->image,
    resample_filter->filter,MagickTrue,resample_filter->exception);
  if (resize_filter == (ResizeFilter *) NULL) {
    (void) ThrowMagickException(resample_filter->exception,GetMagickModule(),
         ModuleError, "UnableToSetFilteringValue",
         "Fall back to Interpolated 'Point' filter");
    resample_filter->filter = PointFilter;
    resample_filter->do_interpolate = MagickTrue;
    return;  /* EWA turned off - nothing more to do */
  }

  /* Get the practical working support for the filter,
   * after any API call blur factors have been accoded for.
   */
#if EWA
  resample_filter->support = GetResizeFilterSupport(resize_filter);
#else
  resample_filter->support = 2.0;  /* fixed support size for HQ-EWA */
#endif

#if FILTER_LUT
  /* Fill the LUT with the weights from the selected filter function */
  { int
       Q;
    double
       r_scale;

    /* Scale radius so the filter LUT covers the full support range */
    r_scale = resample_filter->support*sqrt(1.0/(double)WLUT_WIDTH);
    for(Q=0; Q<WLUT_WIDTH; Q++)
      resample_filter->filter_lut[Q] = (double)
           GetResizeFilterWeight(resize_filter,sqrt((double)Q)*r_scale);

    /* finished with the resize filter */
    resize_filter = DestroyResizeFilter(resize_filter);
  }
#else
  /* save the filter and the scaled ellipse bounds needed for filter */
  resample_filter->filter_def = resize_filter;
  resample_filter->F = resample_filter->support*resample_filter->support;
#endif

  /*
    Adjust the scaling of the default unit circle
    This assumes that any real scaling changes will always
    take place AFTER the filter method has been initialized.
  */
  ScaleResampleFilter(resample_filter, 1.0, 0.0, 0.0, 1.0);

#if 0
  /*
    This is old code kept as a reference only. Basically it generates
    a Gaussian bell curve, with sigma = 0.5 if the support is 2.0

    Create Normal Gaussian 2D Filter Weighted Lookup Table.
    A normal EWA guassual lookup would use   exp(Q*ALPHA)
    where  Q = distance squared from 0.0 (center) to 1.0 (edge)
    and    ALPHA = -4.0*ln(2.0)  ==>  -2.77258872223978123767
    The table is of length 1024, and equates to support radius of 2.0
    thus needs to be scaled by  ALPHA*4/1024 and any blur factor squared

    The it comes from reference code provided by Fred Weinhaus.
  */
  r_scale = -2.77258872223978123767/(WLUT_WIDTH*blur*blur);
  for(Q=0; Q<WLUT_WIDTH; Q++)
    resample_filter->filter_lut[Q] = exp((double)Q*r_scale);
  resample_filter->support = WLUT_WIDTH;
#endif

#if FILTER_LUT
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp single
#endif
  {
    if (IsStringTrue(GetImageArtifact(resample_filter->image,
        "resample:verbose")) != MagickFalse)
      {
        int
          Q;
        double
          r_scale;

        /* Debug output of the filter weighting LUT
          Gnuplot the LUT data, the x scale index has been adjusted
            plot [0:2][-.2:1] "lut.dat" with lines
          The filter values should be normalized for comparision
        */
        printf("#\n");
        printf("# Resampling Filter LUT (%d values) for '%s' filter\n",
                   WLUT_WIDTH, CommandOptionToMnemonic(MagickFilterOptions,
                   resample_filter->filter) );
        printf("#\n");
        printf("# Note: values in table are using a squared radius lookup.\n");
        printf("# As such its distribution is not uniform.\n");
        printf("#\n");
        printf("# The X value is the support distance for the Y weight\n");
        printf("# so you can use gnuplot to plot this cylindrical filter\n");
        printf("#    plot [0:2][-.2:1] \"lut.dat\" with lines\n");
        printf("#\n");

        /* Scale radius so the filter LUT covers the full support range */
        r_scale = resample_filter->support*sqrt(1.0/(double)WLUT_WIDTH);
        for(Q=0; Q<WLUT_WIDTH; Q++)
          printf("%8.*g %.*g\n",
              GetMagickPrecision(),sqrt((double)Q)*r_scale,
              GetMagickPrecision(),resample_filter->filter_lut[Q] );
        printf("\n\n"); /* generate a 'break' in gnuplot if multiple outputs */
      }
    /* Output the above once only for each image, and each setting
    (void) DeleteImageArtifact(resample_filter->image,"resample:verbose");
    */
  }
#endif /* FILTER_LUT */
  return;
}


MagickExport MagickBooleanType ResamplePixelColor(
  ResampleFilter *resample_filter,const double u0,const double v0,
  PixelInfo *pixel,ExceptionInfo *exception)
{
  MagickBooleanType
    status;

  ssize_t u,v, v1, v2, uw, hit;
  double u1;
  double U,V,Q,DQ,DDQ;
  double divisor_c,divisor_m;
  double weight;
  const Quantum *pixels;
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);

  status=MagickTrue;
  /* GetPixelInfo(resample_filter->image,pixel); */
  if ( resample_filter->do_interpolate ) {
    status=InterpolatePixelInfo(resample_filter->image,resample_filter->view,
      resample_filter->interpolate,u0,v0,pixel,resample_filter->exception);
    return(status);
  }

#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "u0=%lf; v0=%lf;\n", u0, v0);
#endif

  /*
    Does resample area Miss the image Proper?
    If and that area a simple solid color - then simply return that color!
    This saves a lot of calculation when resampling outside the bounds of
    the source image.

    However it probably should be expanded to image bounds plus the filters
    scaled support size.
  */
  hit = 0;
  switch ( resample_filter->virtual_pixel ) {
    case BackgroundVirtualPixelMethod:
    case TransparentVirtualPixelMethod:
    case BlackVirtualPixelMethod:
    case GrayVirtualPixelMethod:
    case WhiteVirtualPixelMethod:
    case MaskVirtualPixelMethod:
      if ( resample_filter->limit_reached
           || u0 + resample_filter->Ulimit < 0.0
           || u0 - resample_filter->Ulimit > (double) resample_filter->image->columns-1.0
           || v0 + resample_filter->Vlimit < 0.0
           || v0 - resample_filter->Vlimit > (double) resample_filter->image->rows-1.0
           )
        hit++;
      break;

    case UndefinedVirtualPixelMethod:
    case EdgeVirtualPixelMethod:
      if (    ( u0 + resample_filter->Ulimit < 0.0 && v0 + resample_filter->Vlimit < 0.0 )
           || ( u0 + resample_filter->Ulimit < 0.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows-1.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns-1.0
                && v0 + resample_filter->Vlimit < 0.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns-1.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows-1.0 )
           )
        hit++;
      break;
    case HorizontalTileVirtualPixelMethod:
      if (    v0 + resample_filter->Vlimit < 0.0
           || v0 - resample_filter->Vlimit > (double) resample_filter->image->rows-1.0
           )
        hit++;  /* outside the horizontally tiled images. */
      break;
    case VerticalTileVirtualPixelMethod:
      if (    u0 + resample_filter->Ulimit < 0.0
           || u0 - resample_filter->Ulimit > (double) resample_filter->image->columns-1.0
           )
        hit++;  /* outside the vertically tiled images. */
      break;
    case DitherVirtualPixelMethod:
      if (    ( u0 + resample_filter->Ulimit < -32.0 && v0 + resample_filter->Vlimit < -32.0 )
           || ( u0 + resample_filter->Ulimit < -32.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows+31.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns+31.0
                && v0 + resample_filter->Vlimit < -32.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns+31.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows+31.0 )
           )
        hit++;
      break;
    case TileVirtualPixelMethod:
    case MirrorVirtualPixelMethod:
    case RandomVirtualPixelMethod:
    case HorizontalTileEdgeVirtualPixelMethod:
    case VerticalTileEdgeVirtualPixelMethod:
    case CheckerTileVirtualPixelMethod:
      /* resampling of area is always needed - no VP limits */
      break;
  }
  if ( hit ) {
    /* The area being resampled is simply a solid color
     * just return a single lookup color.
     *
     * Should this return the users requested interpolated color?
     */
    status=InterpolatePixelInfo(resample_filter->image,resample_filter->view,
      IntegerInterpolatePixel,u0,v0,pixel,resample_filter->exception);
    return(status);
  }

  /*
    When Scaling limits reached, return an 'averaged' result.
  */
  if ( resample_filter->limit_reached ) {
    switch ( resample_filter->virtual_pixel ) {
      /*  This is always handled by the above, so no need.
        case BackgroundVirtualPixelMethod:
        case ConstantVirtualPixelMethod:
        case TransparentVirtualPixelMethod:
        case GrayVirtualPixelMethod,
        case WhiteVirtualPixelMethod
        case MaskVirtualPixelMethod:
      */
      case UndefinedVirtualPixelMethod:
      case EdgeVirtualPixelMethod:
      case DitherVirtualPixelMethod:
      case HorizontalTileEdgeVirtualPixelMethod:
      case VerticalTileEdgeVirtualPixelMethod:
        /* We need an average edge pixel, from the correct edge!
           How should I calculate an average edge color?
           Just returning an averaged neighbourhood,
           works well in general, but falls down for TileEdge methods.
           This needs to be done properly!!!!!!
        */
        status=InterpolatePixelInfo(resample_filter->image,
          resample_filter->view,AverageInterpolatePixel,u0,v0,pixel,
          resample_filter->exception);
        break;
      case HorizontalTileVirtualPixelMethod:
      case VerticalTileVirtualPixelMethod:
        /* just return the background pixel - Is there more direct way? */
        status=InterpolatePixelInfo(resample_filter->image,
          resample_filter->view,IntegerInterpolatePixel,-1.0,-1.0,pixel,
          resample_filter->exception);
        break;
      case TileVirtualPixelMethod:
      case MirrorVirtualPixelMethod:
      case RandomVirtualPixelMethod:
      case CheckerTileVirtualPixelMethod:
      default:
        /* generate a average color of the WHOLE image */
        if ( resample_filter->average_defined == MagickFalse ) {
          Image
            *average_image;

          CacheView
            *average_view;

          GetPixelInfo(resample_filter->image,(PixelInfo *)
            &resample_filter->average_pixel);
          resample_filter->average_defined=MagickTrue;

          /* Try to get an averaged pixel color of whole image */
          average_image=ResizeImage(resample_filter->image,1,1,BoxFilter,
            resample_filter->exception);
          if (average_image == (Image *) NULL)
            {
              *pixel=resample_filter->average_pixel; /* FAILED */
              break;
            }
          average_view=AcquireVirtualCacheView(average_image,exception);
          pixels=GetCacheViewVirtualPixels(average_view,0,0,1,1,
            resample_filter->exception);
          if (pixels == (const Quantum *) NULL) {
            average_view=DestroyCacheView(average_view);
            average_image=DestroyImage(average_image);
            *pixel=resample_filter->average_pixel; /* FAILED */
            break;
          }
          GetPixelInfoPixel(resample_filter->image,pixels,
            &(resample_filter->average_pixel));
          average_view=DestroyCacheView(average_view);
          average_image=DestroyImage(average_image);

          if ( resample_filter->virtual_pixel == CheckerTileVirtualPixelMethod )
            {
              /* CheckerTile is a alpha blend of the image's average pixel
                 color and the current background color */

              /* image's average pixel color */
              weight = QuantumScale*((double)
                resample_filter->average_pixel.alpha);
              resample_filter->average_pixel.red *= weight;
              resample_filter->average_pixel.green *= weight;
              resample_filter->average_pixel.blue *= weight;
              divisor_c = weight;

              /* background color */
              weight = QuantumScale*((double)
                resample_filter->image->background_color.alpha);
              resample_filter->average_pixel.red +=
                      weight*resample_filter->image->background_color.red;
              resample_filter->average_pixel.green +=
                      weight*resample_filter->image->background_color.green;
              resample_filter->average_pixel.blue +=
                      weight*resample_filter->image->background_color.blue;
              resample_filter->average_pixel.alpha +=
                      resample_filter->image->background_color.alpha;
              divisor_c += weight;

              /* alpha blend */
              resample_filter->average_pixel.red /= divisor_c;
              resample_filter->average_pixel.green /= divisor_c;
              resample_filter->average_pixel.blue /= divisor_c;
              resample_filter->average_pixel.alpha /= 2; /* 50% blend */

            }
        }
        *pixel=resample_filter->average_pixel;
        break;
    }
    return(status);
  }

  /*
    Initialize weighted average data collection
  */
  hit = 0;
  divisor_c = 0.0;
  divisor_m = 0.0;
  pixel->red = pixel->green = pixel->blue = 0.0;
  if (pixel->colorspace == CMYKColorspace)
    pixel->black = 0.0;
  if (pixel->alpha_trait != UndefinedPixelTrait)
    pixel->alpha = 0.0;

  /*
    Determine the parellelogram bounding box fitted to the ellipse
    centered at u0,v0.  This area is bounding by the lines...
  */
  v1 = (ssize_t)ceil(v0 - resample_filter->Vlimit);  /* range of scan lines */
  v2 = (ssize_t)floor(v0 + resample_filter->Vlimit);

  /* scan line start and width accross the parallelogram */
  u1 = u0 + (v1-v0)*resample_filter->slope - resample_filter->Uwidth;
  uw = (ssize_t)(2.0*resample_filter->Uwidth)+1;

#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "v1=%ld; v2=%ld\n", (long)v1, (long)v2);
  (void) FormatLocaleFile(stderr, "u1=%ld; uw=%ld\n", (long)u1, (long)uw);
#else
# define DEBUG_HIT_MISS 0 /* only valid if DEBUG_ELLIPSE is enabled */
#endif

  /*
    Do weighted resampling of all pixels,  within the scaled ellipse,
    bound by a Parellelogram fitted to the ellipse.
  */
  DDQ = 2*resample_filter->A;
  for( v=v1; v<=v2;  v++ ) {
#if DEBUG_HIT_MISS
    long uu = ceil(u1);   /* actual pixel location (for debug only) */
    (void) FormatLocaleFile(stderr, "# scan line from pixel %ld, %ld\n", (long)uu, (long)v);
#endif
    u = (ssize_t)ceil(u1);        /* first pixel in scanline */
    u1 += resample_filter->slope; /* start of next scan line */

    /* location of this first pixel, relative to u0,v0 */
    U = (double)u-u0;
    V = (double)v-v0;

    /* Q = ellipse quotent ( if Q<F then pixel is inside ellipse) */
    Q = (resample_filter->A*U + resample_filter->B*V)*U + resample_filter->C*V*V;
    DQ = resample_filter->A*(2.0*U+1) + resample_filter->B*V;

    /* get the scanline of pixels for this v */
    pixels=GetCacheViewVirtualPixels(resample_filter->view,u,v,(size_t) uw,
      1,resample_filter->exception);
    if (pixels == (const Quantum *) NULL)
      return(MagickFalse);

    /* count up the weighted pixel colors */
    for( u=0; u<uw; u++ ) {
#if FILTER_LUT
      /* Note that the ellipse has been pre-scaled so F = WLUT_WIDTH */
      if ( Q < (double)WLUT_WIDTH ) {
        weight = resample_filter->filter_lut[(int)Q];
#else
      /* Note that the ellipse has been pre-scaled so F = support^2 */
      if ( Q < (double)resample_filter->F ) {
        weight = GetResizeFilterWeight(resample_filter->filter_def,
             sqrt(Q));    /* a SquareRoot!  Arrggghhhhh... */
#endif

        pixel->alpha  += weight*GetPixelAlpha(resample_filter->image,pixels);
        divisor_m += weight;

        if (pixel->alpha_trait != UndefinedPixelTrait)
          weight *= QuantumScale*((double) GetPixelAlpha(resample_filter->image,pixels));
        pixel->red   += weight*GetPixelRed(resample_filter->image,pixels);
        pixel->green += weight*GetPixelGreen(resample_filter->image,pixels);
        pixel->blue  += weight*GetPixelBlue(resample_filter->image,pixels);
        if (pixel->colorspace == CMYKColorspace)
          pixel->black += weight*GetPixelBlack(resample_filter->image,pixels);
        divisor_c += weight;

        hit++;
#if DEBUG_HIT_MISS
        /* mark the pixel according to hit/miss of the ellipse */
        (void) FormatLocaleFile(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 3\n",
                     (long)uu-.1,(double)v-.1,(long)uu+.1,(long)v+.1);
        (void) FormatLocaleFile(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 3\n",
                     (long)uu+.1,(double)v-.1,(long)uu-.1,(long)v+.1);
      } else {
        (void) FormatLocaleFile(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 1\n",
                     (long)uu-.1,(double)v-.1,(long)uu+.1,(long)v+.1);
        (void) FormatLocaleFile(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 1\n",
                     (long)uu+.1,(double)v-.1,(long)uu-.1,(long)v+.1);
      }
      uu++;
#else
      }
#endif
      pixels+=GetPixelChannels(resample_filter->image);
      Q += DQ;
      DQ += DDQ;
    }
  }
#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "Hit=%ld;  Total=%ld;\n", (long)hit, (long)uw*(v2-v1) );
#endif

  /*
    Result sanity check -- this should NOT happen
  */
  if ( hit == 0 || divisor_m <= MagickEpsilon || divisor_c <= MagickEpsilon ) {
    /* not enough pixels, or bad weighting in resampling,
       resort to direct interpolation */
#if DEBUG_NO_PIXEL_HIT
    pixel->alpha = pixel->red = pixel->green = pixel->blue = 0;
    pixel->red = QuantumRange; /* show pixels for which EWA fails */
#else
    status=InterpolatePixelInfo(resample_filter->image,
      resample_filter->view,resample_filter->interpolate,u0,v0,pixel,
      resample_filter->exception);
#endif
    return status;
  }

  /*
    Finialize results of resampling
  */
  divisor_m = 1.0/divisor_m;
  if (pixel->alpha_trait != UndefinedPixelTrait)
    pixel->alpha = (double) ClampToQuantum(divisor_m*pixel->alpha);
  divisor_c = 1.0/divisor_c;
  pixel->red   = (double) ClampToQuantum(divisor_c*pixel->red);
  pixel->green = (double) ClampToQuantum(divisor_c*pixel->green);
  pixel->blue  = (double) ClampToQuantum(divisor_c*pixel->blue);
  if (pixel->colorspace == CMYKColorspace)
    pixel->black = (double) ClampToQuantum(divisor_c*pixel->black);
  return(MagickTrue);
}


MagickExport void ScaleResampleFilter(ResampleFilter *resample_filter,
  const double dux,const double duy,const double dvx,const double dvy)
{
  double A,B,C,F;

  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickCoreSignature);

  resample_filter->limit_reached = MagickFalse;

  /* A 'point' filter forces use of interpolation instead of area sampling */
  if ( resample_filter->filter == PointFilter )
    return; /* EWA turned off - nothing to do */

#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "# -----\n" );
  (void) FormatLocaleFile(stderr, "dux=%lf; dvx=%lf;   duy=%lf; dvy=%lf;\n",
       dux, dvx, duy, dvy);
#endif

  /* Find Ellipse Coefficents such that
        A*u^2 + B*u*v + C*v^2 = F
     With u,v relative to point around which we are resampling.
     And the given scaling dx,dy vectors in u,v space
         du/dx,dv/dx   and  du/dy,dv/dy
  */
#if EWA
  /* Direct conversion of derivatives into elliptical coefficients
     However when magnifying images, the scaling vectors will be small
     resulting in a ellipse that is too small to sample properly.
     As such we need to clamp the major/minor axis to a minumum of 1.0
     to prevent it getting too small.
  */
#if EWA_CLAMP
  { double major_mag,
           minor_mag,
           major_x,
           major_y,
           minor_x,
           minor_y;

  ClampUpAxes(dux,dvx,duy,dvy, &major_mag, &minor_mag,
                &major_x, &major_y, &minor_x, &minor_y);
  major_x *= major_mag;  major_y *= major_mag;
  minor_x *= minor_mag;  minor_y *= minor_mag;
#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "major_x=%lf; major_y=%lf;  minor_x=%lf; minor_y=%lf;\n",
        major_x, major_y, minor_x, minor_y);
#endif
  A = major_y*major_y+minor_y*minor_y;
  B = -2.0*(major_x*major_y+minor_x*minor_y);
  C = major_x*major_x+minor_x*minor_x;
  F = major_mag*minor_mag;
  F *= F; /* square it */
  }
#else /* raw unclamped EWA */
  A = dvx*dvx+dvy*dvy;
  B = -2.0*(dux*dvx+duy*dvy);
  C = dux*dux+duy*duy;
  F = dux*dvy-duy*dvx;
  F *= F; /* square it */
#endif /* EWA_CLAMP */

#else /* HQ_EWA */
  /*
    This Paul Heckbert's "Higher Quality EWA" formula, from page 60 in his
    thesis, which adds a unit circle to the elliptical area so as to do both
    Reconstruction and Prefiltering of the pixels in the resampling.  It also
    means it is always likely to have at least 4 pixels within the area of the
    ellipse, for weighted averaging.  No scaling will result with F == 4.0 and
    a circle of radius 2.0, and F smaller than this means magnification is
    being used.

    NOTE: This method produces a very blury result at near unity scale while
    producing perfect results for strong minitification and magnifications.

    However filter support is fixed to 2.0 (no good for Windowed Sinc filters)
  */
  A = dvx*dvx+dvy*dvy+1;
  B = -2.0*(dux*dvx+duy*dvy);
  C = dux*dux+duy*duy+1;
  F = A*C - B*B/4;
#endif

#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "A=%lf; B=%lf; C=%lf; F=%lf\n", A,B,C,F);

  /* Figure out the various information directly about the ellipse.
     This information currently not needed at this time, but may be
     needed later for better limit determination.

     It is also good to have as a record for future debugging
  */
  { double alpha, beta, gamma, Major, Minor;
    double Eccentricity, Ellipse_Area, Ellipse_Angle;

    alpha = A+C;
    beta  = A-C;
    gamma = sqrt(beta*beta + B*B );

    if ( alpha - gamma <= MagickEpsilon )
      Major=MagickMaximumValue;
    else
      Major=sqrt(2*F/(alpha - gamma));
    Minor = sqrt(2*F/(alpha + gamma));

    (void) FormatLocaleFile(stderr, "# Major=%lf; Minor=%lf\n", Major, Minor );

    /* other information about ellipse include... */
    Eccentricity = Major/Minor;
    Ellipse_Area = MagickPI*Major*Minor;
    Ellipse_Angle = atan2(B, A-C);

    (void) FormatLocaleFile(stderr, "# Angle=%lf   Area=%lf\n",
         (double) RadiansToDegrees(Ellipse_Angle), Ellipse_Area);
  }
#endif

  /* If one or both of the scaling vectors is impossibly large
     (producing a very large raw F value), we may as well not bother
     doing any form of resampling since resampled area is very large.
     In this case some alternative means of pixel sampling, such as
     the average of the whole image is needed to get a reasonable
     result. Calculate only as needed.
  */
  if ( (4*A*C - B*B) > MagickMaximumValue ) {
    resample_filter->limit_reached = MagickTrue;
    return;
  }

  /* Scale ellipse to match the filters support
     (that is, multiply F by the square of the support)
     Simplier to just multiply it by the support twice!
  */
  F *= resample_filter->support;
  F *= resample_filter->support;

  /* Orthogonal bounds of the ellipse */
  resample_filter->Ulimit = sqrt(C*F/(A*C-0.25*B*B));
  resample_filter->Vlimit = sqrt(A*F/(A*C-0.25*B*B));

  /* Horizontally aligned parallelogram fitted to Ellipse */
  resample_filter->Uwidth = sqrt(F/A); /* Half of the parallelogram width */
  resample_filter->slope = -B/(2.0*A); /* Reciprocal slope of the parallelogram */

#if DEBUG_ELLIPSE
  (void) FormatLocaleFile(stderr, "Ulimit=%lf; Vlimit=%lf; UWidth=%lf; Slope=%lf;\n",
           resample_filter->Ulimit, resample_filter->Vlimit,
           resample_filter->Uwidth, resample_filter->slope );
#endif

  /* Check the absolute area of the parallelogram involved.
   * This limit needs more work, as it is too slow for larger images
   * with tiled views of the horizon.
  */
  if ( (resample_filter->Uwidth * resample_filter->Vlimit)
         > (4.0*resample_filter->image_area)) {
    resample_filter->limit_reached = MagickTrue;
    return;
  }

  /* Scale ellipse formula to directly index the Filter Lookup Table */
  { double scale;
#if FILTER_LUT
    /* scale so that F = WLUT_WIDTH; -- hardcoded */
    scale=(double) WLUT_WIDTH*PerceptibleReciprocal(F);
#else
    /* scale so that F = resample_filter->F (support^2) */
    scale=resample_filter->F*PerceptibleReciprocal(F);
#endif
    resample_filter->A = A*scale;
    resample_filter->B = B*scale;
    resample_filter->C = C*scale;
  }
}